# L11a280

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a280 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{(t(1)-1) (t(2)-1) \left(2 t(2)^2 t(1)^2-t(2) t(1)^2-t(2)^2 t(1)+3 t(2) t(1)-t(1)-t(2)+2\right)}{t(1)^{3/2} t(2)^{3/2}}$ (db) Jones polynomial $-q^{11/2}+3 q^{9/2}-5 q^{7/2}+8 q^{5/2}-12 q^{3/2}+13 \sqrt{q}-\frac{14}{\sqrt{q}}+\frac{12}{q^{3/2}}-\frac{10}{q^{5/2}}+\frac{6}{q^{7/2}}-\frac{3}{q^{9/2}}+\frac{1}{q^{11/2}}$ (db) Signature -1 (db) HOMFLY-PT polynomial $-a^3 z^5-z^5 a^{-3} -3 a^3 z^3-3 z^3 a^{-3} -2 a^3 z-z a^{-3} +a z^7+z^7 a^{-1} +4 a z^5+4 z^5 a^{-1} +5 a z^3+4 z^3 a^{-1} +3 a z+a z^{-1} - a^{-1} z^{-1}$ (db) Kauffman polynomial $a^6 z^4-a^6 z^2+z^7 a^{-5} +3 a^5 z^5-4 z^5 a^{-5} -3 a^5 z^3+4 z^3 a^{-5} +a^5 z-z a^{-5} +3 z^8 a^{-4} +5 a^4 z^6-13 z^6 a^{-4} -5 a^4 z^4+16 z^4 a^{-4} +2 a^4 z^2-5 z^2 a^{-4} +3 z^9 a^{-3} +6 a^3 z^7-10 z^7 a^{-3} -7 a^3 z^5+7 z^5 a^{-3} +4 a^3 z^3+z^3 a^{-3} -a^3 z-z a^{-3} +z^{10} a^{-2} +5 a^2 z^8+4 z^8 a^{-2} -4 a^2 z^6-24 z^6 a^{-2} -2 a^2 z^4+28 z^4 a^{-2} +2 a^2 z^2-9 z^2 a^{-2} +3 a z^9+6 z^9 a^{-1} +a z^7-16 z^7 a^{-1} -11 a z^5+10 z^5 a^{-1} +9 a z^3-z^3 a^{-1} -4 a z-2 z a^{-1} +a z^{-1} + a^{-1} z^{-1} +z^{10}+6 z^8-20 z^6+16 z^4-5 z^2-1$ (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 \
-5-4-3-2-10123456χ
12           11
10          2 -2
8         31 2
6        52  -3
4       73   4
2      65    -1
0     87     1
-2    68      2
-4   46       -2
-6  26        4
-8 14         -3
-10 2          2
-121           -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $r=-5$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-2$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-1$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=0$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{8}$ $r=1$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=2$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=4$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=5$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=6$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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