# L11a35

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a35 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{t(2)^5+2 t(1) t(2)^4-6 t(2)^4-7 t(1) t(2)^3+9 t(2)^3+9 t(1) t(2)^2-7 t(2)^2-6 t(1) t(2)+2 t(2)+t(1)}{\sqrt{t(1)} t(2)^{5/2}}$ (db) Jones polynomial $-3 q^{9/2}+\frac{3}{q^{9/2}}+6 q^{7/2}-\frac{7}{q^{7/2}}-10 q^{5/2}+\frac{10}{q^{5/2}}+14 q^{3/2}-\frac{14}{q^{3/2}}+q^{11/2}-\frac{1}{q^{11/2}}-16 \sqrt{q}+\frac{15}{\sqrt{q}}$ (db) Signature 1 (db) HOMFLY-PT polynomial $a^5 z+a^5 z^{-1} +z a^{-5} -3 a^3 z^3-2 z^3 a^{-3} -5 a^3 z-2 a^3 z^{-1} -z a^{-3} + a^{-3} z^{-1} +2 a z^5+z^5 a^{-1} +5 a z^3-z^3 a^{-1} +6 a z+3 a z^{-1} -4 z a^{-1} -3 a^{-1} z^{-1}$ (db) Kauffman polynomial $z^4 a^{-6} -z^2 a^{-6} +a^5 z^7-4 a^5 z^5+3 z^5 a^{-5} +6 a^5 z^3-3 z^3 a^{-5} -4 a^5 z+z a^{-5} +a^5 z^{-1} +3 a^4 z^8-11 a^4 z^6+5 z^6 a^{-4} +12 a^4 z^4-5 z^4 a^{-4} -4 a^4 z^2+3 z^2 a^{-4} - a^{-4} +3 a^3 z^9-5 a^3 z^7+6 z^7 a^{-3} -11 a^3 z^5-5 z^5 a^{-3} +22 a^3 z^3+z^3 a^{-3} -11 a^3 z-z a^{-3} +2 a^3 z^{-1} + a^{-3} z^{-1} +a^2 z^{10}+8 a^2 z^8+6 z^8 a^{-2} -36 a^2 z^6-5 z^6 a^{-2} +38 a^2 z^4-5 z^4 a^{-2} -14 a^2 z^2+8 z^2 a^{-2} +2 a^2-2 a^{-2} +7 a z^9+4 z^9 a^{-1} -11 a z^7+z^7 a^{-1} -18 a z^5-19 z^5 a^{-1} +35 a z^3+23 z^3 a^{-1} -18 a z-13 z a^{-1} +3 a z^{-1} +3 a^{-1} z^{-1} +z^{10}+11 z^8-35 z^6+27 z^4-6 z^2$ (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
12           1-1
10          2 2
8         41 -3
6        62  4
4       84   -4
2      86    2
0     89     1
-2    67      -1
-4   48       4
-6  36        -3
-8 15         4
-10 2          -2
-121           1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=0$ $i=2$ $r=-6$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}$ $r=-3$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-2$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=-1$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=0$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{8}$ $r=1$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=4$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=5$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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