# L11a521

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a521 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{u^2 v^2 w^3-2 u^2 v^2 w^2-2 u^2 v w^3+4 u^2 v w^2-2 u^2 v w+u^2 w^3-2 u^2 w^2+u^2 w-u v^2 w^3+3 u v^2 w^2-2 u v^2 w+2 u v w^3-5 u v w^2+5 u v w-2 u v+2 u w^2-3 u w+u-v^2 w^2+2 v^2 w-v^2+2 v w^2-4 v w+2 v+2 w-1}{u v w^{3/2}}$ (db) Jones polynomial $q^{10}-3 q^9+7 q^8-11 q^7+16 q^6-17 q^5+18 q^4-15 q^3+12 q^2-7 q+4- q^{-1}$ (db) Signature 4 (db) HOMFLY-PT polynomial $z^4 a^{-8} +3 z^2 a^{-8} + a^{-8} z^{-2} +2 a^{-8} -2 z^6 a^{-6} -8 z^4 a^{-6} -10 z^2 a^{-6} -2 a^{-6} z^{-2} -6 a^{-6} +z^8 a^{-4} +5 z^6 a^{-4} +9 z^4 a^{-4} +8 z^2 a^{-4} + a^{-4} z^{-2} +3 a^{-4} -z^6 a^{-2} -3 z^4 a^{-2} -z^2 a^{-2} + a^{-2}$ (db) Kauffman polynomial $z^4 a^{-12} -z^2 a^{-12} +3 z^5 a^{-11} -2 z^3 a^{-11} +6 z^6 a^{-10} -7 z^4 a^{-10} +5 z^2 a^{-10} - a^{-10} +8 z^7 a^{-9} -10 z^5 a^{-9} +5 z^3 a^{-9} +9 z^8 a^{-8} -17 z^6 a^{-8} +17 z^4 a^{-8} -12 z^2 a^{-8} - a^{-8} z^{-2} +5 a^{-8} +6 z^9 a^{-7} -5 z^7 a^{-7} -12 z^5 a^{-7} +17 z^3 a^{-7} -9 z a^{-7} +2 a^{-7} z^{-1} +2 z^{10} a^{-6} +10 z^8 a^{-6} -43 z^6 a^{-6} +50 z^4 a^{-6} -31 z^2 a^{-6} -2 a^{-6} z^{-2} +11 a^{-6} +11 z^9 a^{-5} -30 z^7 a^{-5} +16 z^5 a^{-5} +7 z^3 a^{-5} -9 z a^{-5} +2 a^{-5} z^{-1} +2 z^{10} a^{-4} +5 z^8 a^{-4} -35 z^6 a^{-4} +41 z^4 a^{-4} -17 z^2 a^{-4} - a^{-4} z^{-2} +5 a^{-4} +5 z^9 a^{-3} -16 z^7 a^{-3} +12 z^5 a^{-3} -z^3 a^{-3} +4 z^8 a^{-2} -15 z^6 a^{-2} +16 z^4 a^{-2} -4 z^2 a^{-2} - a^{-2} +z^7 a^{-1} -3 z^5 a^{-1} +2 z^3 a^{-1}$ (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 \
-3-2-1012345678χ
21           11
19          31-2
17         4  4
15        73  -4
13       94   5
11      87    -1
9     109     1
7    710      3
5   58       -3
3  38        5
1 14         -3
-1 3          3
-31           -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=3$ $i=5$ $r=-3$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=0$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{5}$ $r=1$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=2$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{10}$ $r=3$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=4$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=5$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=6$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=7$ ${\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=8$ ${\mathbb Z}$ ${\mathbb Z}$

### Computer Talk

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