# L11a45

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a45 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{2 (t(1)-1) (t(2)-1) \left(t(2)^2-t(2)+1\right)^2}{\sqrt{t(1)} t(2)^{5/2}}$ (db) Jones polynomial $-q^{7/2}+4 q^{5/2}-9 q^{3/2}+15 \sqrt{q}-\frac{21}{\sqrt{q}}+\frac{22}{q^{3/2}}-\frac{24}{q^{5/2}}+\frac{20}{q^{7/2}}-\frac{14}{q^{9/2}}+\frac{9}{q^{11/2}}-\frac{4}{q^{13/2}}+\frac{1}{q^{15/2}}$ (db) Signature -1 (db) HOMFLY-PT polynomial $a^3 z^7+a z^7-a^5 z^5+3 a^3 z^5+3 a z^5-z^5 a^{-1} -2 a^5 z^3+3 a^3 z^3+3 a z^3-2 z^3 a^{-1} -a^5 z+2 a^3 z-z a^{-1} -a^5 z^{-1} +3 a^3 z^{-1} -2 a z^{-1}$ (db) Kauffman polynomial $-2 a^4 z^{10}-2 a^2 z^{10}-6 a^5 z^9-12 a^3 z^9-6 a z^9-7 a^6 z^8-12 a^4 z^8-14 a^2 z^8-9 z^8-4 a^7 z^7+7 a^5 z^7+15 a^3 z^7-4 a z^7-8 z^7 a^{-1} -a^8 z^6+16 a^6 z^6+31 a^4 z^6+27 a^2 z^6-4 z^6 a^{-2} +9 z^6+9 a^7 z^5+2 a^5 z^5+20 a z^5+12 z^5 a^{-1} -z^5 a^{-3} +2 a^8 z^4-11 a^6 z^4-19 a^4 z^4-10 a^2 z^4+5 z^4 a^{-2} +z^4-5 a^7 z^3+a^3 z^3-13 a z^3-8 z^3 a^{-1} +z^3 a^{-3} -a^8 z^2+2 a^6 z^2+5 a^4 z^2+a^2 z^2-2 z^2 a^{-2} -3 z^2-2 a^5 z-3 a^3 z+a z+2 z a^{-1} -a^6-3 a^4-3 a^2+a^5 z^{-1} +3 a^3 z^{-1} +2 a z^{-1}$ (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 \
-7-6-5-4-3-2-101234χ
8           11
6          3 -3
4         61 5
2        93  -6
0       126   6
-2      1211    -1
-4     1210     2
-6    812      4
-8   612       -6
-10  38        5
-12 16         -5
-14 3          3
-161           -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $r=-7$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-4$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=-3$ ${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=-2$ ${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{12}$ ${\mathbb Z}^{12}$ $r=-1$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{12}$ ${\mathbb Z}^{12}$ $r=0$ ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{12}$ $r=1$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=3$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=4$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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