# L11n25

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n25 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{(u-1) (v-2) (v-1) (2 v-1)}{\sqrt{u} v^{3/2}}$ (db) Jones polynomial $q^{9/2}-4 q^{7/2}+6 q^{5/2}-10 q^{3/2}+12 \sqrt{q}-\frac{12}{\sqrt{q}}+\frac{11}{q^{3/2}}-\frac{9}{q^{5/2}}+\frac{5}{q^{7/2}}-\frac{2}{q^{9/2}}$ (db) Signature -1 (db) HOMFLY-PT polynomial $a^5 z^{-1} +a^3 z^3+z^3 a^{-3} -2 a^3 z-2 a^3 z^{-1} - a^{-3} z^{-1} -a z^5-z^5 a^{-1} -z^3 a^{-1} +2 a z+a z^{-1} + a^{-1} z^{-1}$ (db) Kauffman polynomial $3 a^5 z^3-3 a^5 z+a^5 z^{-1} +a^4 z^6+z^6 a^{-4} +4 a^4 z^4-2 z^4 a^{-4} -3 a^4 z^2+a^4+4 a^3 z^7+4 z^7 a^{-3} -7 a^3 z^5-12 z^5 a^{-3} +14 a^3 z^3+8 z^3 a^{-3} -11 a^3 z+z a^{-3} +2 a^3 z^{-1} - a^{-3} z^{-1} +5 a^2 z^8+5 z^8 a^{-2} -12 a^2 z^6-14 z^6 a^{-2} +19 a^2 z^4+10 z^4 a^{-2} -13 a^2 z^2-4 z^2 a^{-2} +3 a^2+ a^{-2} +2 a z^9+2 z^9 a^{-1} +4 a z^7+4 z^7 a^{-1} -19 a z^5-24 z^5 a^{-1} +22 a z^3+19 z^3 a^{-1} -10 a z-z a^{-1} +a z^{-1} - a^{-1} z^{-1} +10 z^8-28 z^6+27 z^4-14 z^2+2$ (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 \
-4-3-2-1012345χ
10         1-1
8        3 3
6       31 -2
4      73  4
2     53   -2
0    77    0
-2   67     1
-4  35      -2
-6 26       4
-8 3        -3
-102         2
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $r=-4$ ${\mathbb Z}^{2}$ $r=-3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-2$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-1$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=0$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{7}$ $r=1$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=4$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=5$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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