# L10a55

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a55 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{u^2 v^4-4 u^2 v^3+4 u^2 v^2-u^2 v-u v^4+5 u v^3-9 u v^2+5 u v-u-v^3+4 v^2-4 v+1}{u v^2}$ (db) Jones polynomial $q^{9/2}-3 q^{7/2}+6 q^{5/2}-10 q^{3/2}+12 \sqrt{q}-\frac{14}{\sqrt{q}}+\frac{13}{q^{3/2}}-\frac{11}{q^{5/2}}+\frac{7}{q^{7/2}}-\frac{4}{q^{9/2}}+\frac{1}{q^{11/2}}$ (db) Signature -1 (db) HOMFLY-PT polynomial $-a^3 z^5-2 a^3 z^3+z^3 a^{-3} +2 z a^{-3} +a^3 z^{-1} +a z^7+4 a z^5-2 z^5 a^{-1} +5 a z^3-6 z^3 a^{-1} +a z-4 z a^{-1} -a z^{-1}$ (db) Kauffman polynomial $-2 a z^9-2 z^9 a^{-1} -6 a^2 z^8-4 z^8 a^{-2} -10 z^8-8 a^3 z^7-8 a z^7-3 z^7 a^{-1} -3 z^7 a^{-3} -7 a^4 z^6+5 a^2 z^6+10 z^6 a^{-2} -z^6 a^{-4} +23 z^6-4 a^5 z^5+9 a^3 z^5+23 a z^5+19 z^5 a^{-1} +9 z^5 a^{-3} -a^6 z^4+7 a^4 z^4+2 a^2 z^4-6 z^4 a^{-2} +3 z^4 a^{-4} -15 z^4+3 a^5 z^3-a^3 z^3-14 a z^3-18 z^3 a^{-1} -8 z^3 a^{-3} -a^4 z^2-a^2 z^2+z^2 a^{-2} -2 z^2 a^{-4} +3 z^2-2 a^3 z+5 z a^{-1} +3 z a^{-3} -a^2+a^3 z^{-1} +a z^{-1}$ (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
10          1-1
8         2 2
6        41 -3
4       62  4
2      64   -2
0     86    2
-2    67     1
-4   57      -2
-6  37       4
-8 14        -3
-10 3         3
-121          -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $r=-5$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-2$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{5}$ $r=-1$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=0$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{8}$ $r=1$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=4$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=5$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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