# L11a362

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a362 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{t(2)^3 t(1)^4-t(2)^2 t(1)^4+t(2)^4 t(1)^3-4 t(2)^3 t(1)^3+3 t(2)^2 t(1)^3-t(2) t(1)^3-t(2)^4 t(1)^2+3 t(2)^3 t(1)^2-5 t(2)^2 t(1)^2+3 t(2) t(1)^2-t(1)^2-t(2)^3 t(1)+3 t(2)^2 t(1)-4 t(2) t(1)+t(1)-t(2)^2+t(2)}{t(1)^2 t(2)^2}$ (db) Jones polynomial $q^{23/2}-3 q^{21/2}+5 q^{19/2}-8 q^{17/2}+10 q^{15/2}-11 q^{13/2}+10 q^{11/2}-9 q^{9/2}+6 q^{7/2}-4 q^{5/2}+2 q^{3/2}-\sqrt{q}$ (db) Signature 5 (db) HOMFLY-PT polynomial $-z^7 a^{-5} -z^7 a^{-7} +z^5 a^{-3} -4 z^5 a^{-5} -4 z^5 a^{-7} +z^5 a^{-9} +4 z^3 a^{-3} -3 z^3 a^{-5} -4 z^3 a^{-7} +3 z^3 a^{-9} +3 z a^{-3} +z a^{-5} -2 z a^{-7} +z a^{-9} + a^{-5} z^{-1} - a^{-7} z^{-1}$ (db) Kauffman polynomial $z^4 a^{-14} -z^2 a^{-14} +3 z^5 a^{-13} -4 z^3 a^{-13} +z a^{-13} +4 z^6 a^{-12} -4 z^4 a^{-12} +4 z^7 a^{-11} -3 z^5 a^{-11} -2 z^3 a^{-11} +2 z a^{-11} +4 z^8 a^{-10} -7 z^6 a^{-10} +6 z^4 a^{-10} -z^2 a^{-10} +3 z^9 a^{-9} -7 z^7 a^{-9} +9 z^5 a^{-9} -6 z^3 a^{-9} +z a^{-9} +z^{10} a^{-8} +2 z^8 a^{-8} -13 z^6 a^{-8} +16 z^4 a^{-8} -6 z^2 a^{-8} +5 z^9 a^{-7} -19 z^7 a^{-7} +25 z^5 a^{-7} -17 z^3 a^{-7} +6 z a^{-7} - a^{-7} z^{-1} +z^{10} a^{-6} -11 z^6 a^{-6} +16 z^4 a^{-6} -8 z^2 a^{-6} + a^{-6} +2 z^9 a^{-5} -7 z^7 a^{-5} +5 z^5 a^{-5} -2 z^3 a^{-5} +3 z a^{-5} - a^{-5} z^{-1} +2 z^8 a^{-4} -9 z^6 a^{-4} +11 z^4 a^{-4} -4 z^2 a^{-4} +z^7 a^{-3} -5 z^5 a^{-3} +7 z^3 a^{-3} -3 z a^{-3}$ (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          2 2
20         31 -2
18        52  3
16       64   -2
14      54    1
12     56     1
10    45      -1
8   25       3
6  24        -2
4 13         2
2 1          -1
01           1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=4$ $i=6$ $r=-2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=0$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=1$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=2$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=3$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=4$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=5$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=6$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{5}$ $r=7$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=8$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=9$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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