# L10a21

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a21 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{(u-1) (v-1)^3 \left(v^2-v+1\right)}{\sqrt{u} v^{5/2}}$ (db) Jones polynomial $-8 q^{9/2}+12 q^{7/2}-\frac{1}{q^{7/2}}-15 q^{5/2}+\frac{4}{q^{5/2}}+16 q^{3/2}-\frac{8}{q^{3/2}}-q^{13/2}+4 q^{11/2}-16 \sqrt{q}+\frac{11}{\sqrt{q}}$ (db) Signature 1 (db) HOMFLY-PT polynomial $-z^7 a^{-1} +a z^5-4 z^5 a^{-1} +2 z^5 a^{-3} +2 a z^3-6 z^3 a^{-1} +5 z^3 a^{-3} -z^3 a^{-5} +a z-3 z a^{-1} +3 z a^{-3} -z a^{-5} +a z^{-1} - a^{-1} z^{-1}$ (db) Kauffman polynomial $-2 z^9 a^{-1} -2 z^9 a^{-3} -12 z^8 a^{-2} -6 z^8 a^{-4} -6 z^8-7 a z^7-12 z^7 a^{-1} -12 z^7 a^{-3} -7 z^7 a^{-5} -4 a^2 z^6+19 z^6 a^{-2} +5 z^6 a^{-4} -4 z^6 a^{-6} +6 z^6-a^3 z^5+13 a z^5+32 z^5 a^{-1} +31 z^5 a^{-3} +12 z^5 a^{-5} -z^5 a^{-7} +6 a^2 z^4-7 z^4 a^{-2} +4 z^4 a^{-4} +6 z^4 a^{-6} +z^4+a^3 z^3-8 a z^3-25 z^3 a^{-1} -23 z^3 a^{-3} -6 z^3 a^{-5} +z^3 a^{-7} -a^2 z^2-z^2 a^{-2} -3 z^2 a^{-4} -2 z^2 a^{-6} -z^2+2 a z+6 z a^{-1} +6 z a^{-3} +2 z a^{-5} +1-a z^{-1} - a^{-1} z^{-1}$ (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 \
-4-3-2-10123456χ
14          11
12         3 -3
10        51 4
8       73  -4
6      85   3
4     87    -1
2    88     0
0   510      5
-2  36       -3
-4 15        4
-6 3         -3
-81          1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=0$ $i=2$ $r=-4$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-1$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=0$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{8}$ $r=1$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=2$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=3$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=5$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=6$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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