Integrality of Homfly 1-tangle invariants (Q2464755)

Given a framed knot \(K\), decorated with a pattern \(Q\) (in the annulus), there is a satellite \(K*Q\) of \(K\) with integral Laurent Homfly polynomial \(P \in Z[v^{\pm1},z^{\pm1}]\), thus giving for each \(Q\) a 2-variable knot polynomial invariant. By admitting linear combinations of patterns in the Homfly skein of the annulus, and extending coefficients to the ring \(\Lambda\) of Laurent polynomials \(Z[v^{\pm1},s^{\pm1}]\) with denominators \(s^r-s^{-r}\), taking \(z = s-s^{-1}\) gives an invariant \(J(K) = P(K*Q) \in \Lambda\). In the case where \(Q\) is any eigenvector of the meridian map [see \textit{R. J. Hadji} and \textit{H. R. Morton}, Math. Proc. Camb. Philos. Soc. 141, No. 1, 81--100 (2006; Zbl 1108.57005)], the author proves the 1-tangle invariant \(a_K = P(K*Q)/P(U*Q)\) is a 2-variable integer Laurent polynomial in \(Z[v^{\pm1},s^{\pm1}]\). As corollary, the author notes the Homfly polynomial \(P(K*Q_{\lambda,\mu})\) of the satellite \(K*Q_{\lambda,\mu}\) will always factor as \(P(K*Q_{\lambda,\mu}) = a_K(\lambda,\mu) P(U*Q_{\lambda,\mu})\); here \(\lambda, \mu\) are partitions defined in [loc cit.].