# L11a230

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a230 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{2 u^2 v^4-5 u^2 v^3+6 u^2 v^2-2 u^2 v-u v^4+7 u v^3-11 u v^2+7 u v-u-2 v^3+6 v^2-5 v+2}{u v^2}$ (db) Jones polynomial $q^{3/2}-3 \sqrt{q}+\frac{7}{\sqrt{q}}-\frac{12}{q^{3/2}}+\frac{15}{q^{5/2}}-\frac{19}{q^{7/2}}+\frac{18}{q^{9/2}}-\frac{16}{q^{11/2}}+\frac{12}{q^{13/2}}-\frac{7}{q^{15/2}}+\frac{3}{q^{17/2}}-\frac{1}{q^{19/2}}$ (db) Signature -3 (db) HOMFLY-PT polynomial $z^5 a^7+3 z^3 a^7+3 z a^7-z^7 a^5-4 z^5 a^5-6 z^3 a^5-2 z a^5+2 a^5 z^{-1} -z^7 a^3-4 z^5 a^3-7 z^3 a^3-7 z a^3-3 a^3 z^{-1} +z^5 a+3 z^3 a+3 z a+a z^{-1}$ (db) Kauffman polynomial $-z^5 a^{11}+2 z^3 a^{11}-z a^{11}-3 z^6 a^{10}+5 z^4 a^{10}-2 z^2 a^{10}-5 z^7 a^9+7 z^5 a^9-3 z^3 a^9+z a^9-6 z^8 a^8+9 z^6 a^8-8 z^4 a^8+4 z^2 a^8-4 z^9 a^7+6 z^5 a^7-4 z^3 a^7-z a^7-z^{10} a^6-11 z^8 a^6+26 z^6 a^6-24 z^4 a^6+9 z^2 a^6-7 z^9 a^5+6 z^7 a^5+8 z^5 a^5-10 z^3 a^5+4 z a^5-2 a^5 z^{-1} -z^{10} a^4-9 z^8 a^4+21 z^6 a^4-10 z^4 a^4-2 z^2 a^4+3 a^4-3 z^9 a^3-2 z^7 a^3+18 z^5 a^3-18 z^3 a^3+10 z a^3-3 a^3 z^{-1} -4 z^8 a^2+6 z^6 a^2+4 z^4 a^2-8 z^2 a^2+3 a^2-3 z^7 a+8 z^5 a-7 z^3 a+3 z a-a z^{-1} -z^6+3 z^4-3 z^2+1$ (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 \
-8-7-6-5-4-3-2-10123χ
4           1-1
2          2 2
0         51 -4
-2        72  5
-4       96   -3
-6      106    4
-8     89     1
-10    810      -2
-12   48       4
-14  38        -5
-16 15         4
-18 2          -2
-201           1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-4$ $i=-2$ $r=-8$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}$ $r=-5$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-4$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=-3$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=-2$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{10}$ $r=-1$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=0$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{7}$ $r=1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=2$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=3$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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