# L11a100

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a100 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{(u-1) (v-1) \left(v^2-v+1\right)^2 \left(v^2+v+1\right)}{\sqrt{u} v^{7/2}}$ (db) Jones polynomial $-q^{5/2}+3 q^{3/2}-5 \sqrt{q}+\frac{6}{\sqrt{q}}-\frac{10}{q^{3/2}}+\frac{10}{q^{5/2}}-\frac{11}{q^{7/2}}+\frac{10}{q^{9/2}}-\frac{7}{q^{11/2}}+\frac{5}{q^{13/2}}-\frac{3}{q^{15/2}}+\frac{1}{q^{17/2}}$ (db) Signature -3 (db) HOMFLY-PT polynomial $a^3 z^9-a^5 z^7+7 a^3 z^7-a z^7-5 a^5 z^5+17 a^3 z^5-5 a z^5-7 a^5 z^3+17 a^3 z^3-7 a z^3-3 a^5 z+7 a^3 z-4 a z-a^5 z^{-1} +3 a^3 z^{-1} -2 a z^{-1}$ (db) Kauffman polynomial $a^{10} z^4-a^{10} z^2+3 a^9 z^5-4 a^9 z^3+4 a^8 z^6-5 a^8 z^4+a^8 z^2+4 a^7 z^7-5 a^7 z^5+2 a^7 z^3+4 a^6 z^8-8 a^6 z^6+7 a^6 z^4-2 a^6 z^2+a^6+4 a^5 z^9-14 a^5 z^7+23 a^5 z^5-18 a^5 z^3+6 a^5 z-a^5 z^{-1} +2 a^4 z^{10}-4 a^4 z^8-3 a^4 z^6+11 a^4 z^4-8 a^4 z^2+3 a^4+8 a^3 z^9-38 a^3 z^7+63 a^3 z^5-46 a^3 z^3+13 a^3 z-3 a^3 z^{-1} +2 a^2 z^{10}-5 a^2 z^8-4 a^2 z^6+11 a^2 z^4-6 a^2 z^2+3 a^2+4 a z^9-19 a z^7+z^7 a^{-1} +28 a z^5-4 z^5 a^{-1} -19 a z^3+3 z^3 a^{-1} +7 a z-2 a z^{-1} +3 z^8-13 z^6+13 z^4-2 z^2$ (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 \
-7-6-5-4-3-2-101234χ
6           11
4          2 -2
2         31 2
0        32  -1
-2       73   4
-4      55    0
-6     65     1
-8    45      1
-10   36       -3
-12  24        2
-14 13         -2
-16 2          2
-181           -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-4$ $i=-2$ $r=-7$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-4$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-3$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-2$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=-1$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=0$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{7}$ $r=1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=3$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=4$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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