English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

It is well known that any link can be represented by the closure of a braid.The minimum number of strings needed in a braid whose closure represents agiven link is called the braid index of the link and the well knownMorton-Frank-Williams inequality reveals a close relationship between theHOMFLY polynomial of a link and its braid index. In the case that a link isalready presented in a closed braid form, Jaeger derived a special formulationof the HOMFLY polynomial. In this paper, we prove a variant of Jaeger's resultas well as a dual version of it. Unlike Jaeger's original reasoning, whichrelies on representation theory, our proof uses only elementary geometric andcombinatorial observations. Using our variant and its dual version, we providea direct and elementary proof of the fact that the braid index of a link thathas an $n$-string closed braid diagram that is also reduced and alternating, isexactly $n$. Until know this fact was only known as a consequence of a resultdue to Murasugi on fibered links that are star products of elementary toruslinks and of the fact that alternating braids are fibered.

The HOMFLY Polynomial of Links in Closed Braid Form