An expression for the Homflypt polynomial and some applications (Q392815)

This paper is concerned with the two variable, called in this review skein (elsewhere, also by the author, HOMFLY-PT), link polynomial \(P\), for links \(L\) with closed \(n\)-braid representations \(L=\hat\beta\) with \(\beta\in B_n\). The Morton-Franks-Williams (MFW) inequality can be put to state that, for a usual choice of variables \(P(v,z)\), the polynomial \(P(L)\) has at most \(n\) non-zero coefficients in \(v\), in degrees \(w-n+1+2j\), for \(j=0,\dots,n-1\), where \(w\) is the writhe (exponent sum) of \(\beta\). These coefficients are in turn (Laurent) polynomials in \(z\), and are labelled in the paper by \(h_j(z)\). One main result (Theorem 2.1) is an expression for the first three Laurent coefficient polynomials \(h_0,h_1,h_2\) as a function of the other \(h_j\) for \(j\geq 3\), and the Conway polynomial (together with \(w\) and \(n\)).NEWLINENEWLINEThese expressions are used to study the shape of the skein polynomial for \(n\)-braid links. This is particularly useful for \(n=3\), where \(h_j\), \(j\geq 3\), do not occur. The proofs build entirely on the skein relation approach, but in some intricate and laborious way. One highlight, Proposition 2.19, is the positive resolution of a problem encountered and partially settled by the reviewer: does the Jones polynomial of a 3-braid link determine its skein polynomial?NEWLINENEWLINETwo minor remarks:NEWLINENEWLINEThe 3-braid links with \(v\)-span 2 or 0 (i.e., unsharp MWF inequality), requested below equation (37), are identified in the reviewer's paper [``Coefficients and non-triviality of the Jones polynomial'', J. Reine Angew. Math. 657, 1--55 (2011; Zbl 1257.57010)] (and also discussed in the paper cited as [\textit{A. Stoimenow}, Properties of closed 3-braids, {\url arXiv:math/0606435}]; they indeed all come from Birman's construction). In Theorem 2.4.(iii), all `\(J\)' should be `\(j\)'.