Panhandle polynomials of torus links and geometric applications

TL;DR

This paper introduces 'panhandle polynomials' derived from HOMFLY-PT polynomials of torus links, revealing geometric properties and invariants that connect knot theory with contact and braid geometry.

Contribution

It establishes the 'panhandle' shape of HOMFLY-PT polynomials for torus links and extends geometric invariants and properties to links, linking quantum group representations with topological and geometric knot invariants.

Findings

HOMFLY-PT polynomial exhibits a 'panhandle' shape for torus links.

Extension of the Etnyre-Honda result to links using the $ ext{l}$-invariant.

Geometric consequences include relations to braid index and Bennequin surfaces.

Abstract

We use a decomposition of the tensor of the fundamental representation of the quantum group $U_q(\mathfrak{sl}_N)$ and the Rosso-Jones formula to establish a peculiar ``panhandle'' shape of the HOMFLY-PT polynomial of the reverse parallel of torus knots and links. Due to their panhandle-like intrinsic properties, the HOMFLY-PT polynomial is referred to as a ``panhandle polynomial''. With the help of the $\ell$-invariant, this extends to links the Etnyre-Honda result about the arc index and maximal Thurston-Bennequin invariant of torus knots. It has further geometric consequences, related to the braid index, the existence of minimal string Bennequin surfaces for banded and Whitehead doubled links, the Bennequin sharpness problem, and the equivalence of their quasipositivity and strong quasipositivity. We extend these properties to torus links, which relate to the classification of their component-wise Thurston-Bennequin invariants. Finally, we discuss the definition of the $\ell$-invariant for general links.