# L8n7

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L8n7 at Knotilus! L8n7 is $8^4_{2}$ in the Rolfsen table of links.

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{t(1) t(2)-t(1) t(3) t(2)+t(3) t(2)-t(1) t(4) t(2)-t(3)+t(1) t(4)+t(3) t(4)-t(4)}{\sqrt{t(1)} \sqrt{t(2)} \sqrt{t(3)} \sqrt{t(4)}}$ (db) Jones polynomial $-q^{13/2}+q^{11/2}-4 q^{9/2}+q^{7/2}-4 q^{5/2}+2 q^{3/2}-3 \sqrt{q}$ (db) Signature 1 (db) HOMFLY-PT polynomial $- a^{-7} z^{-3} - a^{-7} z^{-1} +3 a^{-5} z^{-3} +3 z a^{-5} +5 a^{-5} z^{-1} -2 z^3 a^{-3} -3 a^{-3} z^{-3} -6 z a^{-3} -7 a^{-3} z^{-1} + a^{-1} z^{-3} +3 z a^{-1} +3 a^{-1} z^{-1}$ (db) Kauffman polynomial $z^5 a^{-7} -4 z^3 a^{-7} + a^{-7} z^{-3} +6 z a^{-7} -4 a^{-7} z^{-1} +z^6 a^{-6} -z^4 a^{-6} -6 z^2 a^{-6} -3 a^{-6} z^{-2} +8 a^{-6} +5 z^5 a^{-5} -16 z^3 a^{-5} +3 a^{-5} z^{-3} +14 z a^{-5} -9 a^{-5} z^{-1} +z^6 a^{-4} +2 z^4 a^{-4} -12 z^2 a^{-4} -6 a^{-4} z^{-2} +15 a^{-4} +4 z^5 a^{-3} -12 z^3 a^{-3} +3 a^{-3} z^{-3} +14 z a^{-3} -9 a^{-3} z^{-1} +3 z^4 a^{-2} -6 z^2 a^{-2} -3 a^{-2} z^{-2} +8 a^{-2} + a^{-1} z^{-3} +6 z a^{-1} -4 a^{-1} z^{-1}$ (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 \
0123456χ
14      11
12       0
10    41 3
8   14  3
6  3    3
41 1    2
243     1
03      3
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=0$ $i=2$ $i=4$ $r=0$ ${\mathbb Z}^{3}$ ${\mathbb Z}^{4}$ ${\mathbb Z}$ $r=1$ ${\mathbb Z}^{3}$ $r=2$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=3$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=4$ ${\mathbb Z}^{4}$ ${\mathbb Z}^{4}$ $r=5$ ${\mathbb Z}$ $r=6$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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