# L11a55

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a55 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{(u-1) (v-1) \left(v^4-4 v^3+7 v^2-4 v+1\right)}{\sqrt{u} v^{5/2}}$ (db) Jones polynomial $-9 q^{9/2}+\frac{1}{q^{9/2}}+14 q^{7/2}-\frac{4}{q^{7/2}}-19 q^{5/2}+\frac{8}{q^{5/2}}+22 q^{3/2}-\frac{14}{q^{3/2}}-q^{13/2}+4 q^{11/2}-22 \sqrt{q}+\frac{18}{\sqrt{q}}$ (db) Signature 1 (db) HOMFLY-PT polynomial $-z^7 a^{-1} +2 a z^5-4 z^5 a^{-1} +2 z^5 a^{-3} -a^3 z^3+5 a z^3-9 z^3 a^{-1} +5 z^3 a^{-3} -z^3 a^{-5} -a^3 z+5 a z-9 z a^{-1} +6 z a^{-3} -z a^{-5} +2 a z^{-1} -4 a^{-1} z^{-1} +3 a^{-3} z^{-1} - a^{-5} z^{-1}$ (db) Kauffman polynomial $-z^{10} a^{-2} -z^{10}-4 a z^9-9 z^9 a^{-1} -5 z^9 a^{-3} -6 a^2 z^8-20 z^8 a^{-2} -9 z^8 a^{-4} -17 z^8-4 a^3 z^7-5 a z^7-z^7 a^{-1} -8 z^7 a^{-3} -8 z^7 a^{-5} -a^4 z^6+12 a^2 z^6+50 z^6 a^{-2} +13 z^6 a^{-4} -4 z^6 a^{-6} +46 z^6+10 a^3 z^5+34 a z^5+45 z^5 a^{-1} +35 z^5 a^{-3} +13 z^5 a^{-5} -z^5 a^{-7} +2 a^4 z^4-5 a^2 z^4-45 z^4 a^{-2} -8 z^4 a^{-4} +5 z^4 a^{-6} -39 z^4-8 a^3 z^3-37 a z^3-58 z^3 a^{-1} -38 z^3 a^{-3} -8 z^3 a^{-5} +z^3 a^{-7} -a^4 z^2+a^2 z^2+17 z^2 a^{-2} +4 z^2 a^{-4} -z^2 a^{-6} +14 z^2+3 a^3 z+15 a z+26 z a^{-1} +18 z a^{-3} +4 z a^{-5} -a^2-3 a^{-2} - a^{-4} -2-2 a z^{-1} -4 a^{-1} z^{-1} -3 a^{-3} z^{-1} - a^{-5} z^{-1}$ (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          3 -3
10         61 5
8        83  -5
6       116   5
4      118    -3
2     1111     0
0    913      4
-2   59       -4
-4  39        6
-6 15         -4
-8 3          3
-101           -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=0$ $i=2$ $r=-5$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-2$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-1$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=0$ ${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{11}$ $r=1$ ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{11}$ ${\mathbb Z}^{11}$ $r=2$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{11}$ ${\mathbb Z}^{11}$ $r=3$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=5$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=6$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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