# L11a96

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a96 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{(t(1)-1) (t(2)-1) \left(2 t(2)^2-7 t(2)+2\right)}{\sqrt{t(1)} t(2)^{3/2}}$ (db) Jones polynomial $-9 q^{9/2}+12 q^{7/2}-\frac{1}{q^{7/2}}-14 q^{5/2}+\frac{2}{q^{5/2}}+14 q^{3/2}-\frac{6}{q^{3/2}}+q^{15/2}-3 q^{13/2}+5 q^{11/2}-12 \sqrt{q}+\frac{9}{\sqrt{q}}$ (db) Signature 1 (db) HOMFLY-PT polynomial $z a^{-7} -2 z^3 a^{-5} -2 z a^{-5} - a^{-5} z^{-1} +z^5 a^{-3} +z^3 a^{-3} +a^3 z+3 z a^{-3} +a^3 z^{-1} +2 a^{-3} z^{-1} +z^5 a^{-1} -2 a z^3-2 a z-z a^{-1} -a z^{-1} - a^{-1} z^{-1}$ (db) Kauffman polynomial $-z^{10} a^{-2} -z^{10} a^{-4} -2 z^9 a^{-1} -5 z^9 a^{-3} -3 z^9 a^{-5} -3 z^8 a^{-2} -4 z^8 a^{-4} -4 z^8 a^{-6} -3 z^8-3 a z^7-2 z^7 a^{-1} +9 z^7 a^{-3} +5 z^7 a^{-5} -3 z^7 a^{-7} -2 a^2 z^6+7 z^6 a^{-2} +16 z^6 a^{-4} +12 z^6 a^{-6} -z^6 a^{-8} +2 z^6-a^3 z^5+3 a z^5+8 z^5 a^{-1} -z^5 a^{-3} +5 z^5 a^{-5} +10 z^5 a^{-7} +3 a^2 z^4-9 z^4 a^{-2} -17 z^4 a^{-4} -9 z^4 a^{-6} +3 z^4 a^{-8} -z^4+3 a^3 z^3+2 a z^3-11 z^3 a^{-1} -15 z^3 a^{-3} -13 z^3 a^{-5} -8 z^3 a^{-7} +7 z^2 a^{-2} +5 z^2 a^{-4} +2 z^2 a^{-6} -z^2 a^{-8} +5 z^2-3 a^3 z-3 a z+8 z a^{-1} +13 z a^{-3} +7 z a^{-5} +2 z a^{-7} -a^2-3 a^{-2} - a^{-4} -2+a^3 z^{-1} +a z^{-1} - a^{-1} z^{-1} -2 a^{-3} z^{-1} - a^{-5} z^{-1}$ (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 \
-4-3-2-101234567χ
16           1-1
14          2 2
12         31 -2
10        62  4
8       63   -3
6      86    2
4     66     0
2    68      -2
0   58       3
-2  14        -3
-4 15         4
-6 1          -1
-81           1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=0$ $i=2$ $r=-4$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=0$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{6}$ $r=1$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=2$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=3$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=5$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=6$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=7$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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