# L11a328

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a328 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{(t(1)-1) (t(2)-1) \left(2 t(2)^2 t(1)^2-t(2) t(1)^2-2 t(2)^2 t(1)+4 t(2) t(1)-2 t(1)-t(2)+2\right)}{t(1)^{3/2} t(2)^{3/2}}$ (db) Jones polynomial $-3 q^{9/2}+\frac{4}{q^{9/2}}+6 q^{7/2}-\frac{8}{q^{7/2}}-11 q^{5/2}+\frac{12}{q^{5/2}}+15 q^{3/2}-\frac{16}{q^{3/2}}+q^{11/2}-\frac{1}{q^{11/2}}-18 \sqrt{q}+\frac{17}{\sqrt{q}}$ (db) Signature 1 (db) HOMFLY-PT polynomial $a^3 z^5+z^5 a^{-3} +2 a^3 z^3+3 z^3 a^{-3} +2 z a^{-3} -a z^7-z^7 a^{-1} -3 a z^5-4 z^5 a^{-1} -a z^3-6 z^3 a^{-1} +2 a z-4 z a^{-1} +a z^{-1} - a^{-1} z^{-1}$ (db) Kauffman polynomial $z^4 a^{-6} -z^2 a^{-6} +a^5 z^7-3 a^5 z^5+3 z^5 a^{-5} +3 a^5 z^3-3 z^3 a^{-5} -a^5 z+z a^{-5} +4 a^4 z^8-14 a^4 z^6+5 z^6 a^{-4} +14 a^4 z^4-4 z^4 a^{-4} -4 a^4 z^2+z^2 a^{-4} +5 a^3 z^9-14 a^3 z^7+7 z^7 a^{-3} +5 a^3 z^5-8 z^5 a^{-3} +6 a^3 z^3+4 z^3 a^{-3} -3 a^3 z-z a^{-3} +2 a^2 z^{10}+6 a^2 z^8+8 z^8 a^{-2} -37 a^2 z^6-13 z^6 a^{-2} +38 a^2 z^4+8 z^4 a^{-2} -10 a^2 z^2-z^2 a^{-2} +11 a z^9+6 z^9 a^{-1} -28 a z^7-6 z^7 a^{-1} +9 a z^5-10 z^5 a^{-1} +11 a z^3+15 z^3 a^{-1} -6 a z-6 z a^{-1} +a z^{-1} + a^{-1} z^{-1} +2 z^{10}+10 z^8-41 z^6+37 z^4-9 z^2-1$ (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 \
-6-5-4-3-2-1012345χ
12           1-1
10          2 2
8         41 -3
6        72  5
4       84   -4
2      107    3
0     910     1
-2    78      -1
-4   59       4
-6  37        -4
-8 15         4
-10 3          -3
-121           1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=0$ $i=2$ $r=-6$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-3$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-2$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=-1$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=0$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{10}$ $r=1$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=4$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=5$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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