# L11a21

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a21 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{(t(1)-1) (t(2)-1) \left(t(2)^2+1\right) \left(2 t(2)^2-3 t(2)+2\right)}{\sqrt{t(1)} t(2)^{5/2}}$ (db) Jones polynomial $q^{23/2}-4 q^{21/2}+9 q^{19/2}-13 q^{17/2}+17 q^{15/2}-18 q^{13/2}+17 q^{11/2}-15 q^{9/2}+9 q^{7/2}-6 q^{5/2}+2 q^{3/2}-\sqrt{q}$ (db) Signature 5 (db) HOMFLY-PT polynomial $-z^7 a^{-5} -z^7 a^{-7} +z^5 a^{-3} -4 z^5 a^{-5} -3 z^5 a^{-7} +z^5 a^{-9} +4 z^3 a^{-3} -6 z^3 a^{-5} -2 z^3 a^{-7} +2 z^3 a^{-9} +5 z a^{-3} -5 z a^{-5} -z a^{-7} +z a^{-9} +2 a^{-3} z^{-1} -2 a^{-5} z^{-1} - a^{-7} z^{-1} + a^{-9} z^{-1}$ (db) Kauffman polynomial $z^4 a^{-14} +4 z^5 a^{-13} -z^3 a^{-13} +9 z^6 a^{-12} -9 z^4 a^{-12} +4 z^2 a^{-12} - a^{-12} +12 z^7 a^{-11} -16 z^5 a^{-11} +7 z^3 a^{-11} -z a^{-11} +10 z^8 a^{-10} -10 z^6 a^{-10} -4 z^4 a^{-10} +4 z^2 a^{-10} +5 z^9 a^{-9} +4 z^7 a^{-9} -22 z^5 a^{-9} +12 z^3 a^{-9} -2 z a^{-9} - a^{-9} z^{-1} +z^{10} a^{-8} +13 z^8 a^{-8} -33 z^6 a^{-8} +19 z^4 a^{-8} -6 z^2 a^{-8} +3 a^{-8} +7 z^9 a^{-7} -11 z^7 a^{-7} -8 z^5 a^{-7} +15 z^3 a^{-7} -4 z a^{-7} - a^{-7} z^{-1} +z^{10} a^{-6} +5 z^8 a^{-6} -21 z^6 a^{-6} +19 z^4 a^{-6} -4 z^2 a^{-6} +2 z^9 a^{-5} -2 z^7 a^{-5} -11 z^5 a^{-5} +20 z^3 a^{-5} -10 z a^{-5} +2 a^{-5} z^{-1} +2 z^8 a^{-4} -7 z^6 a^{-4} +6 z^4 a^{-4} +2 z^2 a^{-4} -3 a^{-4} +z^7 a^{-3} -5 z^5 a^{-3} +9 z^3 a^{-3} -7 z a^{-3} +2 a^{-3} z^{-1}$ (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 \
-2-10123456789χ
24           1-1
22          3 3
20         61 -5
18        73  4
16       106   -4
14      87    1
12     910     1
10    68      -2
8   39       6
6  36        -3
4 15         4
2 1          -1
01           1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=4$ $i=6$ $r=-2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=0$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{3}$ $r=1$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=2$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=3$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=4$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=5$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{10}$ $r=6$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=7$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=8$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=9$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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