# L8n3

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L8n3 at Knotilus! L8n3 is $8^3_{7}$ in the Rolfsen table of links.

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{u v^2 w^2-1}{\sqrt{u} v w}$ (db) Jones polynomial $q^{-3} + q^{-5} + q^{-7} + q^{-9}$ (db) Signature -6 (db) HOMFLY-PT polynomial $a^{10} z^{-2} +a^{10}-z^4 a^8-5 z^2 a^8-2 a^8 z^{-2} -6 a^8+z^6 a^6+6 z^4 a^6+10 z^2 a^6+a^6 z^{-2} +5 a^6$ (db) Kauffman polynomial $a^{12}+a^{10} z^2+a^{10} z^{-2} -3 a^{10}+a^9 z^5-5 a^9 z^3+6 a^9 z-2 a^9 z^{-1} +a^8 z^6-6 a^8 z^4+11 a^8 z^2+2 a^8 z^{-2} -8 a^8+a^7 z^5-5 a^7 z^3+6 a^7 z-2 a^7 z^{-1} +a^6 z^6-6 a^6 z^4+10 a^6 z^2+a^6 z^{-2} -5 a^6$ (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 \
-6-5-4-3-2-10χ
-5      11
-7      11
-9    1  1
-11  1    1
-13  21   1
-151      1
-1721     1
-191      1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-7$ $i=-5$ $i=-3$ $r=-6$ ${\mathbb Z}$ ${\mathbb Z}^{2}$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}_2$ ${\mathbb Z}^{2}$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}$

### Computer Talk

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