# L11a104

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a104 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{u v^5-6 u v^4+12 u v^3-12 u v^2+4 u v+4 v^4-12 v^3+12 v^2-6 v+1}{\sqrt{u} v^{5/2}}$ (db) Jones polynomial $q^{9/2}-5 q^{7/2}+10 q^{5/2}-15 q^{3/2}+20 \sqrt{q}-\frac{23}{\sqrt{q}}+\frac{22}{q^{3/2}}-\frac{19}{q^{5/2}}+\frac{13}{q^{7/2}}-\frac{8}{q^{9/2}}+\frac{3}{q^{11/2}}-\frac{1}{q^{13/2}}$ (db) Signature -1 (db) HOMFLY-PT polynomial $a z^7-2 a^3 z^5+4 a z^5-2 z^5 a^{-1} +a^5 z^3-6 a^3 z^3+9 a z^3-4 z^3 a^{-1} +z^3 a^{-3} +2 a^5 z-9 a^3 z+8 a z-3 z a^{-1} +2 a^5 z^{-1} -4 a^3 z^{-1} +3 a z^{-1} - a^{-1} z^{-1}$ (db) Kauffman polynomial $-2 a^2 z^{10}-2 z^{10}-5 a^3 z^9-12 a z^9-7 z^9 a^{-1} -7 a^4 z^8-12 a^2 z^8-9 z^8 a^{-2} -14 z^8-6 a^5 z^7-4 a^3 z^7+15 a z^7+8 z^7 a^{-1} -5 z^7 a^{-3} -3 a^6 z^6+8 a^4 z^6+27 a^2 z^6+21 z^6 a^{-2} -z^6 a^{-4} +38 z^6-a^7 z^5+10 a^5 z^5+24 a^3 z^5+11 a z^5+8 z^5 a^{-1} +10 z^5 a^{-3} +4 a^6 z^4-3 a^4 z^4-14 a^2 z^4-11 z^4 a^{-2} +z^4 a^{-4} -19 z^4+2 a^7 z^3-10 a^5 z^3-32 a^3 z^3-24 a z^3-7 z^3 a^{-1} -3 z^3 a^{-3} -a^6 z^2-2 a^4 z^2-4 a^2 z^2-z^2 a^{-2} -4 z^2-a^7 z+7 a^5 z+18 a^3 z+13 a z+2 z a^{-1} -z a^{-3} +2 a^4+3 a^2+ a^{-2} +3-2 a^5 z^{-1} -4 a^3 z^{-1} -3 a z^{-1} - a^{-1} z^{-1}$ (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 \
-6-5-4-3-2-1012345χ
10           1-1
8          4 4
6         61 -5
4        94  5
2       116   -5
0      129    3
-2     1112     1
-4    811      -3
-6   511       6
-8  38        -5
-10 16         5
-12 2          -2
-141           1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $r=-6$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}$ $r=-3$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-2$ ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=-1$ ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{11}$ ${\mathbb Z}^{11}$ $r=0$ ${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{11}$ ${\mathbb Z}^{12}$ $r=1$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{11}$ ${\mathbb Z}^{11}$ $r=2$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=3$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=4$ ${\mathbb Z}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=5$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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