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A relation between the HOMFLY-PT and Kauffman polynomials via characters

Andreani Petrou, Shinobu Hikami

Abstract

The HOMFLY-PT and Kauffman polynomials are related to each other for special classes of knots constructed by full twists and Jucys-Murphy twists. The conditions for this relation are articulated in terms of characters of the Birman-Murakami-Wenzl algebra. The latter are the coefficients in the expansion of the Kauffman polynomial involving the quantum dimensions of SO(N + 1). This expansion allows to prove the conjectural 1-1 correspondence between the HOMFLY-PT/Kauffman relation and the Harer-Zagier (HZ) factorisability for a large family of 3-strand knots. However, explicit counterexamples with 4-strands negate one side of the conjecture, i.e. the HOMFLY-PT/Kauffman relation only implies HZ factorisability for knots with braid index four or higher.

A relation between the HOMFLY-PT and Kauffman polynomials via characters

Abstract

The HOMFLY-PT and Kauffman polynomials are related to each other for special classes of knots constructed by full twists and Jucys-Murphy twists. The conditions for this relation are articulated in terms of characters of the Birman-Murakami-Wenzl algebra. The latter are the coefficients in the expansion of the Kauffman polynomial involving the quantum dimensions of SO(N + 1). This expansion allows to prove the conjectural 1-1 correspondence between the HOMFLY-PT/Kauffman relation and the Harer-Zagier (HZ) factorisability for a large family of 3-strand knots. However, explicit counterexamples with 4-strands negate one side of the conjecture, i.e. the HOMFLY-PT/Kauffman relation only implies HZ factorisability for knots with braid index four or higher.
Paper Structure (15 sections, 9 theorems, 105 equations, 4 figures, 3 tables)

This paper contains 15 sections, 9 theorems, 105 equations, 4 figures, 3 tables.

Table of Contents

  1. Keywords.
  2. Acknowledgments.
  3. Introduction
  4. The Dubrovnik polynomial via characters of $SO(N+1)$
  5. 2 strands.
  6. 3 strands.
  7. 4 strands.
  8. 5 strands.
  9. BMW algebra representations and the Kauffman character expansion
  10. 2 strands.
  11. 3 strands.
  12. 4 strands.
  13. The HOMFLY--PT/Kauffman relation and HZ factorisability
  14. Summary and discussion
  15. Declaration of competing interest.

Key Result

Proposition 3.1

The correction terms $\delta$ in Sec. sec:dubrovnik can be obtained in terms of BMW algebra representations byThe sign $\pm1$ comes from an overall factor $(-i)^{-w}$, where $w$ is even for knots.

Figures (4)

  • Figure 1: The BMW algebra generators $\sigma_i$ and $e_i$.
  • Figure 2: The Bratelli diagram, with the dimensions of the representations of $C_m(\alpha,\beta)$ indicated next to each Young diagram.
  • Figure 3: Various oriented diagrams contributing in the state sum (\ref{['jaegerSum']}) for the trefoil knot (left), along with their coefficients $[\mathcal{K}|\sigma]$.
  • Figure 4: Three oriented states with the same coefficient $c_\sigma(z)=[\mathcal{K}|\sigma]=-z^3$.

Theorems & Definitions (19)

  • Proposition 3.1
  • proof
  • Corollary 3.1
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof
  • Proposition 3.4
  • proof
  • Proposition 4.1
  • ...and 9 more