# L11a63

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a63 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{(t(1)-1) (t(2)-1) \left(t(2)^2-3 t(2)+1\right) \left(t(2)^2-t(2)+1\right)}{\sqrt{t(1)} t(2)^{5/2}}$ (db) Jones polynomial $q^{17/2}-4 q^{15/2}+9 q^{13/2}-13 q^{11/2}+17 q^{9/2}-20 q^{7/2}+18 q^{5/2}-16 q^{3/2}+11 \sqrt{q}-\frac{7}{\sqrt{q}}+\frac{3}{q^{3/2}}-\frac{1}{q^{5/2}}$ (db) Signature 3 (db) HOMFLY-PT polynomial $z^3 a^{-7} +z a^{-7} + a^{-7} z^{-1} -2 z^5 a^{-5} -5 z^3 a^{-5} -5 z a^{-5} -3 a^{-5} z^{-1} +z^7 a^{-3} +4 z^5 a^{-3} +8 z^3 a^{-3} +8 z a^{-3} +3 a^{-3} z^{-1} -2 z^5 a^{-1} +a z^3-6 z^3 a^{-1} +2 a z-6 z a^{-1} +a z^{-1} -2 a^{-1} z^{-1}$ (db) Kauffman polynomial $z^4 a^{-10} +4 z^5 a^{-9} -z^3 a^{-9} +9 z^6 a^{-8} -9 z^4 a^{-8} +4 z^2 a^{-8} - a^{-8} +12 z^7 a^{-7} -16 z^5 a^{-7} +9 z^3 a^{-7} -3 z a^{-7} + a^{-7} z^{-1} +10 z^8 a^{-6} -8 z^6 a^{-6} -7 z^4 a^{-6} +5 z^2 a^{-6} -2 a^{-6} +5 z^9 a^{-5} +9 z^7 a^{-5} -39 z^5 a^{-5} +31 z^3 a^{-5} -12 z a^{-5} +3 a^{-5} z^{-1} +z^{10} a^{-4} +17 z^8 a^{-4} -46 z^6 a^{-4} +31 z^4 a^{-4} -7 z^2 a^{-4} +8 z^9 a^{-3} -8 z^7 a^{-3} -28 z^5 a^{-3} +41 z^3 a^{-3} -18 z a^{-3} +3 a^{-3} z^{-1} +z^{10} a^{-2} +10 z^8 a^{-2} -40 z^6 a^{-2} +41 z^4 a^{-2} -13 z^2 a^{-2} +2 a^{-2} +3 z^9 a^{-1} +a z^7-4 z^7 a^{-1} -4 a z^5-13 z^5 a^{-1} +6 a z^3+26 z^3 a^{-1} -4 a z-13 z a^{-1} +a z^{-1} +2 a^{-1} z^{-1} +3 z^8-11 z^6+13 z^4-5 z^2$ (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 \
-4-3-2-101234567χ
18           1-1
16          3 3
14         61 -5
12        73  4
10       106   -4
8      107    3
6     810     2
4    810      -2
2   510       5
0  26        -4
-2 15         4
-4 2          -2
-61           1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=2$ $i=4$ $r=-4$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-1$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=0$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{8}$ $r=1$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=2$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{10}$ $r=3$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{10}$ $r=4$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=5$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=6$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=7$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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