# L10a103

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a103 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{(t(1)-1) (t(2)-1) (t(2) t(1)-t(1)+1) (t(1) t(2)-t(2)+1)}{t(1)^{3/2} t(2)^{3/2}}$ (db) Jones polynomial $-3 q^{9/2}+\frac{1}{q^{9/2}}+6 q^{7/2}-\frac{3}{q^{7/2}}-10 q^{5/2}+\frac{5}{q^{5/2}}+11 q^{3/2}-\frac{9}{q^{3/2}}+q^{11/2}-12 \sqrt{q}+\frac{11}{\sqrt{q}}$ (db) Signature 1 (db) HOMFLY-PT polynomial $-z^7 a^{-1} +2 a z^5-5 z^5 a^{-1} +z^5 a^{-3} -a^3 z^3+7 a z^3-10 z^3 a^{-1} +3 z^3 a^{-3} -2 a^3 z+7 a z-8 z a^{-1} +3 z a^{-3} +a z^{-1} - a^{-1} z^{-1}$ (db) Kauffman polynomial $-a z^9-z^9 a^{-1} -3 a^2 z^8-4 z^8 a^{-2} -7 z^8-3 a^3 z^7-7 a z^7-10 z^7 a^{-1} -6 z^7 a^{-3} -a^4 z^6+6 a^2 z^6+z^6 a^{-2} -5 z^6 a^{-4} +13 z^6+10 a^3 z^5+27 a z^5+28 z^5 a^{-1} +8 z^5 a^{-3} -3 z^5 a^{-5} +3 a^4 z^4+5 z^4 a^{-2} +5 z^4 a^{-4} -z^4 a^{-6} -4 z^4-10 a^3 z^3-27 a z^3-26 z^3 a^{-1} -6 z^3 a^{-3} +3 z^3 a^{-5} -2 a^4 z^2-2 a^2 z^2-3 z^2 a^{-2} -2 z^2 a^{-4} +z^2 a^{-6} +3 a^3 z+10 a z+10 z a^{-1} +2 z a^{-3} -z a^{-5} +1-a z^{-1} - a^{-1} z^{-1}$ (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 \
-5-4-3-2-1012345χ
12          1-1
10         2 2
8        41 -3
6       62  4
4      54   -1
2     76    1
0    67     1
-2   35      -2
-4  26       4
-6 13        -2
-8 2         2
-101          -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=0$ $i=2$ $r=-5$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-2$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-1$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=0$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{7}$ $r=1$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=4$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=5$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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