Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-27T01:58:25.293Z Has data issue: false hasContentIssue false
  • English
  • Français

The braid index of reduced alternating links

Part of: Low-dimensional topology

Published online by Cambridge University Press:  10 January 2019

YUANAN DIAO ,
GÁBOR HETYEI  and
PENGYU LIU
YUANAN DIAO*
Affiliation:
Department of Mathematics and Statistics, University of North Carolina Charlotte, Charlotte, NC 28223, U.S.A.
GÁBOR HETYEI
Affiliation:
Department of Mathematics and Statistics, University of North Carolina Charlotte, Charlotte, NC 28223, U.S.A.
PENGYU LIU
Affiliation:
Department of Mathematics and Statistics, University of North Carolina Charlotte, Charlotte, NC 28223, U.S.A.
*
e-mails: [email protected], [email protected], [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

It is well known that the minimum crossing number of an alternating link equals the number of crossings in any reduced alternating link diagram of the link. This remarkable result is an application of the Jones polynomial. In the case of the braid index of an alternating link, Yamada showed that the minimum number of Seifert circles over all regular projections of a link equals the braid index. Thus one may conjecture that the number of Seifert circles in a reduced alternating diagram of the link equals the braid index of the link, but this turns out to be false. In this paper we prove the next best thing that one could hope for: we characterise exactly those alternating links for which their braid indices equal the numbers of Seifert circles in their corresponding reduced alternating link diagrams. More specifically, we prove that if D is a reduced alternating link diagram of an alternating link L, then b(L), the braid index of L, equals the number of Seifert circles in D if and only if GS(D) contains no edges of weight one. Here GS(D), called the Seifert graph of D, is an edge weighted simple graph obtained from D by identifying each Seifert circle of D as a vertex of GS(D) such that two vertices in GS(D) are connected by an edge if and only if the two corresponding Seifert circles share crossings between them in D and that the weight of the edge is the number of crossings between the two Seifert circles. This result is partly based on the well-known MFW inequality, which states that the a-span of the HOMFLY polynomial of L is a lower bound of 2b(L)−2, as well as the result of Yamada relating the minimum number of Seifert circles over all link diagrams of L to b(L).

MSC classification

Primary: 57M25: Knots and links in $S^3$
Secondary: 57M15: Relations with graph theory 57M27: Invariants of knots and 3-manifolds
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 168 , Issue 3 , May 2020 , pp. 415 - 434
Copyright
Copyright © Cambridge Philosophical Society 2019

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Birman, J. and Menasco, W.. Studying links via closed braids III: classifying links which are closed 3-braids. Pac. J. Math. 161 (1993), 25–113.CrossRefGoogle Scholar
Cromwell, P.. Homogeneous links. J. London Math. Soc. 39 (1989), 535–552.CrossRefGoogle Scholar
Elrifai, E.. Positive braids and Lorenz links. PhD thesis. Liverpool University, (1988).Google Scholar
Franks, J. and Williams, R.. Braids and the Jones polynomial. Trans. Amer. Math. Soc. 303 (1987), 97–108.CrossRefGoogle Scholar
Freyd, P., Yetter, D., Hoste, J., Lickorish, W., Millett, K. and Ocneanu, A.. A new polynomial invariant of knots and link. Bull. Am. Math. Soc. (N.S.) 12 (1985), 239–246.CrossRefGoogle Scholar
Jaeger, F.. Circuit partitions and the HOMFLY polynomial of closed braids. Trans. Am. Math. Soc. 323 (1991), no. 1, 449–463.CrossRefGoogle Scholar
Jones, V.. Hecke algebra representations of braid groups and link polynomials. Ann. Math. 126 (1987), 335–388.CrossRefGoogle Scholar
Kauffman, L.. State models and the Jones polynomial. Topology 26 (1987), no. 3, 395–407.CrossRefGoogle Scholar
Lee, S. and Seo, M.. A formula for the braid index of links. Topol. Appl. 157 (2010), 247–260.CrossRefGoogle Scholar
Liu, P., Diao, Y. and Hetyei, G.. The HOMFLY polynomial of links in closed braid form. Discrete Math. to appear.Google Scholar
Morton, H.. Seifert circles and knot polynomials. Math. Proc. Camb. Phil. Soc. 99 (1986), 107–109.CrossRefGoogle Scholar
Morton, H. and Short, H.. The 2-variable polynomial of cable knots. Math. Proc. Camb. Phil. Soc. 101 (1987), 267–278.CrossRefGoogle Scholar
Murasugi, K.. Jones polynomials and classical conjectures in knot theory. Topology 26 (1987), no. 2, 187–194.CrossRefGoogle Scholar
Murasugi, K.. On the braid index of alternating links. Trans. Am. Math. Soc. 326 (1991), 237–260.CrossRefGoogle Scholar
Murasugi, K. and Przytycki, J.. An index of a graph with applications to knot theory. Mem. Am. Math. Soc. 106 (1993), no. 508.Google Scholar
Nakamura, T.. Positive alternating links are positively alternating. J. Knot Theory Ramif. 9 (2000), 107–112.CrossRefGoogle Scholar
Przytycki, J. and Traczyk, P.. Conway algebras and Skein equivalence of links. Proc. Am. Math. Soc. 100 (1987), 744–748.CrossRefGoogle Scholar
Stoimenow, A.. On the crossing number of positive knots and braids and braid index criteria of Jones and Morton–Williams–Franks. Trans. Am. Math. Soc. 354 (2002), 3927–3954.CrossRefGoogle Scholar
Stoimenow, A.. On the definition of graph index. J. Aust. Math. Soc. 94 (2013), 417–429.CrossRefGoogle Scholar
Stoimenow, A.. Diagram Genus, Generators and Applications (Chapman & Hall/CRC Monographs and Research Notes in Mathematics, 2016).Google Scholar
Tait, P.. On Knots I, II, III. Scientific Papers 1 (Cambridge University Press, London, 1990), 273–347.Google Scholar
Thistlethwaite, M.. A spanning tree expansion of the Jones polynomial. Topology 26 (1987), no. 3, 297–309.CrossRefGoogle Scholar
Traczyk, P.. On the index of graphs: index versus cycle index. J. Knot Theory Ramif. 3 (1994), no. 4, 531–536.CrossRefGoogle Scholar
Traczyk, P.. Reducing braid index by Murasugi–Przytycki operation. J. Knot Theory Ramif. 20 (2011), no. 1, 223–229.CrossRefGoogle Scholar
Yamada, S.. The minimal number of Seifert circles equals the braid index of a link. Invent. Math. 89 (1987), 347–356.CrossRefGoogle Scholar