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

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 Qˆ3(m1,−m2,n,ℓ). 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 Qˆ3(m1,−m2,n,ℓ) 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 Qˆ3(1,−1,n,±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 behavior 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 mth Lucas numbers, denoted hereafter as Lm,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 F12[a,b,c;z].