# L11n246

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n246 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{(u+v) \left(u^2 v^2-2 u^2 v+u^2-2 u v^2+5 u v-2 u+v^2-2 v+1\right)}{u^{3/2} v^{3/2}}$ (db) Jones polynomial $-q^{3/2}+4 \sqrt{q}-\frac{8}{\sqrt{q}}+\frac{10}{q^{3/2}}-\frac{12}{q^{5/2}}+\frac{11}{q^{7/2}}-\frac{10}{q^{9/2}}+\frac{7}{q^{11/2}}-\frac{4}{q^{13/2}}+\frac{1}{q^{15/2}}$ (db) Signature -1 (db) HOMFLY-PT polynomial $a^7 (-z)+3 a^5 z^3+4 a^5 z+a^5 z^{-1} -2 a^3 z^5-6 a^3 z^3-7 a^3 z-a^3 z^{-1} +3 a z^3+3 a z-z a^{-1}$ (db) Kauffman polynomial $-2 a^5 z^9-2 a^3 z^9-5 a^6 z^8-9 a^4 z^8-4 a^2 z^8-4 a^7 z^7-4 a^5 z^7-2 a^3 z^7-2 a z^7-a^8 z^6+12 a^6 z^6+23 a^4 z^6+10 a^2 z^6+11 a^7 z^5+24 a^5 z^5+15 a^3 z^5+2 a z^5+2 a^8 z^4-3 a^6 z^4-15 a^4 z^4-14 a^2 z^4-4 z^4-7 a^7 z^3-20 a^5 z^3-19 a^3 z^3-7 a z^3-z^3 a^{-1} -a^8 z^2-a^6 z^2+2 a^4 z^2+4 a^2 z^2+2 z^2+a^7 z+7 a^5 z+8 a^3 z+3 a z+z a^{-1} +a^4-a^5 z^{-1} -a^3 z^{-1}$ (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 \
-7-6-5-4-3-2-1012χ
4         11
2        3 -3
0       51 4
-2      64  -2
-4     64   2
-6    56    1
-8   56     -1
-10  36      3
-12 14       -3
-14 3        3
-161         -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $r=-7$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-4$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{5}$ $r=-3$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-2$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=-1$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=0$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{5}$ $r=1$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=2$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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