# L11n129

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n129 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{u^2 v^2-2 u^2 v+u^2-3 u v^2+7 u v-3 u+v^2-2 v+1}{u v}$ (db) Jones polynomial $q^{9/2}-\frac{1}{q^{9/2}}-3 q^{7/2}+\frac{2}{q^{7/2}}+4 q^{5/2}-\frac{5}{q^{5/2}}-6 q^{3/2}+\frac{6}{q^{3/2}}+7 \sqrt{q}-\frac{7}{\sqrt{q}}$ (db) Signature -1 (db) HOMFLY-PT polynomial $a^5 z^{-1} +z^3 a^{-3} -3 a^3 z-2 a^3 z^{-1} +z a^{-3} -z^5 a^{-1} +3 a z^3-3 z^3 a^{-1} +5 a z+2 a z^{-1} -4 z a^{-1} - a^{-1} z^{-1}$ (db) Kauffman polynomial $-a z^9-z^9 a^{-1} -2 a^2 z^8-3 z^8 a^{-2} -5 z^8-a^3 z^7-2 z^7 a^{-1} -3 z^7 a^{-3} +8 a^2 z^6+10 z^6 a^{-2} -z^6 a^{-4} +19 z^6+3 a^3 z^5+10 a z^5+18 z^5 a^{-1} +11 z^5 a^{-3} -2 a^4 z^4-15 a^2 z^4-7 z^4 a^{-2} +3 z^4 a^{-4} -23 z^4-a^5 z^3-9 a^3 z^3-21 a z^3-22 z^3 a^{-1} -9 z^3 a^{-3} +2 a^4 z^2+8 a^2 z^2+2 z^2 a^{-2} -z^2 a^{-4} +9 z^2+2 a^5 z+8 a^3 z+12 a z+8 z a^{-1} +2 z a^{-3} -a^2-a^5 z^{-1} -2 a^3 z^{-1} -2 a z^{-1} - a^{-1} z^{-1}$ (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        2 2
6       21 -1
4      42  2
2     32   -1
0    44    0
-2   34     1
-4  23      -1
-6 14       3
-8 1        -1
-101         1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $r=-4$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=-1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=0$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{4}$ $r=1$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=4$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=5$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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