Colored HOMFLY and their differential expansion
A list of available colored HOMFLY (up to representation $[21]$) for ordinary prime knots with
38,
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
$\ \Big(\{x\} = x1/x\Big)$:
$$
\begin{array}{rl}
H_{[1]} = &1 + G_1\cdot \{Aq\}\{A/q\},\\ \\
H_{[2]} = &1 + [2]G_1\cdot\{Aq^2\}\{A/q\} + G_2\cdot\{Aq^3\}\{Aq^2\}\{A/q\}, \\ \\
H_{[3]} = &1 + [3]G_1\cdot\{Aq^3\}\{A/q\}+ [3]G_2\cdot\{Aq^4\}\{Aq^3\}\{A/q\}+G_3\cdot\{Aq^5\}\{Aq^4\}\{Aq^3\}\{A/q\},\\ \\
H_{[2,1]} = &1+ G_1\cdot\Big(\{Aq^3\}\{A/q^3\}+\{Aq^2\}\{A\}+\{A\}\{A/q^2\}\Big) + \\
&+\ \{Aq^2\}\{A/q^2\}\cdot\Big([3]\{A\}\cdot G_2(q=1 ) + \{Aq^3\}\{A/q^3\}\cdot G_3(q=1 )
+ \{q\}^2\cdot G_{21}\Big)
\end{array}
$$

More fundamental (310,
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 $R\longrightarrow \tilde R$).

4strand nonarborescent knots HOMFLY in representation [2]

Some 3strand HOMFLY in representation [3,1]

Some 3strand HOMFLY in representation [2,2]

Some arborescent 1, 2 HOMFLY in representation [2,2], which are not 3strand

Some 3strand HOMFLY in representation [3,3]

Some twist and antiparalleldoublebraid HOMFLY in representation [3,3,3]

More HOMFLY for torus knots
and mutants

[2]colored Jones polynomials

Fundamental
Kauffman polynomials

Universal
adjoint polynomials

Values of HOMFLY at q = root of unity

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.

Resultants of colored HOMFLY and their zeroes.

Concrete values of parameters, associated with particular knots up to 10 crossings, for various families from the paper arXiv:1601.04199 see on the following page.

LMOV invariants for HOMFLY polynomials (see arXiv:1702.06316 for definitions and details).