A characterisation of alternating knot exteriors

Howie, Joshua¹

¹ School of Mathematics and Statistics, University of Melbourne, Melbourne VIC 3010, Australia

Geometry & topology, Tome 21 (2017) no. 4, pp. 2353-2371.

Résumé

We give a topological characterisation of alternating knot exteriors based on the presence of special spanning surfaces. This shows that being alternating is a topological property of the knot exterior and not just a property of diagrams, answering an old question of Fox. We also give a characterisation of alternating link exteriors which have marked meridians. We then describe a normal surface algorithm which can decide if a knot is alternating given a triangulation of its exterior as input.

Détail

DOI : 10.2140/gt.2017.21.2353

Classification : 57M25
Keywords: alternating knot, alternating link, spanning surface, normal surface algorithm

Bibliographie

[1] C Adams, T Kindred, A classification of spanning surfaces for alternating links, Algebr. Geom. Topol. 13 (2013) 2967 | DOI

[2] R J Aumann, Asphericity of alternating knots, Ann. of Math. 64 (1956) 374 | DOI

[3] C L Curtis, S J Taylor, The Jones polynomial and boundary slopes of alternating knots, J. Knot Theory Ramifications 20 (2011) 1345 | DOI

[4] C H Dowker, M B Thistlethwaite, Classification of knot projections, Topology Appl. 16 (1983) 19 | DOI

[5] C M Gordon, Links and their complements, from: "Topology and geometry : commemorating SISTAG" (editors A J Berrick, M C Leung, X Xu), Contemp. Math. 314, Amer. Math. Soc. (2002) 71 | DOI

[6] C M Gordon, J Luecke, Knots are determined by their complements, J. Amer. Math. Soc. 2 (1989) 371 | DOI

[7] J E Greene, Alternating links and definite surfaces, preprint (2015)

[8] W Haken, Theorie der Normalflächen, Acta Math. 105 (1961) 245 | DOI

[9] A E Hatcher, On the boundary curves of incompressible surfaces, Pacific J. Math. 99 (1982) 373 | DOI

[10] J A Howie, Surface-alternating knots and links, PhD thesis, University of Melbourne (2015)

[11] J A Howie, J H Rubinstein, Weakly generalised alternating links,

[12] W Jaco, J H Rubinstein, 0–efficient triangulations of 3–manifolds, J. Differential Geom. 65 (2003) 61 | DOI

[13] W Jaco, J H Rubinstein, Inflations of ideal triangulations, Adv. Math. 267 (2014) 176 | DOI

[14] W Jaco, E Sedgwick, Decision problems in the space of Dehn fillings, Topology 42 (2003) 845 | DOI

[15] W Jaco, J L Tollefson, Algorithms for the complete decomposition of a closed 3–manifold, Illinois J. Math. 39 (1995) 358

[16] W B R Lickorish, An introduction to knot theory, 175, Springer (1997) | DOI

[17] W Menasco, Closed incompressible surfaces in alternating knot and link complements, Topology 23 (1984) 37 | DOI

[18] W Menasco, M Thistlethwaite, The classification of alternating links, Ann. of Math. 138 (1993) 113 | DOI

[19] T Nowik, Dissecting the 2–sphere by immersions, Geom. Dedicata 127 (2007) 37 | DOI

[20] C Petronio, Ideal triangulations of link complements and hyperbolicity equations, Geom. Dedicata 66 (1997) 27 | DOI

[21] J H Rubinstein, An algorithm to recognize the 3–sphere, from: "Proceedings of the International Congress of Mathematicians" (editor S D Chatterji), Birkhäuser (1995) 601

[22] A Thompson, Thin position and the recognition problem for S3, Math. Res. Lett. 1 (1994) 613 | DOI