# L10a58

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a58 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{u^2 v^3-4 u^2 v^2+2 u^2 v+u v^4-5 u v^3+7 u v^2-5 u v+u+2 v^3-4 v^2+v}{u v^2}$ (db) Jones polynomial $\frac{11}{q^{9/2}}-\frac{11}{q^{7/2}}+\frac{8}{q^{5/2}}-\frac{6}{q^{3/2}}-\frac{1}{q^{19/2}}+\frac{2}{q^{17/2}}-\frac{5}{q^{15/2}}+\frac{8}{q^{13/2}}-\frac{10}{q^{11/2}}-\sqrt{q}+\frac{3}{\sqrt{q}}$ (db) Signature -3 (db) HOMFLY-PT polynomial $a^9 z+a^9 z^{-1} -2 a^7 z^3-3 a^7 z-2 a^7 z^{-1} +a^5 z^5+a^5 z^3+a^5 z+2 a^5 z^{-1} +a^3 z^5+a^3 z^3-a^3 z-a^3 z^{-1} -a z^3-a z$ (db) Kauffman polynomial $a^{11} z^5-3 a^{11} z^3+2 a^{11} z+2 a^{10} z^6-4 a^{10} z^4+a^{10} z^2+3 a^9 z^7-6 a^9 z^5+5 a^9 z^3-4 a^9 z+a^9 z^{-1} +3 a^8 z^8-6 a^8 z^6+8 a^8 z^4-5 a^8 z^2+a^7 z^9+5 a^7 z^7-17 a^7 z^5+23 a^7 z^3-11 a^7 z+2 a^7 z^{-1} +6 a^6 z^8-13 a^6 z^6+15 a^6 z^4-7 a^6 z^2+a^6+a^5 z^9+6 a^5 z^7-18 a^5 z^5+20 a^5 z^3-9 a^5 z+2 a^5 z^{-1} +3 a^4 z^8-2 a^4 z^6-3 a^4 z^4+a^4 z^2+4 a^3 z^7-7 a^3 z^5+3 a^3 z^3-3 a^3 z+a^3 z^{-1} +3 a^2 z^6-6 a^2 z^4+2 a^2 z^2+a z^5-2 a z^3+a z$ (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       53  -2
-6      63   3
-8     55    0
-10    56     -1
-12   35      2
-14  25       -3
-16 14        3
-18 1         -1
-201          1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-4$ $i=-2$ $r=-8$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=-5$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-4$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-3$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-2$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=-1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=0$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{4}$ $r=1$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=2$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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