# L11a377

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a377 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{t(2)^4 t(1)^4-t(2)^3 t(1)^4-2 t(2)^4 t(1)^3+5 t(2)^3 t(1)^3-4 t(2)^2 t(1)^3+t(2) t(1)^3+t(2)^4 t(1)^2-5 t(2)^3 t(1)^2+7 t(2)^2 t(1)^2-5 t(2) t(1)^2+t(1)^2+t(2)^3 t(1)-4 t(2)^2 t(1)+5 t(2) t(1)-2 t(1)-t(2)+1}{t(1)^2 t(2)^2}$ (db) Jones polynomial $q^{17/2}-3 q^{15/2}+6 q^{13/2}-10 q^{11/2}+13 q^{9/2}-15 q^{7/2}+14 q^{5/2}-13 q^{3/2}+9 \sqrt{q}-\frac{6}{\sqrt{q}}+\frac{3}{q^{3/2}}-\frac{1}{q^{5/2}}$ (db) Signature 3 (db) HOMFLY-PT polynomial $-z^9 a^{-3} +z^7 a^{-1} -7 z^7 a^{-3} +z^7 a^{-5} +5 z^5 a^{-1} -18 z^5 a^{-3} +5 z^5 a^{-5} +8 z^3 a^{-1} -20 z^3 a^{-3} +8 z^3 a^{-5} +5 z a^{-1} -8 z a^{-3} +4 z a^{-5} + a^{-1} z^{-1} - a^{-3} z^{-1}$ (db) Kauffman polynomial $z^4 a^{-10} -z^2 a^{-10} +3 z^5 a^{-9} -3 z^3 a^{-9} +z a^{-9} +5 z^6 a^{-8} -5 z^4 a^{-8} +2 z^2 a^{-8} +6 z^7 a^{-7} -6 z^5 a^{-7} +z^3 a^{-7} +z a^{-7} +6 z^8 a^{-6} -8 z^6 a^{-6} +3 z^4 a^{-6} +5 z^9 a^{-5} -10 z^7 a^{-5} +11 z^5 a^{-5} -11 z^3 a^{-5} +4 z a^{-5} +2 z^{10} a^{-4} +2 z^8 a^{-4} -16 z^6 a^{-4} +16 z^4 a^{-4} -6 z^2 a^{-4} +9 z^9 a^{-3} -32 z^7 a^{-3} +40 z^5 a^{-3} -27 z^3 a^{-3} +9 z a^{-3} - a^{-3} z^{-1} +2 z^{10} a^{-2} -z^8 a^{-2} -15 z^6 a^{-2} +20 z^4 a^{-2} -7 z^2 a^{-2} + a^{-2} +4 z^9 a^{-1} +a z^7-15 z^7 a^{-1} -4 a z^5+16 z^5 a^{-1} +4 a z^3-8 z^3 a^{-1} -a z+4 z a^{-1} - a^{-1} z^{-1} +3 z^8-12 z^6+13 z^4-4 z^2$ (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 \
-4-3-2-101234567χ
18           1-1
16          2 2
14         41 -3
12        62  4
10       74   -3
8      86    2
6     78     1
4    67      -1
2   48       4
0  25        -3
-2 14         3
-4 2          -2
-61           1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=2$ $i=4$ $r=-4$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-1$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=0$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{6}$ $r=1$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=2$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{8}$ $r=3$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=4$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=5$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=6$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=7$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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