# L11a105

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a105 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{t(1) t(2)^5-2 t(2)^5-4 t(1) t(2)^4+6 t(2)^4+6 t(1) t(2)^3-6 t(2)^3-6 t(1) t(2)^2+6 t(2)^2+6 t(1) t(2)-4 t(2)-2 t(1)+1}{\sqrt{t(1)} t(2)^{5/2}}$ (db) Jones polynomial $q^{11/2}-4 q^{9/2}+7 q^{7/2}-11 q^{5/2}+15 q^{3/2}-16 \sqrt{q}+\frac{15}{\sqrt{q}}-\frac{13}{q^{3/2}}+\frac{9}{q^{5/2}}-\frac{6}{q^{7/2}}+\frac{2}{q^{9/2}}-\frac{1}{q^{11/2}}$ (db) Signature 1 (db) HOMFLY-PT polynomial $a^5 z+2 a^5 z^{-1} +z^5 a^{-3} -3 a^3 z^3+2 z^3 a^{-3} -8 a^3 z-4 a^3 z^{-1} -z^7 a^{-1} +3 a z^5-4 z^5 a^{-1} +10 a z^3-6 z^3 a^{-1} +9 a z-4 z a^{-1} +3 a z^{-1} - a^{-1} z^{-1}$ (db) Kauffman polynomial $-a^2 z^{10}-z^{10}-2 a^3 z^9-6 a z^9-4 z^9 a^{-1} -2 a^4 z^8-4 a^2 z^8-7 z^8 a^{-2} -9 z^8-a^5 z^7+2 a^3 z^7+9 a z^7-2 z^7 a^{-1} -8 z^7 a^{-3} +7 a^4 z^6+18 a^2 z^6+5 z^6 a^{-2} -7 z^6 a^{-4} +23 z^6+5 a^5 z^5+13 a^3 z^5+13 a z^5+16 z^5 a^{-1} +7 z^5 a^{-3} -4 z^5 a^{-5} -6 a^4 z^4-12 a^2 z^4+7 z^4 a^{-2} +7 z^4 a^{-4} -z^4 a^{-6} -7 z^4-9 a^5 z^3-25 a^3 z^3-25 a z^3-11 z^3 a^{-1} +z^3 a^{-3} +3 z^3 a^{-5} -a^4 z^2-3 a^2 z^2-7 z^2 a^{-2} -2 z^2 a^{-4} -7 z^2+7 a^5 z+16 a^3 z+13 a z+3 z a^{-1} -z a^{-3} +2 a^4+3 a^2+ a^{-2} +3-2 a^5 z^{-1} -4 a^3 z^{-1} -3 a z^{-1} - a^{-1} z^{-1}$ (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         41 -3
6        73  4
4       84   -4
2      87    1
0     89     1
-2    57      -2
-4   48       4
-6  25        -3
-8 15         4
-10 1          -1
-121           1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=0$ $i=2$ $r=-6$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=-3$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-2$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-1$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=0$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{8}$ $r=1$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=4$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=5$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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