# L7a3

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L7a3 at Knotilus! L7a3 is $7^2_4$ in the Rolfsen table of links.

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{(t(1)-1) (t(2)-1) \left(t(2)^2+1\right)}{\sqrt{t(1)} t(2)^{3/2}}$ (db) Jones polynomial $3 q^{9/2}-3 q^{7/2}+2 q^{5/2}-3 q^{3/2}+q^{13/2}-2 q^{11/2}+\sqrt{q}-\frac{1}{\sqrt{q}}$ (db) Signature 3 (db) HOMFLY-PT polynomial $z^3 a^{-5} +2 z a^{-5} + a^{-5} z^{-1} -z^5 a^{-3} -4 z^3 a^{-3} -5 z a^{-3} -3 a^{-3} z^{-1} +z^3 a^{-1} +3 z a^{-1} +2 a^{-1} z^{-1}$ (db) Kauffman polynomial $z^2 a^{-8} +2 z^3 a^{-7} +3 z^4 a^{-6} -3 z^2 a^{-6} + a^{-6} +3 z^5 a^{-5} -6 z^3 a^{-5} +4 z a^{-5} - a^{-5} z^{-1} +z^6 a^{-4} +z^4 a^{-4} -6 z^2 a^{-4} +3 a^{-4} +4 z^5 a^{-3} -12 z^3 a^{-3} +9 z a^{-3} -3 a^{-3} z^{-1} +z^6 a^{-2} -2 z^4 a^{-2} -2 z^2 a^{-2} +3 a^{-2} +z^5 a^{-1} -4 z^3 a^{-1} +5 z a^{-1} -2 a^{-1} z^{-1}$ (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 \
-2-1012345χ
14       1-1
12      1 1
10     21 -1
8    11  0
6   12   1
4  21    1
2 13     2
0        0
-21       1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=2$ $i=4$ $r=-2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=0$ ${\mathbb Z}^{3}$ ${\mathbb Z}^{2}$ $r=1$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=3$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=4$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=5$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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