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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: ydiao@uncc.edu, ghetyei@uncc.edu, pliu10@uncc.edu
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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

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