On the \(S^{1}\times S^{2}\) HOMFLY-PT invariant and Legendrian links (Q2848023)

The authors study the \(S^1\times S^2\) HOMFLY-PT invariant, which was first introduced by Gilmer and Zhong. They provide a practical result to compute the HOMFLY-PT invariant of links in \(S^1 \times S^2\).NEWLINENEWLINELet \(S(S^1 \times D^2)\) denote the HOMFLY-PT skein module of \(S^1 \times D^2\) and \(\mathbb Q(a, s)\) the field of rational functions. The main theorem relates the HOMFLY-PT invariant NEWLINE\[NEWLINEF: S(S^1 \times D^2) \to \mathbb Q(a, s)NEWLINE\]NEWLINE and the \textit{two-graded ruling polynomial}. The latter, defined by Chekanov and Pushkar, is an invariant of Legendrian links in \(S^1 \times \mathbb R^2\) with respect to the standard tight contact structure given as \(\ker(dz-ydx)\) where the \(x\)-coordinate is \(S^1\)-valued. An important fact for the main theorem is that the contact framing of a Legendrian link agrees with the blackboard framing of its projection to the \(xy\)-annulus.NEWLINENEWLINELet \(\{ A_\lambda A_{-\mu}\}\) be the Turaev's geometric basis of the skein module \(S(S^1 \times D^2)\), where \(\lambda, \mu\) are partitions. Putting \(z=s-s^{-1}\) any link in \(S^2\times D^2\) is a \(\mathbb Z[a^{\pm 1}, z^{\pm 1}]\)-linear combination of the Turaev-basis up to the HOMFLY-PT skein relations.NEWLINENEWLINEWith respect to the Turaev-basis, the main theorem states that NEWLINE\[NEWLINEF(A_\lambda A_{-\mu}) = R^2_{A_\lambda A_{-\mu}}(z)NEWLINE\]NEWLINE where \(R^2\) denotes the \(2\)-graded ruling polynomial.NEWLINENEWLINESince the ruling polynomial \(R^2\) is a Laurent polynomial in \(z\), as a corollary of the main theorem the authors prove that the HOMFLY-PT polynomial is a Laurent polynomial in \(a\) and \(z\).NEWLINENEWLINEAnother corollary of the main theorem and a result of Chmutov and Goryunov is an inequality for a null-homologous Legendrian link \(L \subset S^1 \times S^2\) NEWLINE\[NEWLINE\text{tb}(L) + | \text{rot}(L) | \leq -\deg_a(\tilde F_L)NEWLINE\]NEWLINE where \(\tilde F_L = a^{-w(L)}F(L)\) is a normalized \(S^1 \times S^2\) HOMFLY-PT invariant using the writhe \(w(L)\) of \(L\).