This paper investigates the relation between colored HOMFLY-PT and Kauffman homology, \({{\,\mathrm{SO}\,}}(N)\) quantum 6j-symbols, and (a, t)-deformed \(F_K\). First, we present a simple rule of grading change which allows us to obtain the [r]-colored quadruply graded Kauffman homology from the \([r^2]\)-colored quadruply graded HOMFLY-PT homology for thin knots. This rule stems from the isomorphism of the representations \((\mathfrak {so}_6,[r]) \cong (\mathfrak {sl}_4,[r^2])\). Also, we find the relationship among A-polynomials of \({{\,\mathrm{SO}\,}}\) and \({{\,\mathrm{SU}\,}}\) type coming from a differential on Kauffman homology. Second, we put forward a closed-form expression of \({{\,\mathrm{SO}\,}}(N)(N\ge 4)\) quantum 6j-symbols for symmetric representations and calculate the corresponding \({{\,\mathrm{SO}\,}}(N)\) fusion matrices for the cases when representations

本文研究了彩色HOMFLY-PT与Kauffman同源性，\（{{\，mathrm {SO} \，}}（N）\）量子6 j符号和（a，  t）变形\（ F_K \）。首先，我们给出一个简单的渐变变化规则，该规则使我们能够从\（[r ^ 2] \）色四重渐变HOMFLY-PT同源性获得薄的结的[ r ]色四重渐变Kauffman同源性。该规则源于表示\（（\ mathfrak {so} _6，[r]）\ cong（\ mathfrak {sl} _4，[r ^ 2]）\）的同构。此外，我们发现\（{{\，\ mathrm {SO} \，}} \）的A-多项式与\（{{\，\ mathrm {SU} \，}} \）类型来自考夫曼同源性的差异。其次，我们为对称表示提出了\（{{\，\ mathrm {SO} \，}}（N）（N \ ge 4）\）量子6 j-符号的闭式表达式，并计算了相应的\ （{{\，\ mathrm {SO} \，}}（N）\）融合矩阵