# L11a233

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a233 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{3 u^2 v^3-7 u^2 v^2+6 u^2 v-2 u^2+2 u v^4-11 u v^3+17 u v^2-11 u v+2 u-2 v^4+6 v^3-7 v^2+3 v}{u v^2}$ (db) Jones polynomial $25 q^{9/2}-23 q^{7/2}+16 q^{5/2}-10 q^{3/2}+q^{21/2}-4 q^{19/2}+10 q^{17/2}-16 q^{15/2}+22 q^{13/2}-26 q^{11/2}+4 \sqrt{q}-\frac{1}{\sqrt{q}}$ (db) Signature 3 (db) HOMFLY-PT polynomial $z^3 a^{-9} + a^{-9} z^{-1} -z^5 a^{-7} +z^3 a^{-7} -2 a^{-7} z^{-1} -3 z^5 a^{-5} -5 z^3 a^{-5} -3 z a^{-5} -z^5 a^{-3} +2 z^3 a^{-3} +4 z a^{-3} + a^{-3} z^{-1} +z^3 a^{-1}$ (db) Kauffman polynomial $z^6 a^{-12} -2 z^4 a^{-12} +z^2 a^{-12} +4 z^7 a^{-11} -8 z^5 a^{-11} +4 z^3 a^{-11} +8 z^8 a^{-10} -18 z^6 a^{-10} +14 z^4 a^{-10} -7 z^2 a^{-10} +2 a^{-10} +8 z^9 a^{-9} -13 z^7 a^{-9} +5 z^5 a^{-9} -4 z^3 a^{-9} +2 z a^{-9} - a^{-9} z^{-1} +3 z^{10} a^{-8} +13 z^8 a^{-8} -43 z^6 a^{-8} +39 z^4 a^{-8} -19 z^2 a^{-8} +5 a^{-8} +17 z^9 a^{-7} -31 z^7 a^{-7} +14 z^5 a^{-7} -z^3 a^{-7} +2 z a^{-7} -2 a^{-7} z^{-1} +3 z^{10} a^{-6} +17 z^8 a^{-6} -48 z^6 a^{-6} +41 z^4 a^{-6} -14 z^2 a^{-6} +3 a^{-6} +9 z^9 a^{-5} -5 z^7 a^{-5} -14 z^5 a^{-5} +18 z^3 a^{-5} -5 z a^{-5} +12 z^8 a^{-4} -20 z^6 a^{-4} +14 z^4 a^{-4} -3 z^2 a^{-4} - a^{-4} +9 z^7 a^{-3} -14 z^5 a^{-3} +10 z^3 a^{-3} -5 z a^{-3} + a^{-3} z^{-1} +4 z^6 a^{-2} -4 z^4 a^{-2} +z^5 a^{-1} -z^3 a^{-1}$ (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 \
-2-10123456789χ
22           1-1
20          3 3
18         71 -6
16        93  6
14       137   -6
12      139    4
10     1213     1
8    1113      -2
6   613       7
4  410        -6
2 17         6
0 3          -3
-21           1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=2$ $i=4$ $r=-2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=0$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{4}$ $r=1$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=2$ ${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{11}$ $r=3$ ${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{12}$ ${\mathbb Z}^{12}$ $r=4$ ${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{13}$ ${\mathbb Z}^{13}$ $r=5$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{13}$ ${\mathbb Z}^{13}$ $r=6$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=7$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=8$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=9$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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