Inner products on the Hecke algebra of the braid group (Q629722)

Given braids \(a,b\) on \(n\) strings, consider the (Laurent polynomial in \(z\)) coefficient of \(v^{\text{w}+n-1}\) in \((-z)^{n-1}\) times the framed Homfly polynomial of the closure of \(ab^*\), where w is the writhe of \(ab^*\) and \(b^*\) is the reverse braid of \(b\). This definition extends to a symmetric bilinear form on the Hecke algebra \(H_n(z)\) associated to the Artin braid group (the symmetry follows from the fact that the closures of \(ab^*\) and \(ba^*\) are isotopic links, since \((ab^*)^*=ba^*\)). Note that \(\text{w}+n-1\) is the possible highest \(v\)-degree of the Homfly polynomial, by the Morton-Franks-Williams (MFW) inequality. The main theorem in the paper states that the set of positive permutation braids on \(n\) strings (called simple braids in the more recent literature) is an orthonormal basis for this product. In particular, the MFW inequality is not sharp for the Homfly polynomial of the closure of a braid if and only if the braid is orthogonal to the identity braid. The proof uses a nice connection with Contact Topology. Given permutations \(\alpha\) and \(\beta\), the author constructs a front projection with exactly \(2n\) cusps of a Legendrian knot that represents the closure of \(\nu_{\alpha} \nu_{\beta}^*\), where \(\nu_{\alpha}\) denotes the negative permutation braid associated to~\(\alpha\) (the author works with negative permutation braids and the analogous ``left'' inner product). The Thurston-Bennequin number of the projection is \(tb=\text{w}-n\), where \(\text{w}\) is the writhe of \(\nu_{\alpha} \nu_{\beta}^*\). It is trivial to see that the ruling polynomial of this front [\textit{D. Fuchs}, J. Geom. Phys. 47, No.~1, 43--65 (2003; Zbl 1028.57005)] is either \(z^{1-n}\) or zero, depending on whether \(\alpha =\beta\) or not. Then, the key result, proved by Rutherford [\textit{D. Rutherford}, Int. Math. Res. Not. 2006, No. 9, Article ID 78591, 15 p. (2006; Zbl 1106.57012)], is the fact that this polynomial is the coefficient of \(v^{tb + 1}\) in the Homfly Polynomial of the oriented link represented by the front. A purely algebraic proof of the main result (not in the paper) can also be given by induction on the writhe of the braid \(b\) by using the skein Homfly relation and properties of the simple braids. The first step in the induction is equivalent to saying that the highest \(v\)-degree of the Homfly polynomial of the closure of a simple braid on \(n\) strings and w crossings is strictly lower than \(\text{w}+n-1\), except if it is the identity braid, in which case \(\text{w}=0\) and the highest \(v\)-degree monomial is \((-z)^{1-n} v^{n-1}\). This can be proved by induction on the number of strings.