## Colored HOMFLY and their differential expansion

A list of available colored HOMFLY (up to representation $$) for ordinary prime knots with 3-8, 9, 10 intersections (mostly arborescent).

The polynomial $H^{\text{knot}}_{\text{representation}} =H^K_R$ appears in the list as $H[R][K]$: for example, HOMFLY in representation $[2,1]$ for the knot $10_{61}$ is denoted by $H[2,1][10,61]$.
The coefficients $G_R$ of the differential expansion are defined as in eq.(54) of   arXiv:1508.02870   $\ \Big(\{x\} = x-1/x\Big)$: $$\begin{array}{rl} H_{} = &1 + G_1\cdot \{Aq\}\{A/q\},\\ \\ H_{} = &1 + G_1\cdot\{Aq^2\}\{A/q\} + G_2\cdot\{Aq^3\}\{Aq^2\}\{A/q\}, \\ \\ H_{} = &1 + G_1\cdot\{Aq^3\}\{A/q\}+ G_2\cdot\{Aq^4\}\{Aq^3\}\{A/q\}+G_3\cdot\{Aq^5\}\{Aq^4\}\{Aq^3\}\{A/q\},\\ \\ \end{array}$$

• More fundamental (3-10, 11, 12 intersections), and symmetric HOMFLY.
Polynomials in antisymmetric representation $[1^r]$ are obtained from those in symmetric representation [r] by the change $A\longrightarrow 1/A$ (in general this corresponds to the transposition of Young diagram $R\longrightarrow \tilde R$).
• 4-strand non-arborescent knots HOMFLY in representation 
• Some 3-strand HOMFLY in representation [3,1]
• Some 3-strand HOMFLY in representation [2,2]
• Some arborescent 1, 2 HOMFLY in representation [2,2], which are not 3-strand
• Some 3-strand HOMFLY in representation [3,3]
• Some twist and antiparallel-double-braid HOMFLY in representation [3,3,3]
• More HOMFLY for torus knots and mutants
• -colored Jones polynomials
• Fundamental Kauffman polynomials
• For some knots the polynomials $G_R$ factorize further: this depends on the knot's defect of the differential expansion (equal to to degree of Alexander polynomial minus one).   Values of defects for knots with up to ten intersections.