# L11n94

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n94 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{(u-1) (v-1) \left(v^2+1\right)}{\sqrt{u} v^{3/2}}$ (db) Jones polynomial $q^{9/2}-2 q^{7/2}+2 q^{5/2}-\frac{1}{q^{5/2}}-4 q^{3/2}+\frac{2}{q^{3/2}}+q^{15/2}-q^{13/2}+3 \sqrt{q}-\frac{3}{\sqrt{q}}$ (db) Signature 3 (db) HOMFLY-PT polynomial $z a^{-7} + a^{-7} z^{-1} -z^3 a^{-5} -4 z a^{-5} -3 a^{-5} z^{-1} +2 z^3 a^{-3} +6 z a^{-3} +3 a^{-3} z^{-1} -z^5 a^{-1} +a z^3-4 z^3 a^{-1} +2 a z-5 z a^{-1} +a z^{-1} -2 a^{-1} z^{-1}$ (db) Kauffman polynomial $-z^9 a^{-1} -z^9 a^{-3} -4 z^8 a^{-2} -2 z^8 a^{-4} -2 z^8-a z^7+2 z^7 a^{-1} +3 z^7 a^{-3} -z^7 a^{-5} -z^7 a^{-7} +21 z^6 a^{-2} +12 z^6 a^{-4} -z^6 a^{-8} +10 z^6+5 a z^5+9 z^5 a^{-1} +6 z^5 a^{-3} +8 z^5 a^{-5} +6 z^5 a^{-7} -29 z^4 a^{-2} -18 z^4 a^{-4} +3 z^4 a^{-6} +5 z^4 a^{-8} -13 z^4-7 a z^3-19 z^3 a^{-1} -18 z^3 a^{-3} -14 z^3 a^{-5} -8 z^3 a^{-7} +13 z^2 a^{-2} +8 z^2 a^{-4} -5 z^2 a^{-6} -5 z^2 a^{-8} +5 z^2+4 a z+11 z a^{-1} +14 z a^{-3} +10 z a^{-5} +3 z a^{-7} -2 a^{-2} +2 a^{-6} + a^{-8} -a z^{-1} -2 a^{-1} z^{-1} -3 a^{-3} z^{-1} -3 a^{-5} z^{-1} - a^{-7} z^{-1}$ (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 \
-4-3-2-101234567χ
16           1-1
14            0
12        111 1
10       21   -1
8      221   1
6     231    0
4    321     2
2   241      1
0  121       0
-2 12         1
-4 1          -1
-61           1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=0$ $i=2$ $i=4$ $r=-4$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}$ $r=1$ ${\mathbb Z}$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=2$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=3$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=4$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=5$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=6$ ${\mathbb Z}$ $r=7$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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