# L11n251

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n251 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{(t(1)-1) (t(2)-1) \left(t(2)^2 t(1)^2-t(2) t(1)^2+t(1)^2+3 t(2) t(1)+t(2)^2-t(2)+1\right)}{t(1)^{3/2} t(2)^{3/2}}$ (db) Jones polynomial $-\frac{3}{q^{9/2}}-2 q^{7/2}+\frac{7}{q^{7/2}}+5 q^{5/2}-\frac{11}{q^{5/2}}-8 q^{3/2}+\frac{11}{q^{3/2}}+\frac{1}{q^{11/2}}+11 \sqrt{q}-\frac{13}{\sqrt{q}}$ (db) Signature -1 (db) HOMFLY-PT polynomial $-a^3 z^5-3 a^3 z^3-4 a^3 z-z a^{-3} +a z^7+5 a z^5-z^5 a^{-1} +10 a z^3-2 z^3 a^{-1} +7 a z-2 z a^{-1} +a z^{-1} - a^{-1} z^{-1}$ (db) Kauffman polynomial $a^6 z^4-a^6 z^2+3 a^5 z^5-2 a^5 z^3+6 a^4 z^6-7 a^4 z^4+4 a^4 z^2+8 a^3 z^7-15 a^3 z^5+3 z^5 a^{-3} +16 a^3 z^3-6 z^3 a^{-3} -8 a^3 z+2 z a^{-3} +5 a^2 z^8+z^8 a^{-2} -3 a^2 z^6+2 z^6 a^{-2} -6 a^2 z^4-9 z^4 a^{-2} +7 a^2 z^2+6 z^2 a^{-2} +a z^9+z^9 a^{-1} +11 a z^7+3 z^7 a^{-1} -30 a z^5-9 z^5 a^{-1} +29 a z^3+5 z^3 a^{-1} -14 a z-4 z a^{-1} +a z^{-1} + a^{-1} z^{-1} +6 z^8-7 z^6-7 z^4+8 z^2-1$ (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 \
-5-4-3-2-101234χ
8         22
6        3 -3
4       52 3
2      63  -3
0     75   2
-2    68    2
-4   55     0
-6  26      4
-8 15       -4
-10 2        2
-121         -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $r=-5$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-2$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-1$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=0$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{7}$ $r=1$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=4$ ${\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$

### Computer Talk

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