# L11n8

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n8 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{(u-1) (v-1)^3}{\sqrt{u} v^{3/2}}$ (db) Jones polynomial $-q^{3/2}+2 \sqrt{q}-\frac{4}{\sqrt{q}}+\frac{4}{q^{3/2}}-\frac{6}{q^{5/2}}+\frac{5}{q^{7/2}}-\frac{4}{q^{9/2}}+\frac{3}{q^{11/2}}-\frac{2}{q^{13/2}}+\frac{1}{q^{15/2}}$ (db) Signature -1 (db) HOMFLY-PT polynomial $a^7 (-z)-a^7 z^{-1} +2 a^5 z^3+5 a^5 z+3 a^5 z^{-1} -a^3 z^5-4 a^3 z^3-7 a^3 z-3 a^3 z^{-1} +2 a z^3+4 a z+2 a z^{-1} -z a^{-1} - a^{-1} z^{-1}$ (db) Kauffman polynomial $-a^5 z^9-a^3 z^9-2 a^6 z^8-4 a^4 z^8-2 a^2 z^8-2 a^7 z^7+a^5 z^7+2 a^3 z^7-a z^7-a^8 z^6+7 a^6 z^6+18 a^4 z^6+10 a^2 z^6+8 a^7 z^5+10 a^5 z^5+6 a^3 z^5+4 a z^5+4 a^8 z^4-3 a^6 z^4-24 a^4 z^4-19 a^2 z^4-2 z^4-7 a^7 z^3-19 a^5 z^3-21 a^3 z^3-10 a z^3-z^3 a^{-1} -3 a^8 z^2-3 a^6 z^2+8 a^4 z^2+11 a^2 z^2+3 z^2+3 a^7 z+12 a^5 z+16 a^3 z+9 a z+2 z a^{-1} +a^8+2 a^6-2 a^2-a^7 z^{-1} -3 a^5 z^{-1} -3 a^3 z^{-1} -2 a z^{-1} - a^{-1} z^{-1}$ (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 \
-7-6-5-4-3-2-1012χ
4         11
2        1 -1
0       31 2
-2      33  0
-4     321  2
-6    34    1
-8   23     -1
-10  131     1
-12 12       -1
-14 1        1
-161         -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-4$ $i=-2$ $i=0$ $r=-7$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-3$ ${\mathbb Z}$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-2$ ${\mathbb Z}_2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}$ $r=1$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=2$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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