# L11n187

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n187 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{t(1) t(2)^4-t(2)^4+2 t(1)^2 t(2)^3-4 t(1) t(2)^3+2 t(2)^3-3 t(1)^2 t(2)^2+5 t(1) t(2)^2-3 t(2)^2+2 t(1)^2 t(2)-4 t(1) t(2)+2 t(2)-t(1)^2+t(1)}{t(1) t(2)^2}$ (db) Jones polynomial $10 q^{9/2}-11 q^{7/2}+9 q^{5/2}-7 q^{3/2}+q^{17/2}-3 q^{15/2}+6 q^{13/2}-9 q^{11/2}+4 \sqrt{q}-\frac{2}{\sqrt{q}}$ (db) Signature 3 (db) HOMFLY-PT polynomial $-2 z^5 a^{-3} -z^5 a^{-5} +2 z^3 a^{-1} -6 z^3 a^{-3} -z^3 a^{-5} +z^3 a^{-7} +4 z a^{-1} -5 z a^{-3} +z a^{-5} +z a^{-7} + a^{-1} z^{-1} - a^{-3} z^{-1}$ (db) Kauffman polynomial $-z^9 a^{-3} -z^9 a^{-5} -z^8 a^{-2} -5 z^8 a^{-4} -4 z^8 a^{-6} -6 z^7 a^{-5} -6 z^7 a^{-7} +10 z^6 a^{-4} +5 z^6 a^{-6} -5 z^6 a^{-8} -3 z^5 a^{-1} +17 z^5 a^{-5} +11 z^5 a^{-7} -3 z^5 a^{-9} +4 z^4 a^{-2} -6 z^4 a^{-4} -3 z^4 a^{-6} +6 z^4 a^{-8} -z^4 a^{-10} +8 z^3 a^{-1} +7 z^3 a^{-3} -15 z^3 a^{-5} -11 z^3 a^{-7} +3 z^3 a^{-9} -z^2 a^{-2} +2 z^2 a^{-4} -2 z^2 a^{-8} +z^2 a^{-10} -5 z a^{-1} -5 z a^{-3} +4 z a^{-5} +4 z a^{-7} - a^{-2} + a^{-1} z^{-1} + a^{-3} z^{-1}$ (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 \
-2-101234567χ
18         1-1
16        2 2
14       41 -3
12      52  3
10     54   -1
8    65    1
6   46     2
4  35      -2
2 25       3
0 2        -2
-22         2
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=2$ $i=4$ $r=-2$ ${\mathbb Z}^{2}$ $r=-1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=0$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}$ $r=1$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=2$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{6}$ $r=3$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=4$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=5$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=6$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=7$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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