# L11a364

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a364 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{t(2)^4 t(1)^4-t(2)^3 t(1)^4-t(2)^4 t(1)^3+3 t(2)^3 t(1)^3-2 t(2)^2 t(1)^3-2 t(2)^3 t(1)^2+3 t(2)^2 t(1)^2-2 t(2) t(1)^2-2 t(2)^2 t(1)+3 t(2) t(1)-t(1)-t(2)+1}{t(1)^2 t(2)^2}$ (db) Jones polynomial $q^{17/2}-2 q^{15/2}+3 q^{13/2}-5 q^{11/2}+6 q^{9/2}-7 q^{7/2}+6 q^{5/2}-6 q^{3/2}+4 \sqrt{q}-\frac{3}{\sqrt{q}}+\frac{2}{q^{3/2}}-\frac{1}{q^{5/2}}$ (db) Signature 3 (db) HOMFLY-PT polynomial $z^7 a^{-5} +6 z^5 a^{-5} +11 z^3 a^{-5} +6 z a^{-5} -z^9 a^{-3} -8 z^7 a^{-3} -23 z^5 a^{-3} -28 z^3 a^{-3} -12 z a^{-3} - a^{-3} z^{-1} +z^7 a^{-1} +6 z^5 a^{-1} +11 z^3 a^{-1} +7 z a^{-1} + a^{-1} z^{-1}$ (db) Kauffman polynomial $-z^{10} a^{-2} -z^{10} a^{-4} -2 z^9 a^{-1} -4 z^9 a^{-3} -2 z^9 a^{-5} +3 z^8 a^{-2} +3 z^8 a^{-4} -2 z^8 a^{-6} -2 z^8-a z^7+10 z^7 a^{-1} +22 z^7 a^{-3} +9 z^7 a^{-5} -2 z^7 a^{-7} +z^6 a^{-2} -z^6 a^{-4} +6 z^6 a^{-6} -2 z^6 a^{-8} +10 z^6+5 a z^5-16 z^5 a^{-1} -45 z^5 a^{-3} -18 z^5 a^{-5} +4 z^5 a^{-7} -2 z^5 a^{-9} -6 z^4 a^{-2} -5 z^4 a^{-4} -8 z^4 a^{-6} +3 z^4 a^{-8} -z^4 a^{-10} -13 z^4-6 a z^3+14 z^3 a^{-1} +43 z^3 a^{-3} +15 z^3 a^{-5} -4 z^3 a^{-7} +4 z^3 a^{-9} +5 z^2 a^{-2} +5 z^2 a^{-4} +z^2 a^{-6} -z^2 a^{-8} +2 z^2 a^{-10} +4 z^2+a z-7 z a^{-1} -14 z a^{-3} -5 z a^{-5} -z a^{-9} - a^{-2} + a^{-1} z^{-1} + a^{-3} z^{-1}$ (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 \
-4-3-2-101234567χ
18           1-1
16          1 1
14         21 -1
12        31  2
10       32   -1
8      43    1
6     34     1
4    33      0
2   24       2
0  12        -1
-2 12         1
-4 1          -1
-61           1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=2$ $i=4$ $r=-4$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=0$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}$ $r=1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{4}$ $r=3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=4$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=5$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=6$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=7$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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