Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Panhandle polynomials of torus links and geometric applications

Andrei Mironov, Hisham Sati, Vivek Kumar Singh, Alexander Stoimenov

TL;DR

The paper investigates reverse 2-cable (panhandle) constructions for torus knots and links, revealing a characteristic panhandle component in the HOMFLY-PT polynomial and deriving explicit closed forms via the Rosso-Jones framework and Adams operations. It provides a cohesive link between polynomial data and geometry by introducing the $\ell$-invariant, which yields algebraic proofs and sharp bounds for arc index, Thurston-Bennequin invariants, and braid indices, and connects these to (strong) quasipositivity and Bennequin surfaces. The Panhandle Theorem for torus knots and its extension to torus links give concrete expressions for the adjoint contribution, enabling detailed study of minimal string strongly quasipositive surfaces and Baker-Motegi-type questions. The results are then applied to torus links, yielding component-wise invariants, refined arc index bounds, and a beaded-grid approach that clarifies when maximal Euler characteristic surfaces are strongly quasipositive. The work also touches Whitehead doubling, proposes a path to generalize the $\ell$-invariant to general links, and outlines how these insights may influence broader questions in contact topology and link theory.

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.

Panhandle polynomials of torus links and geometric applications

TL;DR

The paper investigates reverse 2-cable (panhandle) constructions for torus knots and links, revealing a characteristic panhandle component in the HOMFLY-PT polynomial and deriving explicit closed forms via the Rosso-Jones framework and Adams operations. It provides a cohesive link between polynomial data and geometry by introducing the -invariant, which yields algebraic proofs and sharp bounds for arc index, Thurston-Bennequin invariants, and braid indices, and connects these to (strong) quasipositivity and Bennequin surfaces. The Panhandle Theorem for torus knots and its extension to torus links give concrete expressions for the adjoint contribution, enabling detailed study of minimal string strongly quasipositive surfaces and Baker-Motegi-type questions. The results are then applied to torus links, yielding component-wise invariants, refined arc index bounds, and a beaded-grid approach that clarifies when maximal Euler characteristic surfaces are strongly quasipositive. The work also touches Whitehead doubling, proposes a path to generalize the -invariant to general links, and outlines how these insights may influence broader questions in contact topology and link theory.

Abstract

We use a decomposition of the tensor of the fundamental representation of the quantum group 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 -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 -invariant for general links.
Paper Structure (22 sections, 38 theorems, 192 equations, 4 figures, 5 tables)

This paper contains 22 sections, 38 theorems, 192 equations, 4 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Panhandle polynomials for torus knots and links
  3. Background and Definitions
  4. HOMFLY-PT polynomial of reverse $2$-cable torus knot $T(m,n)$
  5. Proof of the Panhandle theorem for torus knots
  6. Extension to torus links
  7. Geometric properties
  8. Arc index and Thurston-Bennequin invariant
  9. Braid index and braided surfaces
  10. Quasipositivity and strong quasipositivity
  11. Properties of torus knots
  12. Applications to torus links
  13. Invariants
  14. Arc index
  15. Thurston-Bennequin invariants
  16. ...and 7 more sections

Key Result

Theorem 1.1

Let $X = [\underline{M}; M]_v$ denote a polynomial such that Then the HOMFLY-PT polynomial for the reverse $2$- cable torus knot $C_2(T_{m,n},t)$ has the form

Figures (4)

  • Figure 1: Braid representation of torus knots $T(4,3)$.
  • Figure 2: Braid representation of reverse $2$ -cable torus knots $T_{4,5}$.
  • Figure 3: Braid representation of $C_2(T_{2,5},-10)$.
  • Figure 4: Braid representation of reverse two-cable torus knot $C_2(T_{3,4},-8)$.

Theorems & Definitions (99)

  • Theorem 1.1: Panhandle Theorem
  • Theorem 1.2: Panhandle for links
  • Definition 2.1: Torus Knots
  • Example 2.2
  • Definition 2.3: The reverse $2$-cable knots
  • Definition 2.4: HOMFLY-PT PolynomialHOMFLY1985
  • Remark 2.5: Normalizations
  • Definition 2.6: Young tableau diagram representation KRKS
  • Definition 2.7: Composite representation Koike
  • Example 2.8: Fundamental representation
  • ...and 89 more