Colored HOMFLY-PT polynomials of quasi-alternating $3$-braid knots

TL;DR

This paper derives closed-form formulas for colored HOMFLY-PT polynomials of a special class of quasi-alternating 3-braid knots, revealing new connections with enumerative geometry and verifying conjectures in knot theory.

Contribution

It provides explicit formulas for HOMFLY-PT polynomials of twisted generalized hybrid weaving knots and explores their properties, including asymptotic behavior and relations to Lucas numbers.

Findings

Closed-form HOMFLY-PT polynomials for specific 3-braid knots.

Asymptotic trapezoidal behavior of Alexander polynomial coefficients.

Connection between determinants of weaving knots and Lucas numbers.

Abstract

Obtaining a closed-form expression for the colored HOMFLY-PT polynomials of knots from $3$-strand braids carrying arbitrary $SU(N)$ representation is a challenging problem. In this paper, we confine our interest to twisted generalized hybrid weaving knots which we denote hereafter by $\hat{Q}_3(m_1,-m_2,n,\ell)$. This family of knots not only generalizes the well-known class of weaving knots but also contains an infinite family of quasi-alternating knots. Interestingly, we obtain a closed-form expression for the HOMFLY-PT polynomial of $\hat{Q}_3(m_1,-m_2,n,\ell)$ using a modified version of the Reshitikhin-Turaev method. In addition, we compute the exact coefficients of the Jones polynomials and the Alexander polynomials of quasi-alternating knots $\hat{Q}_3(1,-1,n,\pm 1)$. For these homologically-thin knots, such coefficients are known to be the ranks of their Khovanov and link Floer homologies, respectively. We also show that the asymptotic behaviour of the coefficients of the Alexander polynomial is trapezoidal. On the other hand, we compute the $[r]$-colored HOMFLY-PT polynomials of quasi alternating knots for small values of $r$. Remarkably, the study of the determinants of certain twisted weaving knots leads to establish a connection with enumerative geometry related to $m^{th}$ Lucas numbers, denoted hereafter as $L_{m,2n}$. At the end, we verify that the reformulated invariants satisfy Ooguri-Vafa conjecture and we express certain BPS integers in terms of hyper-geometric functions ${}_2 {\bf F}_1\left[a,b, c;z\right]$.