# L10a52

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a52 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{t(1)^2 t(2)^4-2 t(1) t(2)^4+t(2)^4-3 t(1)^2 t(2)^3+6 t(1) t(2)^3-3 t(2)^3+4 t(1)^2 t(2)^2-7 t(1) t(2)^2+4 t(2)^2-3 t(1)^2 t(2)+6 t(1) t(2)-3 t(2)+t(1)^2-2 t(1)+1}{t(1) t(2)^2}$ (db) Jones polynomial $-4 q^{9/2}+\frac{1}{q^{9/2}}+8 q^{7/2}-\frac{4}{q^{7/2}}-12 q^{5/2}+\frac{7}{q^{5/2}}+15 q^{3/2}-\frac{12}{q^{3/2}}+q^{11/2}-16 \sqrt{q}+\frac{14}{\sqrt{q}}$ (db) Signature 1 (db) HOMFLY-PT polynomial $-z^7 a^{-1} +2 a z^5-4 z^5 a^{-1} +z^5 a^{-3} -a^3 z^3+5 a z^3-6 z^3 a^{-1} +2 z^3 a^{-3} -a^3 z+2 a z-3 z a^{-1} +z a^{-3} +a^3 z^{-1} -a z^{-1}$ (db) Kauffman polynomial $-2 a z^9-2 z^9 a^{-1} -5 a^2 z^8-7 z^8 a^{-2} -12 z^8-4 a^3 z^7-9 a z^7-15 z^7 a^{-1} -10 z^7 a^{-3} -a^4 z^6+10 a^2 z^6+2 z^6 a^{-2} -8 z^6 a^{-4} +21 z^6+11 a^3 z^5+34 a z^5+40 z^5 a^{-1} +13 z^5 a^{-3} -4 z^5 a^{-5} +2 a^4 z^4-2 a^2 z^4+11 z^4 a^{-2} +8 z^4 a^{-4} -z^4 a^{-6} -2 z^4-9 a^3 z^3-27 a z^3-27 z^3 a^{-1} -7 z^3 a^{-3} +2 z^3 a^{-5} -a^4 z^2-2 a^2 z^2-6 z^2 a^{-2} -3 z^2 a^{-4} -4 z^2+a^3 z+5 a z+6 z a^{-1} +2 z a^{-3} -a^2+a^3 z^{-1} +a z^{-1}$ (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 \
-5-4-3-2-1012345χ
12          1-1
10         3 3
8        51 -4
6       73  4
4      85   -3
2     87    1
0    79     2
-2   57      -2
-4  38       5
-6 14        -3
-8 3         3
-101          -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=0$ $i=2$ $r=-5$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-2$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{5}$ $r=-1$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=0$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{8}$ $r=1$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=2$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=4$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=5$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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