# L11a371

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a371 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{u^4 v^2-u^4 v+u^3 v^3-3 u^3 v^2+2 u^3 v-u^3+u^2 v^4-3 u^2 v^3+3 u^2 v^2-3 u^2 v+u^2-u v^4+2 u v^3-3 u v^2+u v-v^3+v^2}{u^2 v^2}$ (db) Jones polynomial $9 q^{9/2}-8 q^{7/2}+6 q^{5/2}-4 q^{3/2}+q^{21/2}-2 q^{19/2}+3 q^{17/2}-6 q^{15/2}+7 q^{13/2}-9 q^{11/2}+2 \sqrt{q}-\frac{1}{\sqrt{q}}$ (db) Signature 3 (db) HOMFLY-PT polynomial $-z^5 a^{-3} -z^5 a^{-5} -z^5 a^{-7} +z^3 a^{-1} -2 z^3 a^{-3} -z^3 a^{-5} -3 z^3 a^{-7} +z^3 a^{-9} +2 z a^{-1} +2 z a^{-5} -3 z a^{-7} +2 z a^{-9} + a^{-5} z^{-1} - a^{-7} z^{-1}$ (db) Kauffman polynomial $z^6 a^{-12} -4 z^4 a^{-12} +3 z^2 a^{-12} +2 z^7 a^{-11} -8 z^5 a^{-11} +8 z^3 a^{-11} -3 z a^{-11} +2 z^8 a^{-10} -6 z^6 a^{-10} +2 z^4 a^{-10} +z^2 a^{-10} +2 z^9 a^{-9} -7 z^7 a^{-9} +7 z^5 a^{-9} -z^3 a^{-9} -3 z a^{-9} +z^{10} a^{-8} -2 z^8 a^{-8} -z^6 a^{-8} +5 z^4 a^{-8} -2 z^2 a^{-8} +4 z^9 a^{-7} -18 z^7 a^{-7} +33 z^5 a^{-7} -23 z^3 a^{-7} +8 z a^{-7} - a^{-7} z^{-1} +z^{10} a^{-6} -2 z^8 a^{-6} +8 z^4 a^{-6} -5 z^2 a^{-6} + a^{-6} +2 z^9 a^{-5} -7 z^7 a^{-5} +14 z^5 a^{-5} -12 z^3 a^{-5} +6 z a^{-5} - a^{-5} z^{-1} +2 z^8 a^{-4} -4 z^6 a^{-4} +4 z^4 a^{-4} -3 z^2 a^{-4} +2 z^7 a^{-3} -3 z^5 a^{-3} -z^3 a^{-3} +2 z^6 a^{-2} -5 z^4 a^{-2} +2 z^2 a^{-2} +z^5 a^{-1} -3 z^3 a^{-1} +2 z a^{-1}$ (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 \
-2-10123456789χ
22           1-1
20          1 1
18         21 -1
16        41  3
14       43   -1
12      53    2
10     44     0
8    45      -1
6   24       2
4  24        -2
2 13         2
0 1          -1
-21           1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=2$ $i=4$ $r=-2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=0$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=1$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=3$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=4$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=5$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=6$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{4}$ $r=7$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=8$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=9$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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