Abstract
A knot diagram looks like a two-dimensional drawing of a knotted rubberband. Proving that a given knot diagram can be untangled (that is, is a trivial knot, called an unknot) is one of the most famous problems of knot theory. For a small knot diagram, one can try to find a sequence of untangling moves explicitly, but for a larger knot diagram producing such a proof is difficult, and the produced proofs are hard to inspect and understand. Advanced approaches use algebra, with an advantage that since the proofs are algebraic, a computer can be used to produce the proofs, and, therefore, a proof can be produced even for large knot diagrams. However, such produced proofs are not easy to read and, for larger diagrams, not likely to be human readable at all. We propose a new approach combining advantages of these: the proofs are algebraic and can be produced by a computer, whilst each part of the proof can be represented as a reasonably small knot-like diagram (a new representation as a labeled tangle diagram), which can be easily inspected by a human for the purposes of checking the proof and finding out interesting facts about the knot diagram.
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Notes
- 1.
In the implementation presented in Sect. 6 we did not use word reversion because, although reversing a word may make a proof shorter, it is not easy to implement it in the prover software we used.
- 2.
Another approach that we tested (as presented in Sect. 6) is simply to place two tangle diagrams next to each other without connecting them by an arc; this corresponds, in terms of labelling words, to concatenating the words.
- 3.
System used in experiments: Intel(R) Core(TM) i7-4790 CPU 3.60 Ghz, RAM 32 GB, Windows 7 Enterprise.
References
Burde, G., Heusener, M., Zieschang, H.: Knots. De Gruyter (2013)
Coward, A., Lackenby, M.: An upper bound on reidemeister moves. Am. J. Math. 136(4), 1023–1066 (2014)
Dynnikov, I.A.: Three-page approach to knot theory. Universal semigroup. Funct. Anal. Appl. 34(1), 24–32 (2000)
Fish, A., Lisitsa, A.: Detecting unknots via equational reasoning, I: exploration. In: Watt, S.M., Davenport, J.H., Sexton, A.P., Sojka, P., Urban, J. (eds.) CICM 2014. LNCS (LNAI), vol. 8543, pp. 76–91. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-08434-3_7
Fish, A., Lisitsa, A., Stanovský, D.: A combinatorial approach to knot recognition. In: Horne, R. (ed.) EGC 2015. CCIS, vol. 514, pp. 64–78. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-25043-4_7
Gilbert, N.D., Porter, T.: Knots and Surfaces. Oxford University Press, Oxford (1994)
Kauffman, L.H., Henrich, A.: Unknotting unknots. https://arxiv.org/abs/1006.4176
Kauffman, L.H., Lambropoulou, S.: Hard unknots and collapsing tangles. https://arxiv.org/abs/math/0601525v5
Kawauchi, A.: A Survey of Knot Theory. Birkhäuser, Basel (1996). https://doi.org/10.1007/978-3-0348-9227-8
Lickorish, W.B.R.: An Introduction to Knot Theory, vol. 175. Springer Science & Business Media, New York (2012). https://doi.org/10.1007/978-1-4612-0691-0
Lisitsa, A., Vernitski, A.: Automated reasoning for knot semigroups and \(\pi \)-orbifold groups of knots. In: Blömer, J., Kotsireas, I.S., Kutsia, T., Simos, D.E. (eds.) MACIS 2017. LNCS, vol. 10693, pp. 3–18. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-72453-9_1
Livingston, C.: Knot Theory, vol. 24. Cambridge University Press, Cambridge (1993)
Manturov, V.: Knot Theory. CRC Press, Boca Raton (2004)
McCune, W.: Prover9 and Mace4 (2005–2010). http://www.cs.unm.edu/~mccune/prover9/
Morgan, J.W., Bass, H. (eds.): The Smith Conjecture. Elsevier, New York (1984)
Murasugi, K.: Knot Theory and Its Applications. Springer Science & Business Media, New York (2007). https://doi.org/10.1007/978-0-8176-4719-3
Ochiai, M.: Non-trivial projections of the trivial knot. http://repository.kulib.kyoto-u.ac.jp/dspace/handle/2433/99940
Vernitski, A.: Describing semigroups with defining relations of the form xy = yz and yx = zy and connections with knot theory. Semigroup Forum 95(1), 66–82 (2017)
Winker, S.K.: Quandles, knot invariants, and the n-fold branched cover. Ph.D. thesis, University of Illinois at Chicago (1984)
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Fish, A., Lisitsa, A., Vernitski, A. (2018). Visual Algebraic Proofs for Unknot Detection. In: Chapman, P., Stapleton, G., Moktefi, A., Perez-Kriz, S., Bellucci, F. (eds) Diagrammatic Representation and Inference. Diagrams 2018. Lecture Notes in Computer Science(), vol 10871. Springer, Cham. https://doi.org/10.1007/978-3-319-91376-6_12
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