# L11a282

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a282 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{t(1)^3 t(2)^5-t(1)^2 t(2)^5-2 t(1)^3 t(2)^4+4 t(1)^2 t(2)^4-t(1) t(2)^4+2 t(1)^3 t(2)^3-6 t(1)^2 t(2)^3+5 t(1) t(2)^3-t(2)^3-t(1)^3 t(2)^2+5 t(1)^2 t(2)^2-6 t(1) t(2)^2+2 t(2)^2-t(1)^2 t(2)+4 t(1) t(2)-2 t(2)-t(1)+1}{t(1)^{3/2} t(2)^{5/2}}$ (db) Jones polynomial $-q^{13/2}+3 q^{11/2}-6 q^{9/2}+9 q^{7/2}-13 q^{5/2}+14 q^{3/2}-15 \sqrt{q}+\frac{12}{\sqrt{q}}-\frac{9}{q^{3/2}}+\frac{6}{q^{5/2}}-\frac{3}{q^{7/2}}+\frac{1}{q^{9/2}}$ (db) Signature 1 (db) HOMFLY-PT polynomial $z^9 a^{-1} -a z^7+7 z^7 a^{-1} -z^7 a^{-3} -5 a z^5+18 z^5 a^{-1} -5 z^5 a^{-3} -8 a z^3+21 z^3 a^{-1} -8 z^3 a^{-3} -5 a z+12 z a^{-1} -5 z a^{-3} -a z^{-1} +3 a^{-1} z^{-1} -2 a^{-3} z^{-1}$ (db) Kauffman polynomial $-2 z^{10} a^{-2} -2 z^{10}-5 a z^9-9 z^9 a^{-1} -4 z^9 a^{-3} -5 a^2 z^8+z^8 a^{-2} -4 z^8 a^{-4} -3 a^3 z^7+18 a z^7+35 z^7 a^{-1} +10 z^7 a^{-3} -4 z^7 a^{-5} -a^4 z^6+17 a^2 z^6+7 z^6 a^{-2} +5 z^6 a^{-4} -3 z^6 a^{-6} +17 z^6+9 a^3 z^5-24 a z^5-57 z^5 a^{-1} -17 z^5 a^{-3} +6 z^5 a^{-5} -z^5 a^{-7} +3 a^4 z^4-16 a^2 z^4-18 z^4 a^{-2} +6 z^4 a^{-6} -31 z^4-4 a^3 z^3+17 a z^3+39 z^3 a^{-1} +16 z^3 a^{-3} +2 z^3 a^{-7} -a^4 z^2+7 a^2 z^2+11 z^2 a^{-2} -2 z^2 a^{-6} +17 z^2-6 a z-15 z a^{-1} -8 z a^{-3} -z a^{-7} -a^2-3 a^{-2} -3+a z^{-1} +3 a^{-1} z^{-1} +2 a^{-3} z^{-1}$ (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 \
-5-4-3-2-10123456χ
14           11
12          2 -2
10         41 3
8        63  -3
6       73   4
4      76    -1
2     87     1
0    58      3
-2   47       -3
-4  25        3
-6 14         -3
-8 2          2
-101           -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=0$ $i=2$ $r=-5$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-2$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-1$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=0$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{8}$ $r=1$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=2$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{4}$ $r=5$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=6$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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