# L11n120

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n120 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{t(2)^5-t(1) t(2)^4-3 t(2)^4+2 t(1) t(2)^3+2 t(2)^2-3 t(1) t(2)-t(2)+t(1)}{\sqrt{t(1)} t(2)^{5/2}}$ (db) Jones polynomial $q^{9/2}-2 q^{7/2}+2 q^{5/2}-q^{3/2}-\frac{2}{q^{3/2}}+\frac{2}{q^{5/2}}-\frac{3}{q^{7/2}}+\frac{2}{q^{9/2}}-\frac{1}{q^{11/2}}$ (db) Signature 1 (db) HOMFLY-PT polynomial $a^5 z+a^5 z^{-1} -2 a^3 z^3+z^3 a^{-3} -5 a^3 z-2 a^3 z^{-1} +2 z a^{-3} + a^{-3} z^{-1} +a z^5-z^5 a^{-1} +5 a z^3-5 z^3 a^{-1} +7 a z+3 a z^{-1} -7 z a^{-1} -3 a^{-1} z^{-1}$ (db) Kauffman polynomial $a^5 z^7-5 a^5 z^5+7 a^5 z^3-4 a^5 z+a^5 z^{-1} +2 a^4 z^8-10 a^4 z^6+z^6 a^{-4} +12 a^4 z^4-4 z^4 a^{-4} -4 a^4 z^2+3 z^2 a^{-4} - a^{-4} +a^3 z^9-3 a^3 z^7+2 z^7 a^{-3} -6 a^3 z^5-9 z^5 a^{-3} +16 a^3 z^3+8 z^3 a^{-3} -10 a^3 z-3 z a^{-3} +2 a^3 z^{-1} + a^{-3} z^{-1} +3 a^2 z^8+z^8 a^{-2} -18 a^2 z^6-4 z^6 a^{-2} +27 a^2 z^4-z^4 a^{-2} -13 a^2 z^2+6 z^2 a^{-2} +2 a^2-2 a^{-2} +a z^9-4 a z^7+2 z^7 a^{-1} -6 a z^5-14 z^5 a^{-1} +23 a z^3+22 z^3 a^{-1} -16 a z-13 z a^{-1} +3 a z^{-1} +3 a^{-1} z^{-1} +2 z^8-13 z^6+18 z^4-6 z^2$ (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
10           1-1
8          1 1
6         11 0
4       221  -1
2      111   1
0     242    0
-2    222     2
-4   121      0
-6  221       1
-8 12         1
-10 1          -1
-121           1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $i=2$ $r=-6$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=-3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-1$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=0$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=2$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=3$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=4$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=5$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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