# L7n1

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L7n1 at Knotilus! L7n1 is $7^2_7$ in the Rolfsen table of links.

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{-u v^3-1}{\sqrt{u} v^{3/2}}$ (db) Jones polynomial $-\frac{1}{q^{5/2}}-\frac{1}{q^{9/2}}-\frac{1}{q^{13/2}}+\frac{1}{q^{15/2}}$ (db) Signature -5 (db) HOMFLY-PT polynomial $-a^9 z^{-1} +a^7 z^3+4 a^7 z+3 a^7 z^{-1} -a^5 z^5-5 a^5 z^3-6 a^5 z-2 a^5 z^{-1}$ (db) Kauffman polynomial $a^{10}+a^9 z-a^9 z^{-1} +a^8 z^4-4 a^8 z^2+3 a^8+a^7 z^5-5 a^7 z^3+7 a^7 z-3 a^7 z^{-1} +a^6 z^4-4 a^6 z^2+3 a^6+a^5 z^5-5 a^5 z^3+6 a^5 z-2 a^5 z^{-1}$ (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 \
-5-4-3-2-10χ
-4     11
-6     11
-8   1  1
-10 1    1
-12 21   1
-14      0
-161     -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-6$ $i=-4$ $i=-2$ $r=-5$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}_2$ ${\mathbb Z}^{2}$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}$

### Computer Talk

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