Abstract
Knots are defined as embeddings of a circle in 3-dimensional Euclidean space, but can be faithfully represented by finite structures, such as graphs or words. One of such discrete representations is a Gauss code. In this paper we consider knot transformations in terms of string rewriting systems. We formulate the concept of knot transformations in the context of Gauss word rewriting and present linear lower and upper bounds on the length of knot transformations for the equivalence problem of two knot diagrams reachable by a sequence of Reidemeister moves of type I and II.
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References
Baader, F., Nipkow, T.: Term rewriting and all that. Cambridge Univ. Pr. (1999)
Björklund, H., Schwentick, T.: On Notions of Regularity for Data Languages. In: Csuhaj-Varjú, E., Ésik, Z. (eds.) FCT 2007. LNCS, vol. 4639, pp. 88–99. Springer, Heidelberg (2007)
Coward, A.: Ordering the Reidemeister moves of a classical knot. Algebraic & Geometric Topology 6, 659–671 (2006)
Coward, A., Lackenby, M.: An upper bound on reidemeister moves. Arxiv preprint arXiv:1104.1882 (2011)
Haken, W.: Theorie der Normalfachen, ein Isotopiekriterium fur den Kreisknoten. Journal of Acta Mathematica 105 (1961)
Hass, J., Lagarias, J.C.: The number of Reidemeister moves needed for unknotting. Journal of the American Mathematical Society 14(2), 399–428 (2001)
Hass, J., Nowik, T.: Unknot diagrams requiring a quadratic number of Reidemeister moves to untangle. Discrete and Computational Geometry 44(1), 91–95 (2010)
Hayashi, C.: A lower bound for the number of Reidemeister moves for unknotting. Journal of Knot Theory and its Ramifications 15(3), 313 (2006)
Kauffman, L.H.: Virtual knot theory. Arxiv Preprint Math. GT/, 9811028 (1998)
Lisitsa, A., Potapov, I., Saleh, R.: Planarity of Knots, Register Automata and LogSpace Computability. In: Dediu, A.-H., Inenaga, S., MartÃn-Vide, C. (eds.) LATA 2011. LNCS, vol. 6638, pp. 366–377. Springer, Heidelberg (2011)
Neven, F., Schwentick, T., Vianu, V.: Towards Regular Languages over Infinite Alphabets. In: Sgall, J., Pultr, A., Kolman, P. (eds.) MFCS 2001. LNCS, vol. 2136, pp. 560–572. Springer, Heidelberg (2001)
Newman, M.H.A.: On theories with a combinatorial definition of equivalence. The Annals of Mathematics 43(2), 223–243 (1942)
Reidemeister, K.: Elementare Begründung der Knotentheorie. In: Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, vol. 5, pp. 24–32. Springer (1927)
Suh, C.H.O.: A short proof of the hass–lagarias theorem (2008)
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Saleh, R. (2012). On the Length of Knot Transformations via Reidemeister Moves I and II. In: Finkel, A., Leroux, J., Potapov, I. (eds) Reachability Problems. RP 2012. Lecture Notes in Computer Science, vol 7550. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33512-9_11
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DOI: https://doi.org/10.1007/978-3-642-33512-9_11
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