# L7a7

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L7a7 at Knotilus! L7a7 is $7^3_1$ in the Rolfsen table of links.

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{u v w+u (-v)-u w+2 u-2 v w+v+w-1}{\sqrt{u} \sqrt{v} \sqrt{w}}$ (db) Jones polynomial $q^{-4} -q^3- q^{-3} +3 q^2+4 q^{-2} -3 q-3 q^{-1} +4$ (db) Signature 0 (db) HOMFLY-PT polynomial $a^4 z^{-2} +a^4-2 z^2 a^2-2 a^2 z^{-2} -3 a^2+z^4+2 z^2+ z^{-2} +2-z^2 a^{-2}$ (db) Kauffman polynomial $a^2 z^6+z^6+a^3 z^5+4 a z^5+3 z^5 a^{-1} +a^4 z^4+a^2 z^4+3 z^4 a^{-2} +3 z^4-4 a z^3-3 z^3 a^{-1} +z^3 a^{-3} -3 a^4 z^2-5 a^2 z^2-3 z^2 a^{-2} -5 z^2-3 a^3 z-3 a z+3 a^4+5 a^2+3+2 a^3 z^{-1} +2 a z^{-1} -a^4 z^{-2} -2 a^2 z^{-2} - z^{-2}$ (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 \
-4-3-2-10123χ
7       1-1
5      2 2
3     11 0
1    32  1
-1   34   1
-3  1     1
-5  3     3
-711      0
-91       1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-1$ $i=1$ $r=-4$ ${\mathbb Z}$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=0$ ${\mathbb Z}^{4}$ ${\mathbb Z}^{3}$ $r=1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=2$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=3$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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