# L11a103

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a103 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{3 (t(1)-1) (t(2)-1) \left(t(2)^2-t(2)+1\right)}{\sqrt{t(1)} t(2)^{3/2}}$ (db) Jones polynomial $-q^{9/2}+3 q^{7/2}-6 q^{5/2}+7 q^{3/2}-10 \sqrt{q}+\frac{11}{\sqrt{q}}-\frac{10}{q^{3/2}}+\frac{9}{q^{5/2}}-\frac{7}{q^{7/2}}+\frac{4}{q^{9/2}}-\frac{3}{q^{11/2}}+\frac{1}{q^{13/2}}$ (db) Signature 1 (db) HOMFLY-PT polynomial $-z^3 a^5-z a^5+a^5 z^{-1} +z^5 a^3+2 z^3 a^3-2 a^3 z^{-1} +z^5 a+z^3 a+a z^{-1} +z^5 a^{-1} +2 z^3 a^{-1} +2 z a^{-1} + a^{-1} z^{-1} -z^3 a^{-3} -z a^{-3} - a^{-3} z^{-1}$ (db) Kauffman polynomial $-2 a^4 z^{10}-2 a^2 z^{10}-3 a^5 z^9-8 a^3 z^9-5 a z^9-a^6 z^8+7 a^4 z^8+a^2 z^8-7 z^8+17 a^5 z^7+38 a^3 z^7+14 a z^7-7 z^7 a^{-1} +5 a^6 z^6+20 a^2 z^6-7 z^6 a^{-2} +18 z^6-30 a^5 z^5-51 a^3 z^5-5 a z^5+10 z^5 a^{-1} -6 z^5 a^{-3} -7 a^6 z^4-13 a^4 z^4-29 a^2 z^4+8 z^4 a^{-2} -3 z^4 a^{-4} -12 z^4+17 a^5 z^3+15 a^3 z^3-6 a z^3+3 z^3 a^{-1} +6 z^3 a^{-3} -z^3 a^{-5} +3 a^6 z^2+6 a^4 z^2+10 a^2 z^2+7 z^2+a^5 z+7 a^3 z+6 a z-3 z a^{-1} -3 z a^{-3} -a^4-3 a^2- a^{-2} -2-a^5 z^{-1} -2 a^3 z^{-1} -a z^{-1} + a^{-1} z^{-1} + a^{-3} z^{-1}$ (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 \
-7-6-5-4-3-2-101234χ
10           11
8          2 -2
6         41 3
4        32  -1
2       74   3
0      65    -1
-2     45     -1
-4    56      1
-6   24       -2
-8  25        3
-10 12         -1
-12 2          2
-141           -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=0$ $i=2$ $r=-7$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-4$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-3$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-2$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-1$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=0$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{7}$ $r=1$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=3$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=4$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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