# L11a400

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a400 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{(u-1) (v-1)^2 (w-1)^2 \left(w^2+1\right)}{\sqrt{u} v w^2}$ (db) Jones polynomial $q^7-4 q^6+8 q^5-14 q^4+18 q^3-20 q^2+21 q-16+14 q^{-1} -7 q^{-2} +4 q^{-3} - q^{-4}$ (db) Signature 2 (db) HOMFLY-PT polynomial $-z^8 a^{-2} -5 z^6 a^{-2} +z^6 a^{-4} +2 z^6-a^2 z^4-9 z^4 a^{-2} +3 z^4 a^{-4} +7 z^4-2 a^2 z^2-5 z^2 a^{-2} +2 z^2 a^{-4} +5 z^2+a^2+4 a^{-2} - a^{-4} -4+2 a^2 z^{-2} +4 a^{-2} z^{-2} - a^{-4} z^{-2} -5 z^{-2}$ (db) Kauffman polynomial $2 z^{10} a^{-2} +2 z^{10}+5 a z^9+13 z^9 a^{-1} +8 z^9 a^{-3} +4 a^2 z^8+18 z^8 a^{-2} +13 z^8 a^{-4} +9 z^8+a^3 z^7-14 a z^7-32 z^7 a^{-1} -5 z^7 a^{-3} +12 z^7 a^{-5} -15 a^2 z^6-67 z^6 a^{-2} -24 z^6 a^{-4} +8 z^6 a^{-6} -50 z^6-3 a^3 z^5+7 a z^5+11 z^5 a^{-1} -19 z^5 a^{-3} -16 z^5 a^{-5} +4 z^5 a^{-7} +18 a^2 z^4+68 z^4 a^{-2} +18 z^4 a^{-4} -6 z^4 a^{-6} +z^4 a^{-8} +61 z^4+2 a^3 z^3+a z^3+6 z^3 a^{-1} +15 z^3 a^{-3} +6 z^3 a^{-5} -2 z^3 a^{-7} -7 a^2 z^2-24 z^2 a^{-2} -8 z^2 a^{-4} -23 z^2+5 a z+9 z a^{-1} +5 z a^{-3} +z a^{-5} -3 a^2-2 a^{-2} -4-5 a z^{-1} -9 a^{-1} z^{-1} -5 a^{-3} z^{-1} - a^{-5} z^{-1} +2 a^2 z^{-2} +4 a^{-2} z^{-2} + a^{-4} z^{-2} +5 z^{-2}$ (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 \
-5-4-3-2-10123456χ
15           11
13          3 -3
11         51 4
9        93  -6
7       95   4
5      119    -2
3     109     1
1    813      5
-1   68       -2
-3  310        7
-5 14         -3
-7 3          3
-91           -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=1$ $i=3$ $r=-5$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-2$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{6}$ $r=-1$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=0$ ${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{10}$ $r=1$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{11}$ ${\mathbb Z}^{11}$ $r=2$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=3$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=5$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=6$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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