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Verification of the Jones unknot conjecture up to 22 crossings. (English) Zbl 1386.57016

Summary: We proved by computer enumeration that the Jones polynomial distinguishes the unknot for knots up to 22 crossings. Following an approach by Shuji Yamada, we generated knot diagrams by inserting algebraic tangles into Conway polyhedra, computed their Jones polynomials by a divide-and-conquer method, and tested those with trivial Jones polynomials for unknottedness with the computer program SnapPy. We employed numerous novel strategies for reducing the computation time per knot diagram and the number of knot diagrams to be considered. That made computations up to 21 crossings possible on a single processor desktop computer. We explain these strategies in this paper. We also provide total numbers of algebraic tangles up to 18 crossings and of Conway polyhedra up to 22 vertices. We encountered new unknot diagrams with no crossing-reducing pass moves in our search. We report one such diagram in this paper.

MSC:

57M25 Knots and links in the \(3\)-sphere (MSC2010)
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
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References:

[1] Adams, C., The Knot Book, An Elementary Introduction to the Mathematical Theory of Knots (AMS, 2004). · Zbl 1065.57003
[2] A. Champanerkar and I. Kofman, A survey on the Turaev Genus of Knots, arXiv:1406.1945. · Zbl 1306.57007
[3] Anstee, R. P., Przytycki, J. H. and Rolfsen, D., Knot polynomials and generalized mutation, Topology Appl.32 (1989) 237-249. · Zbl 0638.57006
[4] D. Bar-Natan, Fast Khovanov Homology Computations, arXiv:0606318. · Zbl 1234.57013
[5] Bigelow, S., Does the Jones polynomial detect the unknot?, J. Knot Theory Ramifications11 (2002) 493-505. · Zbl 1003.57017
[6] Boost, C++ libraries, http://www.boost.org.
[7] Boyer, J. M. and Myrvold, W. J., On the edge: Simplified O(n) planarity testing by edge addition, J. Graph Theory Appl.8 (2004) 241-273. · Zbl 1086.05067
[8] Brinkmann, G., Greenberg, S., Greenhill, C., McKay, B. D., Thomas, R. and Wollan, P., Generation of simple quadrangulations of the sphere, Discrete Math.305 (2005) 33-54. · Zbl 1078.05023
[9] Brinkmann, G. and McKay, B. D., Fast generation of planar graphs, MATCH Commun. Math. Comput. Chem.58 (2007) 323-357. · Zbl 1164.68025
[10] Cheng, Z. Y. and Gao, H. Z., Mutation on knots and Whitney’s 2-isomorphism theorem, Acta Math. Sinica, Engl. Ser.29 (2013) 1219-1230. · Zbl 1270.57018
[11] Conway, J. H., An enumeration of knots and links and some of their algebraic properties, in Computational Problems in Abstract Algebra, ed. Leech, J. (Pergamon Press, Oxford, England, 1970), pp. 329-358, http://www.maths.ed.ac.uk/˜aar/papers/conway.pdf.
[12] Dasbach, O. T. and Hougardy, S., Does the Jones polynomial detect unknottedness?Exp. Math.6 (1997) 51-56. · Zbl 0883.57006
[13] J. D’Aurizio, Sum of Catalan numbers, http://math.stackexchange.com/questions/ 903593/sum-of-catalan-numbers.
[14] Eliahou, S., Kauffman, L. H. and Thistlethwaite, M. B., Infinite families of links with trivial Jones polynomial, Topology42 (2001) 155-169. · Zbl 1013.57005
[15] Ganzell, S., Local moves and restrictions on the Jones polynomial, J. Knot Theory Ramifications23 (2014) 1450011. · Zbl 1290.57006
[16] Hoste, J., Thistlethwaite, M. and Weeks, J., The First 1,701,935 Knots, Math. Intell.20(4) (1998) 33-48. · Zbl 0916.57008
[17] Ito, T., A kernel of a braid group representation yields a knot with trivial knot polynomials, Math. Z.280 (2015) 347-353. · Zbl 1323.57009
[18] Jaeger, F., Vertigan, D. L. and Welsh, D. J. A., On the computational complexity of the Jones and Tutte polynomials, Math. Proc. Cambridge Phil. Soc.108 (1990) 35-53. · Zbl 0747.57006
[19] Jones, V. F. R., Ten problems, in Mathematics: Frontiers and Perspectives (American Mathematical Society, Providence, 2000), pp. 79-91. · Zbl 0969.57001
[20] Jones, V. F. R. and Rolfsen, D. P. O., A theorem regarding 4-braids and the \(V = 1\) problem, in Proceedings of the Conference on Quantum Topology (Manhattan, KS, 1993) (World Scientific, 1994), pp. 127-135. · Zbl 0900.20066
[21] Kauffman, L. H., State models and the Jones polynomial, Topology26 (1987) 395-407. · Zbl 0622.57004
[22] Kauffman, L. H. and Lambropoulou, S., On the classification of rational tangles, Adv. Appl. Math.33(2) (2004) 199-237. · Zbl 1057.57006
[23] Knot Atlas, katlas.org/wiki/The_Determinant_and_the_Signature.
[24] I. Kofman and Y. Rong, private communication.
[25] Kredel, H., Evaluation of a Java computer algebra system, Comput. Math.5081 (2008) 121-138. · Zbl 1166.68377
[26] P. B. Kronheimer and T. S. Mrowka, Khovanov homology is an unknot-detector, arXiv:1005.4346. · Zbl 1241.57017
[27] M. Lackenby, The crossing number of composite knots, http://arXiv.org/pdf/0805.4706.pdf. · Zbl 1190.57003
[28] Meringer, M., Fast generation of regular graphs and construction of cages, J. Graph Theory30 (1999) 137-146. · Zbl 0918.05062
[29] Tables of various graph enumerations, including those in this work up 18 vertices, appear in http://www.mathe2.uni-bayreuth.de/markus/reggraphs.html.
[30] Murasugi, K., Jones polynomials and classical conjectures in knot theory, Topology26 (1987) 187-194. · Zbl 0628.57004
[31] Przytycki, J. H., Fundamentals of Kauffman bracket skein modules, Kobe Math. J.16(1) (1999) 45-66, arXiv:math/9809113. · Zbl 0947.57017
[32] Przytycki, J. H., Search for different links with the same Jones’ type polynomials, in Ideas from Graph Theory and Statistical Mechanics, , Vol. 34 (Banach Center Publications, Warszawa1995), pp. 121-148, arXiv:math/0405447. · Zbl 0848.57011
[33] Rolfsen, D., The quest for a knot with trivial Jones polynomial: Diagram surgery and the Temperley-Lieb algebra, in Topics in Knot Theory, ed. Bozhüyük, M. E. (Kluwer Academic Publications, Dordrecht, 1993), pp. 195-210. · Zbl 0833.57002
[34] SnapPy, program for studying the topology and geometry of 3-manifolds, www.math.uic.edu/t3m/SnapPy.
[35] Thistlethwaite, M. B., A spanning tree expansion of the Jones polynomial, Topology26 (1987) 297-309. · Zbl 0622.57003
[36] Thistlethwaite, M. B., Links with trivial Jones polynomial, J. Knot Theory Ramifications10 (2001) 641-643. · Zbl 1001.57022
[37] M. B. Thistlethwaite, private communication.
[38] Yamada, S., How to find knots with unit Jones polynomials, in Proc. Conf. Dedicated to Prof. K. Murasugi for his 70th birthday (Toronto, July 1999) (Springer, 2000), pp. 355-361.
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