# L11a111

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a111 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{(u-1) (v-1) \left(2 v^4-v^3+2 v^2-v+2\right)}{\sqrt{u} v^{5/2}}$ (db) Jones polynomial $q^{23/2}-2 q^{21/2}+4 q^{19/2}-6 q^{17/2}+9 q^{15/2}-10 q^{13/2}+9 q^{11/2}-9 q^{9/2}+6 q^{7/2}-5 q^{5/2}+2 q^{3/2}-\sqrt{q}$ (db) Signature 5 (db) HOMFLY-PT polynomial $z^5 a^{-9} +4 z^3 a^{-9} +4 z a^{-9} +2 a^{-9} z^{-1} -z^7 a^{-7} -5 z^5 a^{-7} -9 z^3 a^{-7} -9 z a^{-7} -4 a^{-7} z^{-1} -z^7 a^{-5} -4 z^5 a^{-5} -3 z^3 a^{-5} +z a^{-5} + a^{-5} z^{-1} +z^5 a^{-3} +4 z^3 a^{-3} +4 z a^{-3} + a^{-3} z^{-1}$ (db) Kauffman polynomial $-z^{10} a^{-6} -z^{10} a^{-8} -2 z^9 a^{-5} -5 z^9 a^{-7} -3 z^9 a^{-9} -2 z^8 a^{-4} -z^8 a^{-8} -3 z^8 a^{-10} -z^7 a^{-3} +6 z^7 a^{-5} +21 z^7 a^{-7} +11 z^7 a^{-9} -3 z^7 a^{-11} +8 z^6 a^{-4} +11 z^6 a^{-6} +12 z^6 a^{-8} +6 z^6 a^{-10} -3 z^6 a^{-12} +5 z^5 a^{-3} -z^5 a^{-5} -35 z^5 a^{-7} -22 z^5 a^{-9} +5 z^5 a^{-11} -2 z^5 a^{-13} -7 z^4 a^{-4} -18 z^4 a^{-6} -22 z^4 a^{-8} -4 z^4 a^{-10} +6 z^4 a^{-12} -z^4 a^{-14} -8 z^3 a^{-3} -2 z^3 a^{-5} +35 z^3 a^{-7} +22 z^3 a^{-9} -4 z^3 a^{-11} +3 z^3 a^{-13} -z^2 a^{-4} +15 z^2 a^{-6} +20 z^2 a^{-8} -4 z^2 a^{-10} -6 z^2 a^{-12} +2 z^2 a^{-14} +5 z a^{-3} -3 z a^{-5} -17 z a^{-7} -10 z a^{-9} -z a^{-11} + a^{-4} -5 a^{-6} -6 a^{-8} + a^{-10} +2 a^{-12} - a^{-3} z^{-1} + a^{-5} z^{-1} +4 a^{-7} z^{-1} +2 a^{-9} z^{-1}$ (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 \
-2-10123456789χ
24           1-1
22          1 1
20         31 -2
18        31  2
16       63   -3
14      43    1
12     56     1
10    44      0
8   25       3
6  34        -1
4 14         3
2 1          -1
01           1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=4$ $i=6$ $r=-2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=0$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{3}$ $r=1$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=2$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=3$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=4$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=5$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=6$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=7$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=8$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=9$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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