# L10a43

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a43 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{u v^5-4 u v^4+8 u v^3-8 u v^2+4 u v+4 v^4-8 v^3+8 v^2-4 v+1}{\sqrt{u} v^{5/2}}$ (db) Jones polynomial $-9 q^{9/2}+13 q^{7/2}-\frac{1}{q^{7/2}}-16 q^{5/2}+\frac{3}{q^{5/2}}+17 q^{3/2}-\frac{8}{q^{3/2}}-q^{13/2}+5 q^{11/2}-16 \sqrt{q}+\frac{11}{\sqrt{q}}$ (db) Signature 1 (db) HOMFLY-PT polynomial $-z^7 a^{-1} +a z^5-5 z^5 a^{-1} +2 z^5 a^{-3} +3 a z^3-12 z^3 a^{-1} +5 z^3 a^{-3} -z^3 a^{-5} +5 a z-12 z a^{-1} +5 z a^{-3} +3 a z^{-1} -5 a^{-1} z^{-1} +2 a^{-3} z^{-1}$ (db) Kauffman polynomial $z^5 a^{-7} +5 z^6 a^{-6} -6 z^4 a^{-6} + a^{-6} +9 z^7 a^{-5} -15 z^5 a^{-5} +5 z^3 a^{-5} +7 z^8 a^{-4} -4 z^6 a^{-4} -9 z^4 a^{-4} +4 z^2 a^{-4} +2 z^9 a^{-3} +15 z^7 a^{-3} +a^3 z^5-34 z^5 a^{-3} -2 a^3 z^3+18 z^3 a^{-3} +a^3 z-5 z a^{-3} +2 a^{-3} z^{-1} +12 z^8 a^{-2} +3 a^2 z^6-13 z^6 a^{-2} -4 a^2 z^4-10 z^4 a^{-2} +a^2 z^2+17 z^2 a^{-2} -5 a^{-2} +2 z^9 a^{-1} +6 a z^7+12 z^7 a^{-1} -12 a z^5-31 z^5 a^{-1} +14 a z^3+29 z^3 a^{-1} -10 a z-16 z a^{-1} +3 a z^{-1} +5 a^{-1} z^{-1} +5 z^8-z^6-11 z^4+14 z^2-5$ (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         4 -4
10        51 4
8       84  -4
6      85   3
4     98    -1
2    78     -1
0   510      5
-2  36       -3
-4  5        5
-613         -2
-81          1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=0$ $i=2$ $r=-4$ ${\mathbb Z}$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{3}$ $r=-2$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-1$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=0$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{7}$ $r=1$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=2$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=3$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=4$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=5$ ${\mathbb Z}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=6$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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