# L11a356

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a356 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{u^4 v^3-u^4 v^2+u^3 v^4-3 u^3 v^3+4 u^3 v^2-2 u^3 v-u^2 v^4+4 u^2 v^3-7 u^2 v^2+4 u^2 v-u^2-2 u v^3+4 u v^2-3 u v+u-v^2+v}{u^2 v^2}$ (db) Jones polynomial $-\frac{8}{q^{9/2}}-q^{7/2}+\frac{10}{q^{7/2}}+3 q^{5/2}-\frac{13}{q^{5/2}}-6 q^{3/2}+\frac{13}{q^{3/2}}+\frac{1}{q^{15/2}}-\frac{2}{q^{13/2}}+\frac{4}{q^{11/2}}+9 \sqrt{q}-\frac{12}{\sqrt{q}}$ (db) Signature -1 (db) HOMFLY-PT polynomial $a^3 z^7+a z^7-a^5 z^5+5 a^3 z^5+4 a z^5-z^5 a^{-1} -4 a^5 z^3+9 a^3 z^3+4 a z^3-3 z^3 a^{-1} -4 a^5 z+6 a^3 z-a z-2 z a^{-1} +a^3 z^{-1} -a z^{-1}$ (db) Kauffman polynomial $a^8 z^6-4 a^8 z^4+4 a^8 z^2+2 a^7 z^7-7 a^7 z^5+7 a^7 z^3-2 a^7 z+2 a^6 z^8-4 a^6 z^6+a^6 z^2+2 a^5 z^9-5 a^5 z^7+9 a^5 z^5-12 a^5 z^3+3 a^5 z+a^4 z^{10}-a^4 z^6+2 a^4 z^4-4 a^4 z^2+5 a^3 z^9-16 a^3 z^7+30 a^3 z^5+z^5 a^{-3} -27 a^3 z^3-2 z^3 a^{-3} +10 a^3 z-a^3 z^{-1} +a^2 z^{10}+3 a^2 z^8-11 a^2 z^6+3 z^6 a^{-2} +16 a^2 z^4-6 z^4 a^{-2} -6 a^2 z^2+z^2 a^{-2} +a^2+3 a z^9-4 a z^7+5 z^7 a^{-1} +a z^5-12 z^5 a^{-1} +3 a z^3+9 z^3 a^{-1} +2 a z-3 z a^{-1} -a z^{-1} +5 z^8-12 z^6+12 z^4-4 z^2$ (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 \
-7-6-5-4-3-2-101234χ
8           11
6          2 -2
4         41 3
2        52  -3
0       74   3
-2      76    -1
-4     66     0
-6    58      3
-8   35       -2
-10  15        4
-12 13         -2
-14 1          1
-161           -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $r=-7$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-3$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-2$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{6}$ $r=-1$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=0$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{7}$ $r=1$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=3$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=4$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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