# L11n83

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n83 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{2 (t(1)-1) (t(2)-1) \left(t(2)^2-t(2)+1\right)}{\sqrt{t(1)} t(2)^{3/2}}$ (db) Jones polynomial $-\sqrt{q}+\frac{2}{\sqrt{q}}-\frac{5}{q^{3/2}}+\frac{6}{q^{5/2}}-\frac{8}{q^{7/2}}+\frac{8}{q^{9/2}}-\frac{8}{q^{11/2}}+\frac{5}{q^{13/2}}-\frac{3}{q^{15/2}}+\frac{2}{q^{17/2}}$ (db) Signature -3 (db) HOMFLY-PT polynomial $-a^9 z^{-1} -a^7 z^3+2 a^7 z^{-1} +a^5 z^5+2 a^5 z^3+a^5 z-a^5 z^{-1} +a^3 z^5+2 a^3 z^3+a^3 z+a^3 z^{-1} -a z^3-2 a z-a z^{-1}$ (db) Kauffman polynomial $3 a^{10} z^4-8 a^{10} z^2+3 a^{10}+a^9 z^7-a^9 z^5-a^9 z^3+a^9 z-a^9 z^{-1} +2 a^8 z^8-8 a^8 z^6+19 a^8 z^4-20 a^8 z^2+7 a^8+a^7 z^9-a^7 z^7-a^7 z^5+5 a^7 z^3+2 a^7 z-2 a^7 z^{-1} +4 a^6 z^8-13 a^6 z^6+22 a^6 z^4-14 a^6 z^2+4 a^6+a^5 z^9-2 a^5 z^5+3 a^5 z^3+a^5 z-a^5 z^{-1} +2 a^4 z^8-3 a^4 z^6+2 a^4 z^4-2 a^4 z^2+2 a^3 z^7-a^3 z^5-6 a^3 z^3+3 a^3 z-a^3 z^{-1} +2 a^2 z^6-4 a^2 z^4+a^2+a z^5-3 a z^3+3 a z-a z^{-1}$ (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 \
-7-6-5-4-3-2-1012χ
2         11
0        1 -1
-2       41 3
-4      43  -1
-6     42   2
-8    44    0
-10   44     0
-12  14      3
-14 24       -2
-16 1        1
-182         -2
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-4$ $i=-2$ $r=-7$ ${\mathbb Z}^{2}$ $r=-6$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-5$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-3$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=0$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{4}$ $r=1$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=2$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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