# L8a20

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L8a20 at Knotilus! L8a20 is $8^3_{4}$ in the Rolfsen table of links.

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{(w-1) \left(u v w-u v-2 u w-2 v w-w^2+w\right)}{\sqrt{u} \sqrt{v} w^{3/2}}$ (db) Jones polynomial $q^4-2 q^3+5 q^2-5 q+6-5 q^{-1} +5 q^{-2} -2 q^{-3} + q^{-4}$ (db) Signature 0 (db) HOMFLY-PT polynomial $a^4+ a^{-4} -2 a^2 z^2+a^2 z^{-2} -2 z^2 a^{-2} + a^{-2} z^{-2} +z^4-2 z^{-2} -2$ (db) Kauffman polynomial $a^4 z^4+z^4 a^{-4} -2 a^4 z^2-2 z^2 a^{-4} +a^4+ a^{-4} +2 a^3 z^5+2 z^5 a^{-3} -2 a^3 z^3-2 z^3 a^{-3} +3 a^2 z^6+3 z^6 a^{-2} -5 a^2 z^4-5 z^4 a^{-2} +5 a^2 z^2+5 z^2 a^{-2} +a^2 z^{-2} + a^{-2} z^{-2} -4 a^2-4 a^{-2} +a z^7+z^7 a^{-1} +5 a z^5+5 z^5 a^{-1} -12 a z^3-12 z^3 a^{-1} +8 a z+8 z a^{-1} -2 a z^{-1} -2 a^{-1} z^{-1} +6 z^6-12 z^4+14 z^2+2 z^{-2} -9$ (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 \
-4-3-2-101234χ
9        11
7       21-1
5      3  3
3     22  0
1    43   1
-1   34    1
-3  22     0
-5  3      3
-712       -1
-91        1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-1$ $i=1$ $r=-4$ ${\mathbb Z}$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{2}$ $r=-2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=0$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{4}$ $r=1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=3$ ${\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=4$ ${\mathbb Z}$ ${\mathbb Z}$

### Computer Talk

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