# L11a365

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a365 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{\left(t(2)^2 t(1)^2-t(2) t(1)^2+t(2) t(1)-t(1)+1\right) \left(t(1)^2 t(2)^2-t(1) t(2)^2+t(1) t(2)-t(2)+1\right)}{t(1)^2 t(2)^2}$ (db) Jones polynomial $q^{9/2}-2 q^{7/2}+3 q^{5/2}-5 q^{3/2}+6 \sqrt{q}-\frac{8}{\sqrt{q}}+\frac{7}{q^{3/2}}-\frac{7}{q^{5/2}}+\frac{5}{q^{7/2}}-\frac{3}{q^{9/2}}+\frac{2}{q^{11/2}}-\frac{1}{q^{13/2}}$ (db) Signature -1 (db) HOMFLY-PT polynomial $a^3 z^7+6 a^3 z^5+11 a^3 z^3+6 a^3 z+a^3 z^{-1} -a z^9-8 a z^7+z^7 a^{-1} -23 a z^5+6 z^5 a^{-1} -28 a z^3+11 z^3 a^{-1} -13 a z+6 z a^{-1} -a z^{-1}$ (db) Kauffman polynomial $a^7 z^5-3 a^7 z^3+a^7 z+2 a^6 z^6-6 a^6 z^4+3 a^6 z^2+2 a^5 z^7-5 a^5 z^5+2 a^5 z^3+2 a^4 z^8-6 a^4 z^6+z^6 a^{-4} +6 a^4 z^4-4 z^4 a^{-4} -a^4 z^2+3 z^2 a^{-4} +2 a^3 z^9-9 a^3 z^7+2 z^7 a^{-3} +18 a^3 z^5-8 z^5 a^{-3} -14 a^3 z^3+7 z^3 a^{-3} +6 a^3 z-z a^{-3} -a^3 z^{-1} +a^2 z^{10}-3 a^2 z^8+2 z^8 a^{-2} +a^2 z^6-7 z^6 a^{-2} +9 a^2 z^4+5 z^4 a^{-2} -6 a^2 z^2-z^2 a^{-2} +a^2+4 a z^9+2 z^9 a^{-1} -22 a z^7-9 z^7 a^{-1} +48 a z^5+16 z^5 a^{-1} -42 a z^3-16 z^3 a^{-1} +14 a z+6 z a^{-1} -a z^{-1} +z^{10}-3 z^8+z^6+6 z^4-6 z^2$ (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 \
-6-5-4-3-2-1012345χ
10           1-1
8          1 1
6         21 -1
4        31  2
2       32   -1
0      53    2
-2     34     1
-4    44      0
-6   24       2
-8  13        -2
-10 12         1
-12 1          -1
-141           1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $r=-6$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{4}$ $r=-1$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=0$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{5}$ $r=1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=3$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=4$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=5$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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