# L11a44

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a44 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{(t(1)-1) (t(2)-1) \left(t(2)^4-4 t(2)^3+8 t(2)^2-4 t(2)+1\right)}{\sqrt{t(1)} t(2)^{5/2}}$ (db) Jones polynomial $q^{11/2}-4 q^{9/2}+10 q^{7/2}-17 q^{5/2}+20 q^{3/2}-24 \sqrt{q}+\frac{23}{\sqrt{q}}-\frac{19}{q^{3/2}}+\frac{14}{q^{5/2}}-\frac{8}{q^{7/2}}+\frac{3}{q^{9/2}}-\frac{1}{q^{11/2}}$ (db) Signature 1 (db) HOMFLY-PT polynomial $-z^7 a^{-1} +3 a z^5-4 z^5 a^{-1} +z^5 a^{-3} -3 a^3 z^3+8 a z^3-9 z^3 a^{-1} +2 z^3 a^{-3} +a^5 z-5 a^3 z+9 a z-8 z a^{-1} +3 z a^{-3} +a^5 z^{-1} -3 a^3 z^{-1} +4 a z^{-1} -2 a^{-1} z^{-1}$ (db) Kauffman polynomial $-2 a^2 z^{10}-2 z^{10}-4 a^3 z^9-13 a z^9-9 z^9 a^{-1} -3 a^4 z^8-9 a^2 z^8-17 z^8 a^{-2} -23 z^8-a^5 z^7+7 a^3 z^7+20 a z^7-5 z^7 a^{-1} -17 z^7 a^{-3} +10 a^4 z^6+41 a^2 z^6+26 z^6 a^{-2} -10 z^6 a^{-4} +67 z^6+4 a^5 z^5+8 a^3 z^5+24 a z^5+50 z^5 a^{-1} +26 z^5 a^{-3} -4 z^5 a^{-5} -12 a^4 z^4-42 a^2 z^4-8 z^4 a^{-2} +7 z^4 a^{-4} -z^4 a^{-6} -46 z^4-6 a^5 z^3-24 a^3 z^3-49 a z^3-49 z^3 a^{-1} -18 z^3 a^{-3} +6 a^4 z^2+14 a^2 z^2-z^2 a^{-2} -3 z^2 a^{-4} +10 z^2+4 a^5 z+16 a^3 z+26 a z+21 z a^{-1} +7 z a^{-3} -a^4-3 a^2- a^{-2} -2-a^5 z^{-1} -3 a^3 z^{-1} -4 a z^{-1} -2 a^{-1} z^{-1}$ (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 \
-6-5-4-3-2-1012345χ
12           1-1
10          3 3
8         71 -6
6        103  7
4       107   -3
2      1410    4
0     1112     1
-2    812      -4
-4   611       5
-6  28        -6
-8 16         5
-10 2          -2
-121           1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=0$ $i=2$ $r=-6$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-3$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=-2$ ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=-1$ ${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{11}$ ${\mathbb Z}^{11}$ $r=0$ ${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{12}$ ${\mathbb Z}^{14}$ $r=1$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{10}$ $r=2$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{10}$ $r=3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=4$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=5$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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