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arXiv 2202.09169hep-thmath-phmath.MP
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拟交错$3$ -辫结的着色HOMFLY-PT多项式

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

Nafaa Chbili, Vivek Kumar Singh

备注：
39 pages, version appearing in Nuclear Physics B
发表信息：
Nuclear Physics B 980 (2022) 115800
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摘要：

从携带任意$SU(N)$表示的$3$ -股编织物中获得结的有色HOMFLY-PT多项式的封闭形式表达式是一个具有挑战性的问题。在本文中，我们将我们的兴趣局限于扭曲的广义混合编织结，我们表示为$\hat{Q}_3(m_1,-m_2,n,\ell)$。这类纽结不仅推广了著名的编织纽结，而且还包含了一个准交替纽结的无限族。有趣的是，我们得到一个封闭形式的表达式的HOMFLY-PT多项式的$\hat{Q}_3(m_1,-m_2,n,\ell)$使用修改后的版本的Reshitikhin-Turaev方法。此外，我们还计算了拟交错纽结$\hat{Q}_3(1,-1,n,\pm 1)$的Jones多项式和Alexander多项式的精确系数。对于这些同源薄结，这样的系数被称为是他们的Khovanov和链接Floer同源性的秩，分别。我们还表明，亚历山大多项式的系数的渐近行为是梯形的。另一方面，我们计算的$[r]$着色HOMFLY-PT多项式的准交替结的小值的$r$。值得注意的是，对某些扭曲编织结的决定因素的研究导致建立了与$m^{th}$卢卡斯数相关的枚举几何学的联系，以下表示为$L_{m,2n}$。最后，我们证明了重新表述的不变量满足Ooguri-Vafa猜想，并将某些BPS整数表示为超几何函数${}_2 {\bf F}_1\left[a,b, c;z\right]$。

英文摘要：

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 ${}_2 {\bf F}_1\left[a,b, c;z\right]$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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