论文标题
彩色的homfly-pt ptimials quasi-alternating $ 3 $编织结
Colored HOMFLY-PT polynomials of quasi-alternating $3$-braid knots
论文作者
论文摘要
从$ 3 $ strand的辫子中获得带有任意$ su（n）$表示的颜色的homfly-pt多项式的闭合形式表达式是一个具有挑战性的问题。在本文中，我们将我们的兴趣限制在扭曲的广义混合编织结中，我们以$ \ hat {q} _3（m_1，-m_2，n，n，\ ell）$表示。这个结家族不仅概括了众所周知的编织结阶段，而且还包含一个无限的准结节家族。有趣的是，我们使用reshitikhin-turaev方法的修改版本获得了$ \ hat {q} _3（m_1，-m_2，n，\ ell）$的homfly-pt多项式的封闭形式表达式。此外，我们计算了琼斯多项式的确切系数和准代理结的亚历山大多项式$ \ hat {q} _3（1，-1，n，n，\ pm 1）$。对于这些同源性的结，这种系数分别是其Khovanov和Link Floer同源性的等级。我们还表明，亚历山大多项式的系数的渐近行为是梯形。另一方面，我们计算出$ [r] $ - 彩色的homfly-pt pt quasi交替结的多项式，以$ r $的小值。值得注意的是，对某些扭曲结的决定因素的研究导致建立了与$ m^{th} $ lucas数字相关的枚举几何形状的联系，此后称为$ l_ {m {m，2n} $。最后，我们验证了重新制定的不变式满足了Ooguri-Vafa的猜想，并且我们在超几何函数方面表达了某些BPS整数$ {} _ 2 {\ bf f} _1 _1 \ left [a，b，b，c; z; z \ right] $。
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 ${}_2 {\bf F}_1\left[a,b, c;z\right]$.
