The Unknotting Problem

Abstract

This paper tells the story of knots and the search to detect their knottedness.

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

1 Introduction

We say that a closed loop embedded in three dimensional space is knotted if there is no continuous deformation of the loop, through embeddings, that changes it to a circle embedded in a plane. The fundamental problem in knot theory is to determine whether a closed loop embedded in three dimensional space is knotted. The loop in Fig. 1 is a trefoil knot, the simplest knot. View Fig. 2. Can you tell whether this loop is knotted or not? It requires a special intuition for topology to just look at a loop and know if it is really knotted.

Fig. 1

The trefoil knot

Fig. 2

Knotted?

In order to analyze the knottedness of a knot, a mathematical representation is required. In this paper we shall use the method of knot and link diagrams, and the equivalence relation generated by the Reidemeister moves (see Fig. 3 for an illustration of these moves). Knot diagrams are graphs with extra structure that encode the embedding type of the knot. Each diagram is a pictorial representation of the knot, and so appeals to the intuition of the viewer. The Reidemeister moves are a set of simple combinatorial moves, proved in the 1920s to capture the notion of topological equivalence of knots and links in three dimensional space. Single applications of these moves can leave the diagram with the same number of crossings (places where a weaving of two segments occurs), or increase or decrease the number of crossings. Some unknottings can be accomplished without increasing the number of crossings in the diagram. We call such unknot diagrams easy since the fact that they are unknotted can be determined by a finite search for simplifying moves. However, there are culprit diagrams [28, 56, 58, 60] that require moves that increase the number of crossings before the diagram can be simplified to an unknotted circle with no crossings. It is the structure of such culprits that is the subject matter of Sect. 3 of this paper.

Fig. 3

Reidemeister moves

This paper is an exploration of the theme of detecting knottedness. This exposition is an outgrowth of a series of lectures [30] that I gave in Tokyo at the Knots 96 conference in the summer of 1996. The present exposition goes considerably farther than those lectures, as in the intervening time there has been much progress on this problem. In particular there is a beautiful combinatorial solution to the problem of detecting whether a knot diagram is knotted, due to Dynnikov [10] that we sketch in Sect. 3 of this paper.

The problem of detecting the unknot has been investigated by many people. The three dimensional work of Haken and Hemion [33] solves these problems in principle, giving definite algorithms to tell whether two knots are equivalent, or whether a given knot is unknotted. The algorithms are given in terms of the structure of a triangulation of the complement of the knot or link, and these algorithms are unwieldy. Nevertheless, the major problem that we have discussed is solved by the Haken-Hemion method. This result did not end the field of study of knot invariants. In fact, it spurred it on and the methods that we have discussed in this paper are just the tip of the iceberg of the revolutions that have engulfed the theory of knots and links since the early 1980s. For example, the papers by Birman and Hirsch [54] and Birman and Moody [55] study the unknotting problem. More recently it has been shown that both Khovanov Homology [17] (a generalization of the Jones polynomial) and Heegard Floer Homology (a generalization of the Alexander polynomial) detect the unknot. Heegaard Floer Homology not only detects the unknot, but can be used to calculate the least genus of an orientable spanning surface for any knot. This is an outstanding result. The reader can examine the paper by Manolescu et al. [62] for more information. In that work, the Heegard Floer homology is expressed via a chain complex that is associated to a rectangular diagram of just the type that Dynnikov uses.

This paper is organized as follows. Section 2 discusses the Reidemeister moves and the combinatorial approach to the theory of knots and links. We sketch the proof of Reidemeister’s Theorem that expresses equivalence of knots and links in terms of diagram moves (the Reidemeister moves). We give examples of knot diagrams that are unknotted but must be made more complicated in order to be undone by a sequence of Reidemeister moves. If it were not for this phenomenon, the problem of detecting an unknotted diagram would be over at once. One would only need to try to simplify the diagram by Reidemeister moves (that is, to try to reduce the number of crossings in the diagram). This is a finite search and that would be an algorithm to determine the knottedness of the diagram. Section 2 explains the solution by Dynnikov to this conundrum of unknotting by moves. We reformulate knot theory in terms of arc-diagrams and explain how Dynnikov constructs moves on these diagrams that can simplify an unknotted knot diagram. In this way, the problem of detecting the unknot is solved. There are many questions still remaining in the combinatorial domain, and the end of Sect. 3 discusses them. We give upper bounds on the number of crossings needed to unknot an unknotted diagram (work with Henrich [28] based on Dynnikov’s work) and upper bounds on the number of Reidemeister moves needed for unknotting. Section 4 turns to the question of detection of knotting by the Jones polynomial and gives examples of links that cannot be detected by the Jones polynomial. It is still an unsolved problem whether the Jones polynomial detects the unknot. Section 5 is an introduction to Vassiliev invariants. We include this section in order to give the reader a flavor of the wider context the includes the Jones polynomial with invariants that are related to physics and to Lie algebras. We include a description of the relationship of Vassiliev invariants with Witten’s approach to knot invariants via quantum field theory. We end this chapter with a statement of the remarkable problem to distinguish knots from their reverses. There are knots that are not isotopic to the curve obtained by reversing the orientation of the loop. At this writing it is not known if Vassiliev invariants can detect reversibility.

There are many more problems about the detection of knottedness. We have not touched on the question of distinguishing distinct knots in this paper. It is possible that if two knots are not isotopic, then there are Vassiliev invariants that distinguish them. But that is another country.

2 Reidemeister Moves

Reidemeister [39] discovered a simple set of moves on link diagrams that captures the concept of ambient isotopy of knots in three-dimensional space. There are three basic Reidemeister moves. Reidemeister’s theorem states that two diagrams represent ambient isotopic knots (or links) if and only if there is a sequence of Reidemeister moves taking one diagram to the other. The Reidemeister moves are illustrated in Fig. 3.

Reidemeister’s three moves are interpreted as performed on a larger diagram in which the small diagram shown is a literal part. Each move is performed without disturbing the rest of the diagram. Note that this means that each move occurs, up to topological deformation, just as it is shown in the diagrams in Fig. 3. There are no extra lines in the local diagrams. For example, the equivalence (A) in Fig. 4 is not an instance of a single first Reidemeister move. Taken literally, it factors into a move II followed by a move I.

Fig. 4

Factorable move, move zero

Diagrams are always subject to topological deformations in the plane that preserve the structure of the crossings. These deformations could be designated as “Move Zero”. See Fig. 4.

A few exercises with the Reidemeister moves are in order. First of all, view the diagram in Fig. 5. It is unknotted and you can have a good time finding a sequence of Reidemeister moves that will do the trick. Diagrams of this type are produced by tracing a curve and always producing an undercrossing at each return crossing. This type of knot is called a standard unknot. Of course we see clearly that a standard unknot is unknotted by just pulling on it, since it has the same structure as a coil of rope that is wound down onto a flat surface.

Fig. 5

Standard unknot

Can one recognize unknots by simply looking for sequences of Reidemeister moves that undo them? This would be easy if it were not for the case that there are examples of unknots that require some moves that increase the number of crossings before they can be subsequently decreased. Such an culprit is illustrated in Fig. 6.

Fig. 6

A culprit

It is generally not so easy to recognise unknots. However, here is a tip: Look for macro moves of the type shown in Fig. 7. In a macro move, we identify an arc that passes entirely under some piece of the diagram (or entirely over) and shift this part of the arc, keeping it under (or over) during the shift. In Fig. 7, we illustrate a macro move on an arc that passes under a piece of the diagram that is indicated by arcs going into a circular region. A more general macro move is possible where the moving arc moves underneath one layer of diagram, and at the same time, over another layer of diagram. Macro moves often allow a reduction in the number of crossings even though the number of crossings will increase during a sequence of Reidemeister moves that generates the macro move.

Fig. 7

Macro move

As shown in Fig. 7, the macro-move includes as a special case both the second and the third Reidemeister moves, and it is not hard to verify that a macro move can be generated by a sequence of type II and type III Reidemeister moves. It is easy to see that the type I moves can be left to the end of any deformation. The demon of Fig. 6 is easily demolished by macro moves, and from the point of view of macro moves the diagram never gets more complicated.

Let’s say that a knot can be reduced by a set of moves if it can be transformed by these moves to the unknotted circle diagram through diagrams that never have more crossings than the original diagram. Then we have shown that there are diagrams representing the unknot that cannot be reduced by the Reidemeister moves. On the other hand, I do not know whether unknotted diagrams can always be reduced by (appropriately generalized) macro moves in conjunction with the first Reidemeister move. If this were true it would give a combinatorial way to recognise the unknot.

Remark.

In fact, there is a combinatorial way to recognise the unknot based on a diagrams and moves. In [10] I. A. Dynnikov finds just such a result, using piecewise linear knot diagrams with all ninety degree angles in the diagrams, and all arcs in the diagram either horizontal or vertical. We shall discuss Dynnikov’s work in Sect. 3 of this paper.

2.1 Reidemeister’s Theorem

We now indicate how Reidemeister proved his Theorem.

An embedding of a knot or link in three-dimensional space is said to be piecewise linear if it consists in a collection of straight line segments joined end to end. Reidemeister started with a single move in three-dimensional space for piecewise linear knots and links. Consider a point in the complement of the link, and an edge in the link such that the surface of the triangle formed by the end points of that edge and the new point is not pierced by any other edge in the link. Then one can replace the given edge on the link by the other two edges of the triangle, obtaining a new link that is ambient isotopic to the original link. Conversely, one can remove two consecutive edges in the link and replace them by a new edge that goes directly from initial to final points, whenever the triangle spanned by the two consecutive edges is not pierced by any other edge of the link. This triangle replacement constitutes Reidemeister’s three-dimensional move. See Fig. 8. It can be shown that two piecewise linear knots or links are ambient isotopic in three-dimensional space if and only if there is a sequence of Reidemeister triangle moves from one to the other. This will not be proved here. At the time when Reidemeister wrote his book, equivalence via three-dimensional triangle moves was taken as the definition of topological equivalence of links.

Fig. 8

Triangle move

It can also be shown that tame knots and links have piecewise linear representatives in their ambient isotopy class. It is sufficient for our purposes to work with piecewise linear knots and links. Reidemeister’s planar moves then follow from an analysis of the shadows projected into the plane by Reidemeister triangle moves in space. Figure 9 gives a hint of this analysis. The result is a reformulation of the three-dimensional problems of knot theory to a combinatorial game in the plane.

Fig. 9

Shadows

To go beyond the hint in Fig. 9 to a complete proof that Reidemeister’s planar moves suffice involves preliminary remarks about subdivision. The simplest subdivision that one wants to be able to perform on a piecewise linear link is the placement of a new vertex at an interior point of an edge—so that edge becomes two edges in the subdivided link. Figure 10 shows how to accomplish this subdivision via triangle moves.

Fig. 10

Subdivision of an edge

Any triangle move can be factored into a sequence of smaller triangle moves corresponding to a simplicial subdivision of that triangle. This is obvious, since the triangles in the subdivision of the large triangle that is unpierced by the link are themselves unpierced by the link.

To understand how the Reidemeister triangle move behaves on diagrams it is sufficient to consider a projection of the link in which the triangle is projected to a non-singular triangle in the plane. Of course, there may be many arcs of the link also projected upon the interior of the projected triangle. However, by using subdivision, we can assume that the cases of the extra arcs are as shown in Fig. 11. In Fig. 11 we have also shown how each of these cases can be accomplished by (combinations of) the three Reidemeister moves. This proves that a projection of a single triangle move can be accomplished by a sequence of Reidemeister diagram moves.

Fig. 11

Projections of triangle moves

A piecewise linear isotopy consists in a finite sequence of triangle moves. There exists a direction in three-dimensional space that makes a non-zero angle with each of theses triangles and is in general position with the link diagram. Projecting to the plane along this direction makes it possible to perform the entire ambient isotopy in the language of projected triangle moves. Now apply the results of the previous paragraph and we conclude

Reidemeister’s Theorem If two links are piecewise linearly equivalent (ambient isotopic), then there is a sequence of Reidemeister diagram moves taking a projection of one link to a projection of the other.

Note that the proof tells us that the two diagrams can be obtained from one spatial projection direction for the entire spatial isotopy. It is obvious that diagrams related by Reidemeister moves represent ambient isotopic links. Reidemeister’s Theorem gives a complete combinatorial description of the topology of knots and links in three-dimensional space.

3 Dynnikov’s Solution of the Problem of Knot Detection

We now discuss a powerful result proven by Dynnikov in [10]. Dynnikov uses, a diagram called an arc-presentation for the knot. We define this below, and show how one can detect the unknot using moves that preserve this type of presentation. This section is based on our paper [28].

Definition 1.

An arc–presentation of a knot is a knot diagram comprised of horizontal and vertical line segments such that at each crossing in the diagram, the horizontal arc passes under the vertical arc. Furthermore, we require that no two edges in an arc–diagram are colinear. Two arc–presentations are combinatorially equivalent if they are isotopic in the plane via an ambient isotopy of the form h(x, y) = (f(x), g(y)). The complexity c(L) of an arc–presentation is the number of vertical arcs in the diagram.

We say more generally that a link diagram is rectangular if it has only vertical and horizontal edges. In Fig. 12 we give an example of a rectangular diagram that is an arc–presentation and another example of a rectangular diagram that is not an arc–presentation.

Fig. 12

The picture on the left is an example of an arc–presentation of a trefoil. The picture on the right is an example that is not an arc–presentation (since not all horizontal arcs pass under vertical arcs)

Note that a rectangular diagram can naturally be drawn on a rectangular grid. If we start with such a grid and represent rectangular diagrams on the grid we have called these knots mosaic knots and used them to define a notion of quantum knot. See [61] for more about quantum knots. For now, we focus our attention on arc–presentations.

Proposition 1 (Dynnikov).

Every knot has an arc–presentation. Any two arc–presentations of the same knot can be related to each other by a finite sequence of elementary moves , pictured in Figs.  13 and 14.

Fig. 13

Elementary (de)stabilization moves. Stabilization moves increase the complexity of the arc–presentation while destabilization moves decrease the complexity

Fig. 14

Some examples of exchange moves. Other allowed exchange moves include switching the heights of two horizontal arcs that lie in distinct halves of the diagram

The proof of this proposition is elementary, based on the Reidemeister moves. One shows that each Reidemeister move can be represented by (a sequence of) elementary moves. See [10]. We will show how to convert a usual knot diagram to an arc–presentation in the next few paragraphs, making use of the concept of Morse diagrams of knots.

Definition.

A knot diagram is in Morse form if it has

• 1. no horizontal lines,

• 2. no inflection points,

• 3. a single singularity at each height, and

• 4. each crossing is oriented to create a 45 degree angle with the vertical axis.

We note that converting an arbitrary knot diagram into a diagram in Morse form requires no Reidemeister moves, only ambient isotopies of the plane. More information about Morse diagrams can be found in [59]. See Figs. 15, 16 and 17 for an illustration of the process of conversion of a knot diagram to an arc diagram.

Fig. 15

A Morse diagram of a knot and a corresponding rectangular diagram

Fig. 16

Rotating a crossing to convert a rectangular diagram into an arc–presentation

Fig. 17

Converting a rectangular diagram into an arc–presentation by rotating a crossing

In Fig. 18 we show the example found by Goeritz [56] in 1934 of a knot diagram that is unknotted, but requires Reidemeister moves that create more crossings before it can be simplified. In Fig. 19 we show another example of this same type. We shall refer to the latter as the “culprit” and analyse it below.

Fig. 18

Goeritz unknot

Fig. 19

The culprit

Here is Dynnikov’s solution to the problem of recognizing the unknot.

Theorem 1 (Dynnikov).

If L is an arc–presentation of the unknot, then there exists a finite sequence of exchange and destabilization moves

$$\displaystyle{L \rightarrow L_{1} \rightarrow L_{2} \rightarrow \cdots \rightarrow L_{m}}$$

such than L _m is trivial.

What is particularly interesting about this result is that the unknot can be simplified without increasing the complexity of the arc–presentation, that is, without the use of stabilization moves. This means that a finite search will reveal a diagram to be unknotted if that is the case. Furthermore, if we apply Dynnikov’s method to a knotted knot, it will stop on a diagram that is not a planar circle. Thus Dynnikov’s diagrammatic method can detect the unknot.

We can go further and ask how large a diagram is needed to unknot a knot by Reidemeister moves, and how many Reidemeister moves are needed.

Theorem 2 ([28]).

Suppose K is a diagram (in Morse form) of the unknot with crossing number cr(K) and number of maxima b(K). Then, for every i, the crossing number cr(K _i ) is no more than (M − 2) ² where $M = 2b(K) + cr(K)$ and K = K ₀ ,K ₁ ,K ₂ ,…,K _N is a sequence of knot diagrams such that K _i+1 is obtained from K _i by a single Reidemeister move and K _N is a trivial diagram of the unknot.

To find our upper bound on the number of Reidemeister moves, we must first specify an upper bound on the number m of exchange and destabilization moves required to trivialize an arc–presentation. This bound will depend on the complexity c(L) = n of the arc–diagram L. We also must provide an upper bound on the number of Reidemeister moves required for a destabilization or exchange move.

In [10], Dynnikov provides the following bounds on the number of combinatorially distinct arc–presentations of complexity n.

Proposition 2.

Let N(n) denote the number of combinatorially distinct arc–presentations of complexity n. Then the following inequality holds.

$$\displaystyle{N(n) \leq \frac{1} {2}n[(n - 1)!]^{2}}$$

Using this count on the number of distinct arc–presentations of a given size, we can find a bound (albeit a large one) on the number of arc–presentation moves we need. This is simply by virtue of the fact that any reasonable sequence of moves will contain mutually distinct arc–presentations that don’t exceed the complexity of the original, and there are a limited number of such diagrams. With this we obtain

Theorem 3 ([28]).

Suppose K is a diagram (in Morse form) of the unknot with crossing number cr(K) and number of maxima b(K). Let $M = 2b(K) + cr(K)$ . Then the number of Reidemeister moves required to unknot K is less than or equal to

$$\displaystyle{\sum _{i=2}^{M}\frac{1} {2}i[(i - 1)!]^{2}(M - 2).}$$

We have provided several upper bounds regarding the complexity of the Reidemeister sequence required to simplify an unknot. The bound that Dynnikov’s work helps us obtain for the number of Reidemeister moves required to unknot an unknot is superexponential. Using a different technique, Hass and Lagarias were able to find a bound that is exponential in the crossing number of the diagram [57]. They prove the following result.

Theorem 4 (Hass and Lagarias).

There is a positive constant c ₁ , such that for each n ≥ 1, any unknotted knot diagram $\mathcal{D}$ with n crossings can be transformed to the trivial knot diagram using at most $2^{c_{1}n}$ Reidemeister moves. In fact one can take c ₁= 10 ¹¹.

Hass and Lagarias use the same technique to find an exponential bound for the number of crossings required for unknotting. For bounds of this second sort, the one presented here is a comparatively sharper estimate. Our bound on the number of Reidemeister moves required to unknot an unknot eventually becomes larger than the previously known bound from the above Theorem, but it does remain significantly smaller for knots with up to $10^{10^{10} }$ crossings. We can see this by assuming that the number of maxima in a Morse diagram of a knot is approximately the same size as the crossing number (in practice the number of maxima is significantly smaller than the crossing number). We can estimate the size of our bound by computing the quantity

$$\displaystyle{\frac{M - 2} {2} (M!)^{2}.}$$

This larger estimate remains smaller than the bound proposed by Hass and Lagarias until the knots have $10^{10^{10} }$ crossings. It is possible that these methods may be improved to find smaller unknotting bounds for unknot diagrams. In fact, polynomial bounds are now shown by Lackenby [34].

Let us return to the Culprit. Recall that hard unknots are difficult to unknot by virtue of the fact that no simplifying type I or type II Reidemeister moves and no type III moves are available. In Fig. 20, we picture a Morse diagram of the Culprit, a corresponding arc-presentation and a sequence of Dynnikov moves that simplify it to a standard unknot. Bounds for the size of diagrams that are needed using Reidemeister’s moves can be deduced using Dynnikov’s work. See [10, 28]. In fact, we [28] derive a quadratic upper bound on the crossing number of diagrams in an unknotting sequence. A similar result can be found in [10]. We saw that the Culprit may be unknotted with ten Reidemeister moves in Fig. 21 (see also [60]). The maximum crossing number of all diagrams in the given Reidemeister sequence is 12, two more than the number of crossings in the Culprit. On the other hand, we can compute our upper bound on the number of crossings required for unknotting as follows. Since the crossing number cr(K) = 10 and the number of maxima in the diagram is b(K) = 5, we see that $M = cr(K) + 2b(K) = 20$. Thus, our bound is $(M - 2)^{2} = 18^{2} = 324.$ There is room for improvement!

Fig. 20

Undoing the culprit by Reidemeister moves

Fig. 21

Undoing the culprit by Dynnikov moves

We can also use M to find our bound for the number of Reidemeister moves required to unknot the Culprit.

$$\displaystyle{\sum _{i=2}^{M}\frac{1} {2}i[(i - 1)!]^{2}(M - 2) = 9\sum _{ i=2}^{20}i[(i - 1)!]^{2}.}$$

The largest term in this expression is roughly 10³⁵, unfortunately quite a bit larger than ten.

In this section we have sketched results given in more detail in [10, 28], showing Dynnikov’s remarkable solution to the unknotting problem. Remarkable as this solution is, we are not happy since we believe that better bounds on the number of Reidemeister moves needed to unknot a knot are surely possible, and better bounds on the complexity of diagrams is also possible. Could there be a simple algorithm in the form of a calculation from a diagram that would tell if a knot was knotted? This is the subject of the next section where we discuss the Jones polynomial.

4 The Bracket Polynomial and the Jones Polynomial

Now that we have exhibited a solution to problem of the detection of the unknot, we turn to some unsolved problems related to other methods of detection. It is an open problem whether there exist classical knots (single component loops) that are knotted and yet have unit Jones polynomial [14]. In other words, it is an open problem whether the Jones polynomial can detect all knots. There do exist families of links whose linkedness is undetectable by the Jones polynomial [47, 48]. It is the purpose of this section of the paper to give a summary of some of the information that is known in this arena. We begin with a sketch of ways to calculate the bracket polynomial model of the Jones polynomial, and then discuss how to construct classical links that are undetectable by the Jones polynomial.

The bracket polynomial [20–23, 25] model for the Jones polynomial [14–16, 52] is described by the expansion

(1)

and we have

$$\displaystyle\begin{array}{rcl} \langle K\,\bigcirc \rangle = (-A^{2} - A^{-2})\langle K\rangle & &{}\end{array}$$

(2)

(3)

(4)

A state S of a link diagram K is obtained by choosing a smoothing for each crossing in the diagram and labelling that smoothing with either A or A ⁻¹ according to the convention indicated in the bracket expansion above. Then, given a state S, one has the evaluation < K | S > equal to the product of the labels at the smoothings, and one has the evaluation | | S | | equal to the number of loops in the state. One then has the formula

$$\displaystyle{< K >=\varSigma _{S} < K\vert S > d^{\vert \vert S\vert \vert -1}}$$

where the summation runs over the states S of the diagram K, and $d = -A^{2} - A^{-2}.$ This state summation is invariant under all classical and virtual moves except the first Reidemeister move. The bracket polynomial is normalized to an invariant f _K (A) of all the moves by the formula $f_{K}(A) = (-A^{3})^{-w(K)} < K >$ where w(K) is the writhe of the (now) oriented diagram K. The writhe is the sum of the orientation signs ( ± 1) of the crossings of the diagram. The Jones polynomial, V _K (t) is given in terms of this model by the formula

$$\displaystyle{V _{K}(t) = f_{K}(t^{-1/4}).}$$

The state sum is part of a wider approach to invariants of knots and links that we do not concentrate upon in this paper. First of all, the Alexander polynomial [53] was the first polynomial invariant of knots and links. It was not until 1981 that the Alexander polynomial was reformulated as a state summation [18–20]. In 1983 Jones discovered his polynomial and showed that it satisfied a skein relation similar to that for the Alexander-Conway polynomial [9]. Along with this there arose relations with statistical mechanics [5] in the work of Jones. The bracket model was the first direct relationship of the Jones polynomial with statistical mechanics. In the wake of the discovery of the Jones polynomial came more skein polynomials, particularly the Homflypt polynomial [20, 35] named after the people who discovered it—Hoste, Ocneanu, Freyd, Lickorish, Yetter, Przytycki and Trawczk, and the Kauffman Polynomial [24]. These invariants are more powerful than the Jones polynomial, but it is still conjectural that they detect the unknot. After the skein polynomials, came more algebraic state sums based on the work of Yang and Baxter in statistical mechanics, and then arose relationships with Lie algebras and gauge theoretic physics. We shall sketch some of these developments in the sections to follow. It remains to be seen how powerful all these new invariants are as detectors of knottedness, but it is known that certain families of these invariants do detect knottedness and those results are found in the relationships of the knot theory with physics

It remains on open problem whether the Jones polynomial can detect the unknot. We can make the conjecture as follows:

Knot Detection Conjecture If K is a knot diagram of one component and V _K (t) = 1, then K is equivalent by Reidemeister moves to the unknot.

This knot detection conjecture is false for links. View Fig. 22. Here we have a version of a link L discovered by Thistlethwaite [47] in December 2000. One can verify that this link is indeed non-trivial, but it has the same Jones polynomial as the unlink of two circles. In [48] we produce infinite families of distinct links that appear to be unlinked to the Jones polynomial.

Fig. 22

Thistethwaite’s link

4.1 Present Status of Links Not Detectable by the Jones Polynomial

In this section we give a quick review of the status of our work

A tangle (2-tangle) consists in an embedding of two arcs in a three-ball (and possibly some circles embedded in the interior of the three-ball) such that the endpoints of the arcs are on the boundary of the three-ball. One usually depicts the arcs as crossing the boundary transversely so that the tangle is seen as the embedding in the three-ball augmented by four segments emanating from the ball, each from the intersection of the arcs with the boundary. These four segments are the exterior edges of the tangle, and are used for operations that form new tangles and new knots and links from given tangles. Two tangles in a given three-ball are said to be topologically equivalent if there is an ambient isotopy from one to the other in the given three-ball, fixing the intersections of the tangles with the boundary.

It is customary to illustrate tangles with a diagram that consists in a box (within which are the arcs of the tangle) and with the exterior edges emanating from the box in the NorthWest (NW), NorthEast (NE), SouthWest (SW) and SouthEast (SE) directions. Given tangles T and S, one defines the sum, denoted T + S by placing the diagram for S to the right of the diagram for T and attaching the NE edge of T to the NW edge of S, and the SE edge of T to the SW edge of S. The resulting tangle T + S has exterior edges corresponding to the NW and SW edges of T and the NE and SE edges of S. There are two ways to create links associated to a tangle T. The numerator T ^N is obtained, by attaching the (top) NW and NE edges of T together and attaching the (bottom) SW and SE edges together. The denominator T ^D is obtained, by attaching the (left side) NW and SW edges together and attaching the (right side) NE and SE edges together. We denote by [0] the tangle with only unknotted arcs (no embedded circles) with one arc connecting, within the three-ball, the (top points) NW intersection point with the NE intersection point, and the other arc connecting the (bottom points) SW intersection point with the SE intersection point. A ninety degree turn of the tangle [0] produces the tangle [∞] with connections between NW and SW and between NE and SE. One then can prove the basic formula for any tangle T

$$\displaystyle{< T >=\alpha _{T} < [0] > +\beta _{T} < [\infty ] >}$$

where α _T and β _T are well-defined polynomial invariants (of regular isotopy) of the tangle T. From this formula one can deduce that

$$\displaystyle{< T^{N} >=\alpha _{ T}d +\beta _{T}}$$

and

$$\displaystyle{< T^{D} >=\alpha _{ T} +\beta _{T}d.}$$

We define the bracket vector of T to be the ordered pair (α _T , β _T ) and denote it by br(T), viewing it as a column vector so that br(T)^t = (α _T , β _T ) where v ^t denotes the transpose of the vector v. With this notation the two formulas above for the evaluation for numerator and denominator of a tangle become the single matrix equation

$$\displaystyle{\left [\begin{array}{c} < T^{N} > \\ < T^{D} > \end{array} \right ] = \left [\begin{array}{cc} d&1\\ 1 &d \end{array} \right ]br(T).}$$

We then use this formalism to express the bracket polynomial for our examples. The class of examples that we considered are each denoted by H(T, U) where T and U are each tangles and H(T, U) is a satellite of the Hopf link that conforms to the pattern shown in Fig. 23, formed by clasping together the numerators of the tangles T and U. Our method is based on a transformation H(T, U) ⟶ H(T, U)^ω, whereby the tangles T and U are cut out and reglued by certain specific homeomorphisms of the tangle boundaries. This transformation can be specified by a modification described by a specific rational tangle and its mirror image. Like mutation, the transformation ω preserves the bracket polynomial. However, it is more effective than mutation in generating examples, as a trivial link can be transformed to a prime link, and repeated application yields an infinite sequence of inequivalent links.

Fig. 23

Hopf link satellite H(T, U)

Specifically, the transformation H(T, U)^ω is given by the formula

$$\displaystyle{H(T,U)^{\omega } = H(T^{\omega },U^{\bar{\omega }})}$$

where the tangle operations T ^ω and $U^{\bar{\omega }})$ are as shown in Fig. 24. By direct calculation, there is a matrix M such that

$$\displaystyle{< H(T,U) >= br(T)^{t}Mbr(U)}$$

and there is a matrix Ω such that

$$\displaystyle{br(T^{\omega }) =\varOmega br(T)}$$

and

$$\displaystyle{br(T^{\bar{\omega }}) =\varOmega ^{-1}br(T).}$$

One verifies the identity

$$\displaystyle{\varOmega ^{t}M\varOmega ^{-1} = M}$$

from which it follows that < H(T, U) > ^ω = < H(T, U) > . This completes the sketch of our method for obtaining links that whose linking cannot be seen by the Jones polynomial. Note that the link constructed as $H(T^{\omega },U^{\bar{\omega }})$ in Fig. 25 has the same Jones polynomial as an unlink of two components. This shows how the first example found by Thistlethwaite fits into our construction.

Fig. 24

The omega operations

Fig. 25

Applying omega operations to an unlink

5 Vassiliev Invariants and Invariants of Rigid Vertex Graphs

If V (K) is a (Laurent polynomial valued, or more generally—commutative ring valued) invariant of knots, then it can be naturally extended to an invariant of rigid vertex graphs by defining the invariant of graphs in terms of the knot invariant via an “unfolding” of the vertex. That is, we can regard the vertex as a “black box” and replace it by any tangle of our choice. Rigid vertex motions of the graph preserve the contents of the black box, and hence entail ambient isotopies of the link obtained by replacing the black box by its contents. Invariants of knots and links that are evaluated on these replacements are then automatically rigid vertex invariants of the corresponding graphs. If we set up a collection of multiple replacements at the vertices with standard conventions for the insertions of the tangles, then a summation over all possible replacements can lead to a graph invariant with new coefficients corresponding to the different replacements. In this way each invariant of knots and links implicates a large collection of graph invariants. See [25, 26].

The simplest tangle replacements for a 4-valent vertex are the two crossings, positive and negative, and the oriented smoothing. Let V (K) be any invariant of knots and links. Extend V to the category of rigid vertex embeddings of 4-valent graphs by the formula (see Fig. 26)

$$\displaystyle{V (K_{{\ast}}) = aV (K_{+}) + bV (K_{-}) + cV (K_{0})}$$

Fig. 26

Graphical vertex formulas

Here K _∗ indicates an embedding with a transversal 4-valent vertex. This formula means that we define V (G) for an embedded 4-valent graph G by taking the sum

$$\displaystyle{V (G) =\sum _{S}a^{i_{+}(S)}b^{i_{-}(S)}c^{i_{0}(S)}V (S)}$$

with the summation over all knots and links S obtained from G by replacing a node of G with either a crossing of positive or negative type, or with a smoothing (denoted 0). Here i ₊(S) denotes the number of positive crossings in the replacement, i ₋(S) the number of negative crossings in the replacement, and i ₀(S) the number of smoothings in the replacement. It is not hard to see that if V (K) is an ambient isotopy invariant of knots, then, this extension is a rigid vertex isotopy invariant of graphs. In rigid vertex isotopy the cyclic order at the vertex is preserved, so that the vertex behaves like a rigid disk with flexible strings attached to it at specific points. See the previous section.

There is a rich class of graph invariants that can be studied in this manner. The Vassiliev Invariants [4, 6, 50] constitute the important special case of these graph invariants where $a = +1$, $b = -1$ and c = 0. Thus V (G) is a Vassiliev invariant if

$$\displaystyle{V (K_{{\ast}}) = V (K_{+}) - V (K_{-}).}$$

Call this formula the exchange identity for the Vassiliev invariant V. V is said to be of finite type k if V (G) = 0 whenever | G |  > k where | G | denotes the number of 4-valent nodes in the graph G. The notion of finite type is of paramount significance in studying these invariants. One reason for this is the following basic Lemma.

Lemma.

If a graph G has exactly k nodes, then the value of a Vassiliev invariant v _k of type k on G, v _k (G), is independent of the embedding of G.

Proof.

The different embeddings of G can be represented by link diagrams with some of the 4-valent vertices in the diagram corresponding to the nodes of G. It suffices to show that the value of v _k (G) is unchanged under switching of a crossing. However, the exchange identity for v _k shows that this difference is equal to the evaluation of v _k on a graph with k + 1 nodes and hence is equal to zero. This completes the proof.

The upshot of this Lemma is that Vassiliev invariants of type k are intimately involved with certain abstract evaluations of graphs with k nodes. In fact, there are restrictions (the four-term relations) on these evaluations demanded by the topology (we shall articulate these restrictions shortly) and it follows from results of Kontsevich [4] that such abstract evaluations actually determine the invariants. The invariants derived from classical Lie algebras are all built from Vassiliev invariants of finite type. All this is directly related to Witten’s functional integral [52].

Definition.

Let v _k be a Vassiliev invariant of type k. The top row of v _k is the set of values that v _k assigns to the set of (abstract) 4-valent graphs with k nodes. If we concentrate on Vassiliev invariants of knots, then these graphs are all obtained by marking 2k points on a circle, and choosing a pairing of the 2k points. The pairing can be indicated by drawing a circle and connecting the paired points with arcs. Such a diagram is called a chord diagram. Some examples are indicated in Fig. 27.

Fig. 27

Chord diagrams

Note that a top row diagram cannot contain any isolated pairings since this would correspond to a difference of local curls on the corresponding knot diagram (and these curls, being isotopic, yield the same Vassiliev invariants.

The Four-Term Relation (Compare [45].) Consider a single embedded graphical node in relation to another embedded arc, as illustrated in Fig. 28. The arc underlies the lines incident to the node at four points and can be slid out and isotoped over the top so that it overlies the four nodes. One can also switch the crossings one-by-one to exchange the arc until it overlies the node. Each of these four switchings gives rise to an equation, and the left-hand sides of these equations will add up to zero, producing a relation corresponding to the right-hand sides. Each term in the right-hand side refers to the value of the Vassiliev invariant on a graph with two nodes that are neighbors to each other.

Fig. 28

The four term relation

There is a corresponding 4-term relation for chord diagrams. This is the 4-term relation for the top row. In chord diagrams the relation takes the form shown at the bottom of Fig. 28. Here we have illustrated only those parts of the chord diagram that are relevant to the two nodes in question (indicated by two pairs of points on the circle of the chord diagram). The form of the relation shows the points on the chord diagram that are immediate neighbors. These are actually neighbors on any chord diagram that realizes this form. Otherwise there can be many other pairings present in the situation.

As an example, consider the possible chord diagrams for a Vassiliev invariant of type 3. There are two possible diagrams as shown in Fig. 29. One of these has the projected pattern of the trefoil knot and we shall call it the trefoil graph. These diagrams satisfy the 4-term relation. This shows that one diagram must have twice the evaluation of the other. Hence it suffices to know the evaluation of one of these two diagrams to know the top row of a Vassiliev invariant of type 3. We can take this generator to be the trefoil graph

Fig. 29

The four term relation for a type three invariant

Now one more exercise: Consider any Vassiliev invariant v and let’s determine its value on the trefoil graph as in Fig. 30.

Fig. 30

Trefoil graph

The value of this invariant on the trefoil graph is equal to the difference between its values on the trefoil knot and its mirror image. Therefore any Vassiliev invariant that assigns a non-zero value to the trefoil graph can tell the difference between the trefoil knot and its mirror image.

Example.

This example shows how the original Jones polynomial is composed of Vassiliev invariants of finite type. Let V _K (t) denote the original Jones polynomial [14]. Recall the oriented state expansion for the Jones polynomial [27] with the basic formulas (δ is the loop value.)

$$\displaystyle\begin{array}{rcl} & V _{K_{+}} = -t^{1/2}V _{K_{0}} - tV _{K_{\infty }} & {}\\ & V _{K_{-}} = -t^{-1/2}V _{K_{0}} - t^{-1}V _{K_{\infty }}.& {}\\ & \delta = -(t^{1/2} + t^{-1/2}). & {}\\ \end{array}$$

Let t = e ^x. Then

$$\displaystyle\begin{array}{rcl} & V _{K_{+}} = -e^{x/2}V _{K_{0}} - e^{x}V _{K_{\infty }} & {}\\ & V _{K_{-}} = -e^{-x/2}V _{K_{0}} - e^{-x}V _{K_{\infty }}.& {}\\ & \delta = -(e^{x/2} + e^{-x/2}). & {}\\ \end{array}$$

Thus

$$\displaystyle{V _{K_{{\ast}}} = V _{K_{+}} - V _{K_{-}} = -2sinh(x/2)V _{K_{0}} - 2sinh(x)V _{K_{\infty }}.}$$

Thus x divides $V _{K_{{\ast}}}$, and therefore x ^k divides V _G whenever G is a graph with at least k nodes. Letting

$$\displaystyle{V _{G}(e^{x}) =\sum _{ k=0}^{\infty }v_{ k}(G)x^{k},}$$

we see that this condition implies that v _k (G) vanishes whenever G has more than k nodes. Hence the coefficients of the powers of x in the expansion of V _K (e ^x ) are Vassiliev invariants of finite type! This result was first observed by Birman and Lin [6] by a different argument.

Let’s look a little deeper and see the structure of the top row for the Vassiliev invariants related to the Jones polynomial. By our previous remarks the top row evaluations correspond to the leading terms in the power series expansion. Since

$$\displaystyle\begin{array}{rcl} & \delta = -(e^{x/2} + e^{-x/2}) = -2 + [higher],& {}\\ & -e^{x/2} + e^{-x/2} = -x + [higher], & {}\\ & -e^{x} + e^{-x} = -2x + [higher], & {}\\ \end{array}$$

it follows that the top rows for the Jones polynomial are computed by the recursion formulas

$$\displaystyle\begin{array}{rcl} & v(K_{{\ast}}) = -v(K_{0}) - 2V (K_{\infty })& {}\\ & v([loop]) = -2. & {}\\ \end{array}$$

The reader can easily check that this recursion formula for the top rows of the Jones polynomial implies that v ₃ takes the value 24 on the trefoil graph and hence it is the Vassiliev invariant of type 3 in the Jones polynomial that first detects the difference between the trefoil knot and its mirror image.

This example gives a good picture of the general phenomenon of how the Vassiliev invariants become building blocks for other invariants. In the case of the Jones polynomial, we already know how to construct the invariant and so it is possible to get a lot of information about these particular Vassiliev invariants by looking directly at the Jones polynomial. This, in turn, gives insight into the structure of the Jones polynomial itself.

5.1 Lie Algebra Weights

Consider the diagrammatic relation shown in Fig. 31. Call it (after Bar-Natan [4]) the STU relation.

Fig. 31

The STU relation

Lemma.

STU implies the 4-term relation.

Proof.

View Fig. 32.

Fig. 32

A diagrammatic proof

STU is the smile of the Cheshire cat. That smile generalizes the idea of a Lie algebra. Take a (matrix) Lie algebra with generators T ^a. Then

$$\displaystyle{T^{a}T^{b} - T^{b}T^{a} = if_{ abc}T^{c}}$$

expresses the closure of the Lie algebra under commutators. Translate this equation into diagrams as shown in Fig. 33, and see that this translation is STU with Lie algebraic clothing!

Fig. 33

Algebraic clothing

Here the structure tensor of the Lie algebra has been assumed (for simplicity) to be invariant under cyclic permutation of the indices. This invariance means that our last Lemma applies to this Lie algebraic interpretation of STU. The upshot is that we can manufacture weight systems for graphs that satisfy the 4-term relation by replacing paired points on the chord diagram by an insertion of T ^a in one point of the pair and a corresponding insertion of T ^a at the other point in the pair and summing over all a. The result of all such insertions on a given chord diagram is a big sum of specific matrix products along the circle of the diagram, each of which (being a circular product) is interpreted as a trace.

Let’s say this last matter more precisely: Regard a graph with k nodes as obtained by identifying k pairs of points on a circle. Thus a code such as 1212 taken in cyclic order specifies such a graph by regarding the points 1, 2, 1, 2 as arrayed along a circle with the first and second 1’s and 2’s identified to form the graph. Define,for a code a ₁ a ₂ … a _m

$$\displaystyle{wt(a_{1}a_{2}\ldots a_{m}) = trace(T^{a_{1} }T^{a_{2} }T^{a_{1} }\ldots T^{a_{m} })}$$

where the Einstein summation convention is in place for the double appearances of indices on the right-hand side. This gives the weight system.

The weight system described by the above procedure satisfies the 4-term relation, but does not necessarily satisfy the vanishing condition for isolated pairings. This is because the framing compensation for converting an invariant of regular isotopy to ambient isotopy has not yet been introduced. We will show how to do this in the course of the discussion in the next paragraph. The main point to make here is that by starting with the idea of extending an invariant of knots to a Vassiliev invariant of embedded graphs and searching out the conditions on graph evaluation demanded by the topology, we have inevitably entered the domain of relations between Lie algebras and link invariants. Since the STU relation does not demand Lie algebras for its satisfaction we see that the landscape is wider than the Lie algebra context, but it is not yet understood how big is the class of link invariants derived from Lie algebras.

In fact, we can line up this weight system with the formalism related to the knot diagram by writing the Lie algebra insertions back on the 4-valent graph. We then get a Casimir insertion at the node. See Fig. 34.

Fig. 34

Weight system and Casimir insertion

To get the framing compensation, note that an isolated pairing corresponds to the trace of the Casimir. Let γ denote this trace. See Fig. 34.

$$\displaystyle{\gamma = tr(\sum _{a}T^{a}T^{a})}$$

Let D be the trace of the identity. Then it is easy to see that we must compensate the given weight system by subtracting (γ∕D) multiplied by the result of dropping the identification of the two given points. We can diagram this by drawing two crossed arcs without a node drawn to bind them. Then the modified recursion formula becomes as shown in Fig. 35.

Fig. 35

Modified recursion formula

For example, in the case of SU(N) we have D = N, $\gamma = (N^{2} - 1)/2$ so that we get the transformation shown in Fig. 35, including the use of the Fierz identity.

For N = 2 the final formula of Fig. 35 is,up to a multiple, exactly the top row formula that we deduced for the Jones polynomial from its combinatorial structure.

5.2 Witten’s Functional Integral and Vassiliev Invariants

In [52] Edward Witten proposed a formulation of a class of 3-manifold invariants as generalized Feynman integrals taking the form Z(M) where

$$\displaystyle{Z(M) =\int dAexp[(ik/4\pi )S(M,A)].}$$

Here M denotes a 3-manifold without boundary and A is a gauge field (also called a gauge potential or gauge connection) defined on M. The gauge field is a one-form on a trivial G-bundle over M with values in a representation of the Lie algebra of G. The group G corresponding to this Lie algebra is said to be the gauge group. In this integral the “action” S(M, A) is taken to be the integral over M of the trace of the Chern-Simons three-form $CS = AdA + (2/3)AAA$. (The product is the wedge product of differential forms.)

Z(M) integrates over all gauge fields modulo gauge equivalence (see [2] for a discussion of the definition and meaning of gauge equivalence.)

The formalism and internal logic of Witten’s integral supports the existence of a large class of topological invariants of 3-manifolds and associated invariants of knots and links in these manifolds.

The invariants associated with this integral have been given rigorous combinatorial descriptions [31, 35, 41, 49, 51], but questions and conjectures arising from the integral formulation are still outstanding (see for example [3, 11–13, 40]). Specific conjectures about this integral take the form of just how it involves invariants of links and 3-manifolds, and how these invariants behave in certain limits of the coupling constant k in the integral. Many conjectures of this sort can be verified through the combinatorial models. On the other hand, the really outstanding conjecture about the integral is that it exists! At the present time there is no measure theory or generalization of measure theory that supports it. It is a fascinating exercise to take the speculation seriously, suppose that it does really work like an integral and explore the formal consequences. Here is a formal structure of great beauty. It is also a structure whose consequences can be verified by a remarkable variety of alternative means. Perhaps in the course of the exploration there will appear a hint of the true nature of this form of integration.

We now look at the formalism of the Witten integral in more detail and see how it involves invariants of knots and links corresponding to each classical Lie algebra. In order to accomplish this task, we need to introduce the Wilson loop. The Wilson loop is an exponentiated version of integrating the gauge field along a loop K in three-space that we take to be an embedding (knot) or a curve with transversal self-intersections. For this discussion, the Wilson loop will be denoted by the notation W _K (A) = < K | A > to denote the dependence on the loop K and the field A. It is usually indicated by the symbolism tr(Pexp(∫ _K A)). Thus

$$\displaystyle{W_{K}(A) =< K\vert A >= tr(Pexp(\int _{K}A)).}$$

Here the P denotes path ordered integration—we are integrating and exponentiating matrix valued functions, and so must keep track of the order of the operations. The symbol tr denotes the trace of the resulting matrix.

With the help of the Wilson loop functional on knots and links, Witten writes down a functional integral for link invariants in a 3-manifold M:

$$\displaystyle\begin{array}{rcl} & Z(M,K) =\int dAexp[(ik/4\pi )S(M,A)]tr(Pexp(\int _{K}A))& {}\\ & =\int dAexp[(ik/4\pi )S] < K\vert A >. & {}\\ \end{array}$$

Here S(M, A) is the Chern-Simons Lagrangian, as in the previous discussion.

We abbreviate S(M, A) as S and write < K | A > for the Wilson loop. Unless otherwise mentioned, the manifold M will be the three-dimensional sphere S ³

An analysis of the formalism of this functional integral reveals quite a bit about its role in knot theory. This analysis depends upon key facts relating the curvature of the gauge field to both the Wilson loop and the Chern-Simons Lagrangian. The idea for using the curvature in this way is due to Smolin [43, 44] (see also [38]). To this end, let us recall the local coordinate structure of the gauge field A(x), where x is a point in three-space. We can write A(x) = A _a ^k(x)T ^a dx _k where the index a ranges from 1 to m with the Lie algebra basis {T ¹, T ², T ³, …, T ^m}. The index k goes from 1 to 3. For each choice of a and k, A _a ^k(x) is a smooth function defined on three-space. In A(x) we sum over the values of repeated indices. The Lie algebra generators T ^a are matrices corresponding to a given representation of the Lie algebra of the gauge group G. We assume some properties of these matrices as follows:

• 1. [T ^a, T ^b] = if _abc T ^c where $[x,y] = xy - yx$, and f _abc (the matrix of structure constants) is totally antisymmetric. There is summation over repeated indices.

• 2. $tr(T^{a}T^{b}) =\delta ^{ab}/2$ where δ ^ab is the Kronecker delta (δ ^ab = 1 if a = b and zero otherwise).

We also assume some facts about curvature. (The reader may enjoy comparing with the exposition in [27]. But note the difference of conventions on the use of i in the Wilson loops and curvature definitions.) The first fact is the relation of Wilson loops and curvature for small loops:

Fact 1.

The result of evaluating a Wilson loop about a very small planar circle around a point x is proportional to the area enclosed by this circle times the corresponding value of the curvature tensor of the gauge field evaluated at x. The curvature tensor is written

$$\displaystyle{F_{a}^{rs}(x)T^{a}dx_{ r}dy_{s}.}$$

It is the local coordinate expression of AdA + AA. 

Application of Fact 1 Consider a given Wilson line < K | S > . Ask how its value will change if it is deformed infinitesimally in the neighborhood of a point x on the line. Approximate the change according to Fact 1, and regard the point x as the place of curvature evaluation. Let δ < K | A > denote the change in the value of the line. δ < K | A > is given by the formula

$$\displaystyle{\delta < K\vert A >= dx_{r}dx_{s}F_{a}^{rs}(x)T^{a} < K\vert A >.}$$

This is the first order approximation to the change in the Wilson line.

In this formula it is understood that the Lie algebra matrices T ^a are to be inserted into the Wilson line at the point x, and that we are summing over repeated indices. This means that each T ^a < K | A > is a new Wilson line obtained from the original line < K | A > by leaving the form of the loop unchanged, but inserting the matrix T ^a into that loop at the point x. A Lie algebra generator is diagrammed by a little box with a single index line and two input/output lines which correspond to its role as a matrix (hence as mappings of a vector space to itself). See Fig. 36.

Fig. 36

Wilson loop insertion

Remark.

In thinking about the Wilson line < K | A > = tr(Pexp(∫ _K A)), it is helpful to recall Euler’s formula for the exponential:

$$\displaystyle{e^{x} = lim_{ n\rightarrow \infty }(1 + x/n)^{n}.}$$

The Wilson line is the limit, over partitions of the loop K, of products of the matrices (1 + A(x)) where x runs over the partition. Thus we can write symbolically,

$$\displaystyle{< K\vert A >=\prod _{x\in K}(1 + A(x)) =\prod _{x\in K}(1 + A_{a}^{k}(x)T^{a}dx_{ k}).}$$

It is understood that a product of matrices around a closed loop connotes the trace of the product. The ordering is forced by the one-dimensional nature of the loop. Insertion of a given matrix into this product at a point on the loop is then a well-defined concept. If T is a given matrix then it is understood that T < K | A > denotes the insertion of T into some point of the loop. In the case above, it is understood from context in the formula

$$\displaystyle{dx_{r}dx_{s}F_{a}^{rs}(x)T^{a} < K\vert A >}$$

that the insertion is to be performed at the point x indicated in the argument of the curvature.

Remark.

The previous remark implies the following formula for the variation of the Wilson loop with respect to the gauge field:

$$\displaystyle{\delta < K\vert A > /\delta (A_{a}^{k}(x)) = dx_{ k}T^{a} < K\vert A >.}$$

Varying the Wilson loop with respect to the gauge field results in the insertion of an infinitesimal Lie algebra element into the loop.

Proof.

$$\displaystyle\begin{array}{rcl} & \delta < K\vert A > /\delta (A_{a}^{k}(x)) & {}\\ & =\delta \prod _{y\in K}(1 + A_{a}^{k}(y)T^{a}dy_{k})/\delta (A_{a}^{k}(x)) & {}\\ & =\prod _{y<x\in K}(1 + A_{a}^{k}(y)T^{a}dy_{k})[T^{a}dx_{k}]\prod _{y>x\in K}(1 + A_{a}^{k}(y)T^{a}dy_{k})& {}\\ & = dx_{k}T^{a} < K\vert A >. & {}\\ \end{array}$$

Fact 2.

The variation of the Chern-Simons Lagrangian S with respect to the gauge potential at a given point in three-space is related to the values of the curvature tensor at that point by the following formula:

$$\displaystyle{F_{a}^{rs}(x) =\epsilon _{ rst}\delta S/\delta (A_{a}^{t}(x)).}$$

Here ε _abc is the epsilon symbol for three indices, i.e. it is + 1 for positive permutations of 123 and − 1 for negative permutations of 123 and zero if any two indices are repeated.

With these facts at hand we are prepared to determine how the Witten integral behaves under a small deformation of the loop K. 

In accord with the theme of this paper, we shall use a system of abstract tensor diagrams to look at the differential algebra related to the functional integral. The translation to diagrams is accomplished with the aid of Figs. 37 and 38. In Fig. 37 we give diagrammatic equivalents for the component parts of our machinery. Tensors become labelled boxes. Indices become lines emanating from the boxes. Repeated indices that we intend to sum over become lines from one box to another. (The eye can immediately apprehend the repeated indices and the tensors where they are repeated.) Note that we use a capital D with lines extending from the top and the bottom for the partial derivative with respect to the gauge field, a capital W with a link diagrammatic subscript for the Wilson loop, a cubic vertex for the three-index epsilon, little triangles with emanating arcs for the differentials of the space variables.

Fig. 37

Notation

Fig. 38

Derivation

The Lie algebra generators are little boxes with single index lines and two input/output lines which correspond to their roles as matrices (hence as mappings of a vector space to itself). The Lie algebra generators are, in all cases of our calculation, inserted into the Wilson line either through the curvature tensor or through insertions related to differentiating the Wilson line.

In Fig. 38 we give the diagrammatic calculation of the change of the functional integral corresponding to a tiny change in the Wilson loop. The result is a double insertion of Lie Algebra generators into the line, coupled with the presence of a volume form that will vanish if the deformation does not twist in three independent directions. This shows that the functional integral is formally invariant under regular isotopy since the regular isotopy moves are changes in the Wilson line that happen entirely in a plane. One does not expect the integral to be invariant under a Reidemeister move of type one, and it is not. This framing compensation can be determined by the methods that we are discussing [32], but we will not go into the details of those calculations here.

In Fig. 39 we show the application of the calculation in Fig. 38 to the case of switching a crossing. The same formula applies, with a different interpretation, to the case where x is a double point of transversal self-intersection of a loop K, and the deformation consists in shifting one of the crossing segments perpendicularly to the plane of intersection so that the self-intersection point disappears. In this case, one T ^a is inserted into each of the transversal crossing segments so that T ^a T ^a < K | A > denotes a Wilson loop with a self-intersection at x and insertions of T ^a at x +ε ₁ and x +ε ₂ where ε ₁ and ε ₂ denote small displacements along the two arcs of K that intersect at x. In this case, the volume form is nonzero, with two directions coming from the plane of movement of one arc, and the perpendicular direction is the direction of the other arc. The reason for the insertion into the two lines is a direct consequence of the calculational form of Fig. 38: The first insertion is in the moving line, due to curvature. The second insertion is the consequence of differentiating the self-touching Wilson line. Since this line can be regarded as a product, the differentiation occurs twice at the point of intersection, and it is the second direction that produces the non-vanishing volume form.

Fig. 39

Crossing switch

Up to the choice of our conventions for constants, the switching formula is, as shown in Fig. 39,

$$\displaystyle\begin{array}{rcl} & Z(K_{+}) - Z(K_{-}) = (4\pi i/k)\int dAexp[(ik/4\pi )S]T^{a}T^{a} < K_{{\ast}{\ast}}\vert A >& {}\\ & = (4\pi i/k)Z(T^{a}T^{a}K_{{\ast}{\ast}}). & {}\\ \end{array}$$

The key point is to notice that the Lie algebra insertion for this difference is exactly what we did to make the weight systems for Vassiliev invariants (without the framing compensation). Thus the formalism of the Witten functional integral takes us directly to these weight systems in the case of the classical Lie algebras. The functional integral is central to the structure of the Vassiliev invariants.

5.3 Combinatorial Constructions for Vassiliev Invariants

Perhaps the most remarkable thing about this story of the structure of the Vassiliev invariants is the way that Lie algebras are so naturally involved in the structure of the weight systems. This shows the remarkably close nature of the combinatorial structure of Lie algebras and the combinatorics of knots and links via the Reidemeister moves. A really complete story about the Vassiliev invariants at this combinatorial level would produce their existence on the basis of the weight systems with entirely elementary arguments.

As we have already mentioned, one can prove that a given set of weights for the top row, satisfying the abstract four-term relation does imply that there exists a Vassiliev invariant of finite type n realizing these weights for graphs with n nodes. Proofs of this result either use analysis [1, 4] or non-trivial algebra [4, 7]. There is no known elementary combinatorial proof of the existence of Vassiliev invariants for given top rows.

Of course quantum link invariants (see Sect. 4 of these lectures.) do give combinatorial constructions for large classes of link invariants. These constructions rest on solutions to the Yang-Baxter equations, and it is not known how to describe the subset of finite type Vassiliev invariants that are so produced.

It is certainly helpful to look at the structure of Vassiliev invariants that arise from already-defined knot invariants. If V (K) is an already defined invariant of knots (and possibly links), then its extension to a Vassiliev invariant is calculated on embedded graphs G by expanding each graphical vertex into a difference by resolving the vertex into a positive crossing and a negative crossing. If we know that V (K) is of finite type n and G has n nodes then we can take any embedding of G that is convenient, and calculate V (G) in terms of all the knots that arise in resolving the nodes of this chosen embedding. This is a finite collection of knots. Since there is a finite collection of 4-valent graphs with n nodes, it follows that the top row evaluation for the invariant V (K) is determined by the values of V (K) on a finite collection of knots. Instead of asking for the values of the Vassiliev invariant on a top row, we can ask for this set of knots and the values of the invariant on this set of knots. A minimal set of knots that can be used to generate a given Vassiliev invariant will be called a knots basis for the invariant. Thus we have shown that the set consisting of the unknot, the right-handed trefoil and the left handed-trefoil is a knots basis for a Vassiliev invariant of type 3. See [36] for more information about this point of view.

A tantalizing combinatorial approach to Vassiliev invariants is due to Polyak and Viro [37]. They give explicit formulas for the second, third and fourth Vassiliev invariants and conjecture that their method will work for Vassiliev invariants of all orders. The method is as follows.

First one makes a new representation for oriented knots by taking Gauss diagrams. A Gauss diagram is a diagrammatic representation of the classical Gauss code of the knot. The Gauss code is obtained from the oriented knot diagram by first labelling each crossing with a naming label (such as 1,2,…) and also indicating the crossing type ( + 1 or − 1). Then choose a basepoint on the knot diagram and begin walking along the diagram, recording the name of the crossings encountered, their sign and whether the walk takes you over or under that crossing. For example, if you go under crossing 1 whose sign is + then you will record o + 1. Thus the Gauss code of the positive trefoil diagram is

$$\displaystyle{(o1+)(u2+)(o3+)(u1+)(o2+)(u3+).}$$

For prime knots the Gauss code is sufficient information to reconstruct the knot diagram. See [29] for a sketch of the proof of this result and for other references.

To form a Gauss diagram from a Gauss code, take an oriented circle with a basepoint chosen on the circle. Walk along the circle marking it with the labels for the crossings in the order of the Gauss code. Now draw chords between the points on the circle that have the same label. Orient each chord from overcrossing site to undercrossing site. Mark each chord with + 1 or − 1 according to the sign of the corresponding crossing in the Gauss code. The resulting labelled and basepointed graph is the Gauss diagram for the knot. See Fig. 40 for examples.

Fig. 40

Gauss diagrams

The Gauss diagram is deliberately formulated to have the structure of a chord diagram (as we have discussed for the weight systems for Vassiliev invariants). If G(K) is the Gauss diagram for a knot K, and D is an oriented (i.e. the chords as well as the circle in the diagram are oriented) chord diagram, let | G(K) | denote the number of chords in G(K) and | D | denote the number of chords in D. If | D | ≤ | G(K) | then we may consider oriented embeddings of D in G(K). For a given embedding i: D ⟶ G(K) define

$$\displaystyle{< i(D)\vert G(K) >= sign(i)}$$

where sign(i) denotes the product of the signs of the chords in G(K) ∩ i(D). Now suppose that C is a collection of oriented chord diagrams, each with n chords, and that

$$\displaystyle{eval: C\longrightarrow R}$$

is an evaluation mapping on these diagrams that satisfies the four-term relation at level n. Then we can define

$$\displaystyle{< D\vert K >=\sum _{i:D\longrightarrow G(K)} < i(D)\vert G(K) >}$$

and

$$\displaystyle{v(K) =\sum _{D\in C} < D\vert K > eval(D).}$$

For appropriate oriented chord subsets this definition can produce Vassiliev invariants v(K) of type n. For example, in the case of the Vassiliev invariant of type three taking value 0 on the unknot and value 1 on the right-handed trefoil, − 1 on the left-handed trefoil, Polyak and Viro give the specific formula

$$\displaystyle{v_{3}(K) =< A\vert K > +(1/2) < B\vert K >}$$

where A denotes the trefoil chord diagram as we described it in Sect. 3 and B denotes the three-chord diagram consisting of two parallel chords pierced by a third chord. In Fig. 41 we show the specific orientations for the chord diagrams A and B. The key to this construction is in the choice of orientations for the chord diagrams in C = { A, B}. It is a nice exercise in translation of the Reidemeister moves to Gauss diagrams to see that v ₃(K) is indeed a knot invariant.

Fig. 41

Oriented chord diagrams for v ₃

It is possible that all Vassiliev invariants can be constructed by a method similar to the formula v(K) = ∑ _D ∈ C  < D | K > eval(D). This remains to be seen.

5.4 Invertibility and the knot 8₁₇

It is an open problem whether there are Vassiliev invariants that can detect the difference between a knot and its reverse (The reverse of an oriented knot is obtained by flipping the orientation.). The smallest instance of a non-invertible knot is the knot 8₁₇ depicted in Fig. 42. Hale Trotter [46] was the first person to give proofs that some knots are non-invertible. We have not discussed his methods in this paper.

Fig. 42

Tangle decomposition of 8₁₇

Thus, at the time of this writing there is no known Vassiliev invariant that can detect the non-invertibility of 8₁₇. On the other hand, the tangle decomposition shown in Fig. 42 can be used in conjunction with the results of Siebenmann and Bonahon [42] and the formulations of John Conway [8] to show this non-invertibility. These tangle decomposition methods use higher level information about the diagrams than is easy to encode in Vassiliev invariants. The purpose of this section is to underline this discrepancy between different levels in the combinatorial topology.