# L11a368

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a368 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{u^4 v^3-u^4 v^2+u^3 v^4-3 u^3 v^3+4 u^3 v^2-u^3 v-u^2 v^4+4 u^2 v^3-5 u^2 v^2+4 u^2 v-u^2-u v^3+4 u v^2-3 u v+u-v^2+v}{u^2 v^2}$ (db) Jones polynomial $q^{3/2}-3 \sqrt{q}+\frac{5}{\sqrt{q}}-\frac{8}{q^{3/2}}+\frac{10}{q^{5/2}}-\frac{12}{q^{7/2}}+\frac{11}{q^{9/2}}-\frac{10}{q^{11/2}}+\frac{7}{q^{13/2}}-\frac{4}{q^{15/2}}+\frac{2}{q^{17/2}}-\frac{1}{q^{19/2}}$ (db) Signature -3 (db) HOMFLY-PT polynomial $z^5 a^7+4 z^3 a^7+4 z a^7+a^7 z^{-1} -z^7 a^5-5 z^5 a^5-9 z^3 a^5-7 z a^5-a^5 z^{-1} -z^7 a^3-4 z^5 a^3-4 z^3 a^3-z a^3+z^5 a+3 z^3 a+z a$ (db) Kauffman polynomial $a^{11} z^5-3 a^{11} z^3+2 a^{11} z+2 a^{10} z^6-5 a^{10} z^4+3 a^{10} z^2+2 a^9 z^7-2 a^9 z^5-2 a^9 z^3+a^9 z+2 a^8 z^8-2 a^8 z^6+a^8 z^4-a^8 z^2+2 a^7 z^9-5 a^7 z^7+13 a^7 z^5-15 a^7 z^3+7 a^7 z-a^7 z^{-1} +a^6 z^{10}-3 a^6 z^6+10 a^6 z^4-7 a^6 z^2+a^6+5 a^5 z^9-18 a^5 z^7+31 a^5 z^5-24 a^5 z^3+9 a^5 z-a^5 z^{-1} +a^4 z^{10}+2 a^4 z^8-13 a^4 z^6+17 a^4 z^4-6 a^4 z^2+3 a^3 z^9-8 a^3 z^7+5 a^3 z^5-a^3 z^3+4 a^2 z^8-13 a^2 z^6+10 a^2 z^4-2 a^2 z^2+3 a z^7-10 a z^5+7 a z^3-a z+z^6-3 z^4+z^2$ (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 \
-8-7-6-5-4-3-2-10123χ
4           1-1
2          2 2
0         31 -2
-2        52  3
-4       64   -2
-6      64    2
-8     56     1
-10    56      -1
-12   25       3
-14  25        -3
-16 13         2
-18 1          -1
-201           1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-4$ $i=-2$ $r=-8$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=-5$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-4$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-3$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-2$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=-1$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=0$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{5}$ $r=1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=2$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=3$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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