# L11n307

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n307 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{-3 u v^2 w^2+3 u v^2 w-u v^2-u v w^3+3 u v w^2-2 u v w+u w^3-u w^2+v^2 w^2-v^2 w+2 v w^3-3 v w^2+v w+w^4-3 w^3+3 w^2}{\sqrt{u} v w^2}$ (db) Jones polynomial $-q^{10}+3 q^9-6 q^8+8 q^7-9 q^6+11 q^5-8 q^4+8 q^3-4 q^2+2 q$ (db) Signature 2 (db) HOMFLY-PT polynomial $-z^4 a^{-4} -3 z^4 a^{-6} +2 z^2 a^{-2} +2 z^2 a^{-4} -9 z^2 a^{-6} +4 z^2 a^{-8} + a^{-2} +5 a^{-4} -12 a^{-6} +7 a^{-8} - a^{-10} +2 a^{-4} z^{-2} -5 a^{-6} z^{-2} +4 a^{-8} z^{-2} - a^{-10} z^{-2}$ (db) Kauffman polynomial $z^7 a^{-11} -4 z^5 a^{-11} +6 z^3 a^{-11} -4 z a^{-11} + a^{-11} z^{-1} +3 z^8 a^{-10} -12 z^6 a^{-10} +14 z^4 a^{-10} -6 z^2 a^{-10} - a^{-10} z^{-2} +3 a^{-10} +2 z^9 a^{-9} -z^7 a^{-9} -21 z^5 a^{-9} +36 z^3 a^{-9} -19 z a^{-9} +5 a^{-9} z^{-1} +10 z^8 a^{-8} -39 z^6 a^{-8} +48 z^4 a^{-8} -32 z^2 a^{-8} -4 a^{-8} z^{-2} +15 a^{-8} +2 z^9 a^{-7} +5 z^7 a^{-7} -38 z^5 a^{-7} +52 z^3 a^{-7} -33 z a^{-7} +9 a^{-7} z^{-1} +7 z^8 a^{-6} -24 z^6 a^{-6} +35 z^4 a^{-6} -37 z^2 a^{-6} -5 a^{-6} z^{-2} +20 a^{-6} +7 z^7 a^{-5} -20 z^5 a^{-5} +25 z^3 a^{-5} -18 z a^{-5} +5 a^{-5} z^{-1} +3 z^6 a^{-4} +z^4 a^{-4} -8 z^2 a^{-4} -2 a^{-4} z^{-2} +8 a^{-4} +z^5 a^{-3} +3 z^3 a^{-3} +3 z^2 a^{-2} - a^{-2}$ (db)

### Khovanov Homology

The coefficients of the monomials $t^rq^j$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$).
 \ r \ j \
0123456789χ
21         1-1
19        2 2
17       41 -3
15      42  2
13     65   -1
11    53    2
9   36     3
7  55      0
5  4       4
324        -2
12         2
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=1$ $i=3$ $r=0$ ${\mathbb Z}^{2}$ ${\mathbb Z}^{2}$ $r=1$ ${\mathbb Z}^{4}$ $r=2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{5}$ $r=3$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=4$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=5$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=6$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{4}$ $r=7$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=8$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=9$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory. See A Sample KnotTheory Session.