# L11n231

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n231 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{u^3 v^3-2 u^3 v^2+u^3 v-u^2 v^3+5 u^2 v^2-3 u^2 v-3 u v^2+5 u v-u+v^2-2 v+1}{u^{3/2} v^{3/2}}$ (db) Jones polynomial $-q^{15/2}+3 q^{13/2}-6 q^{11/2}+8 q^{9/2}-9 q^{7/2}+8 q^{5/2}-8 q^{3/2}+5 \sqrt{q}-\frac{3}{\sqrt{q}}+\frac{1}{q^{3/2}}$ (db) Signature 3 (db) HOMFLY-PT polynomial $z^7 a^{-3} -z^5 a^{-1} +5 z^5 a^{-3} -z^5 a^{-5} -3 z^3 a^{-1} +9 z^3 a^{-3} -3 z^3 a^{-5} -2 z a^{-1} +7 z a^{-3} -3 z a^{-5} + a^{-3} z^{-1} - a^{-5} z^{-1}$ (db) Kauffman polynomial $-z^9 a^{-3} -z^9 a^{-5} -3 z^8 a^{-2} -5 z^8 a^{-4} -2 z^8 a^{-6} -3 z^7 a^{-1} -4 z^7 a^{-3} -2 z^7 a^{-5} -z^7 a^{-7} +7 z^6 a^{-2} +12 z^6 a^{-4} +4 z^6 a^{-6} -z^6+10 z^5 a^{-1} +20 z^5 a^{-3} +10 z^5 a^{-5} -6 z^4 a^{-4} -6 z^4 a^{-6} -3 z^4 a^{-8} +3 z^4-9 z^3 a^{-1} -20 z^3 a^{-3} -13 z^3 a^{-5} -3 z^3 a^{-7} -z^3 a^{-9} -2 z^2 a^{-2} -z^2 a^{-4} +z^2 a^{-6} +2 z^2 a^{-8} -2 z^2+3 z a^{-1} +9 z a^{-3} +6 z a^{-5} +z a^{-7} +z a^{-9} + a^{-4} - a^{-3} z^{-1} - a^{-5} z^{-1}$ (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 \
-3-2-10123456χ
16         11
14        2 -2
12       41 3
10      53  -2
8     43   1
6    45    1
4   44     0
2  25      3
0 13       -2
-2 2        2
-41         -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=2$ $i=4$ $r=-3$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=0$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{4}$ $r=1$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=2$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{4}$ $r=5$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=6$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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