# L11n232

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n232 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{u^3 \left(-v^2\right)+2 u^3 v-u^3+u^2 v^2-u^2 v+u^2+u v^3-u v^2+u v-v^3+2 v^2-v}{u^{3/2} v^{3/2}}$ (db) Jones polynomial $2 q^{9/2}-4 q^{7/2}+4 q^{5/2}-\frac{1}{q^{5/2}}-5 q^{3/2}+\frac{3}{q^{3/2}}-q^{11/2}+4 \sqrt{q}-\frac{4}{\sqrt{q}}$ (db) Signature -1 (db) HOMFLY-PT polynomial $-z^5 a^{-1} +a z^3-3 z^3 a^{-1} +2 z^3 a^{-3} +a z-2 z a^{-1} +4 z a^{-3} -z a^{-5} + a^{-3} z^{-1} - a^{-5} z^{-1}$ (db) Kauffman polynomial $z^7 a^{-5} -5 z^5 a^{-5} +8 z^3 a^{-5} -5 z a^{-5} + a^{-5} z^{-1} +2 z^8 a^{-4} -9 z^6 a^{-4} +10 z^4 a^{-4} -2 z^2 a^{-4} - a^{-4} +z^9 a^{-3} -z^7 a^{-3} -11 z^5 a^{-3} +18 z^3 a^{-3} -8 z a^{-3} + a^{-3} z^{-1} +4 z^8 a^{-2} -17 z^6 a^{-2} +17 z^4 a^{-2} +a^2 z^2-4 z^2 a^{-2} +z^9 a^{-1} +a z^7-z^7 a^{-1} -4 a z^5-10 z^5 a^{-1} +5 a z^3+15 z^3 a^{-1} -2 a z-5 z a^{-1} +2 z^8-8 z^6+7 z^4-z^2$ (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 \
-2-10123456χ
12        11
10       1 -1
8      31 2
6     22  0
4    32   1
2  122    1
0  33     0
-2 12      1
-4 2       -2
-61        1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $i=2$ $r=-2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=0$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}$ ${\mathbb Z}$ $r=1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=4$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}$ $r=5$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=6$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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