# L11n127

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n127 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{u^2 v^2-2 u^2 v+u^2-u v^2+u v-u+v^2-2 v+1}{u v}$ (db) Jones polynomial $-q^{5/2}+2 q^{3/2}-3 \sqrt{q}+\frac{3}{\sqrt{q}}-\frac{4}{q^{3/2}}+\frac{3}{q^{5/2}}-\frac{3}{q^{7/2}}+\frac{2}{q^{9/2}}-\frac{1}{q^{11/2}}$ (db) Signature 1 (db) HOMFLY-PT polynomial $a^5 z+a^5 z^{-1} -2 a^3 z^3-5 a^3 z-2 a^3 z^{-1} +a z^5+4 a z^3-z^3 a^{-1} +5 a z+2 a z^{-1} -2 z a^{-1} - a^{-1} z^{-1}$ (db) Kauffman polynomial $-a^3 z^9-a z^9-2 a^4 z^8-4 a^2 z^8-2 z^8-a^5 z^7+2 a^3 z^7+2 a z^7-z^7 a^{-1} +10 a^4 z^6+20 a^2 z^6+10 z^6+5 a^5 z^5+9 a^3 z^5+9 a z^5+5 z^5 a^{-1} -13 a^4 z^4-26 a^2 z^4-13 z^4-7 a^5 z^3-20 a^3 z^3-20 a z^3-7 z^3 a^{-1} +5 a^4 z^2+10 a^2 z^2+5 z^2+4 a^5 z+12 a^3 z+12 a z+4 z a^{-1} -a^2-a^5 z^{-1} -2 a^3 z^{-1} -2 a z^{-1} - a^{-1} z^{-1}$ (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 \
-6-5-4-3-2-1012χ
6        11
4       1 -1
2      21 1
0     22  0
-2    221  1
-4   23    1
-6  121    0
-8 12      1
-10 1       -1
-121        1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $i=2$ $r=-6$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-2$ ${\mathbb Z}$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=1$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=2$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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