# L10a83

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a83 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{2 t(2)^2 t(1)^2-5 t(2) t(1)^2+2 t(1)^2-5 t(2)^2 t(1)+9 t(2) t(1)-5 t(1)+2 t(2)^2-5 t(2)+2}{t(1) t(2)}$ (db) Jones polynomial $-\sqrt{q}+\frac{3}{\sqrt{q}}-\frac{6}{q^{3/2}}+\frac{9}{q^{5/2}}-\frac{12}{q^{7/2}}+\frac{12}{q^{9/2}}-\frac{12}{q^{11/2}}+\frac{9}{q^{13/2}}-\frac{6}{q^{15/2}}+\frac{3}{q^{17/2}}-\frac{1}{q^{19/2}}$ (db) Signature -3 (db) HOMFLY-PT polynomial $z a^9-2 z^3 a^7-z a^7+a^7 z^{-1} +z^5 a^5-2 z a^5-a^5 z^{-1} +z^5 a^3+z^3 a^3-z^3 a-z a$ (db) Kauffman polynomial $a^{11} z^5-2 a^{11} z^3+a^{11} z+3 a^{10} z^6-6 a^{10} z^4+3 a^{10} z^2+4 a^9 z^7-6 a^9 z^5+2 a^9 z^3-a^9 z+3 a^8 z^8-6 a^8 z^4+3 a^8 z^2+a^7 z^9+7 a^7 z^7-14 a^7 z^5+7 a^7 z^3+a^7 z-a^7 z^{-1} +6 a^6 z^8-6 a^6 z^6+a^6+a^5 z^9+7 a^5 z^7-14 a^5 z^5+7 a^5 z^3+a^5 z-a^5 z^{-1} +3 a^4 z^8-6 a^4 z^4+3 a^4 z^2+4 a^3 z^7-6 a^3 z^5+2 a^3 z^3-a^3 z+3 a^2 z^6-6 a^2 z^4+3 a^2 z^2+a z^5-2 a z^3+a z$ (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 \
-8-7-6-5-4-3-2-1012χ
2          11
0         2 -2
-2        41 3
-4       63  -3
-6      63   3
-8     66    0
-10    66     0
-12   36      3
-14  36       -3
-16 14        3
-18 2         -2
-201          1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-4$ $i=-2$ $r=-8$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}$ $r=-5$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-4$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=-3$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=-2$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=-1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=0$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{4}$ $r=1$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=2$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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