# L11n31

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n31 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{(t(1)-1) (t(2)-1) \left(2 t(2)^2-3 t(2)+2\right)}{\sqrt{t(1)} t(2)^{3/2}}$ (db) Jones polynomial $-\sqrt{q}+\frac{3}{\sqrt{q}}-\frac{6}{q^{3/2}}+\frac{7}{q^{5/2}}-\frac{10}{q^{7/2}}+\frac{9}{q^{9/2}}-\frac{9}{q^{11/2}}+\frac{6}{q^{13/2}}-\frac{3}{q^{15/2}}+\frac{2}{q^{17/2}}$ (db) Signature -3 (db) HOMFLY-PT polynomial $-a^9 z^{-1} -z^3 a^7+a^7 z^{-1} +z^5 a^5+2 z^3 a^5+3 z a^5+2 a^5 z^{-1} +z^5 a^3+z^3 a^3-2 z a^3-2 a^3 z^{-1} -z^3 a-z a$ (db) Kauffman polynomial $3 a^{10} z^4-8 a^{10} z^2+4 a^{10}+a^9 z^7-a^9 z^5+a^9 z^3-2 a^9 z-a^9 z^{-1} +2 a^8 z^8-7 a^8 z^6+18 a^8 z^4-21 a^8 z^2+9 a^8+a^7 z^9+a^7 z^7-8 a^7 z^5+17 a^7 z^3-7 a^7 z-a^7 z^{-1} +5 a^6 z^8-14 a^6 z^6+20 a^6 z^4-11 a^6 z^2+4 a^6+a^5 z^9+4 a^5 z^7-16 a^5 z^5+22 a^5 z^3-11 a^5 z+2 a^5 z^{-1} +3 a^4 z^8-4 a^4 z^6-a^4 z^4+3 a^4 z^2-2 a^4+4 a^3 z^7-8 a^3 z^5+4 a^3 z^3-5 a^3 z+2 a^3 z^{-1} +3 a^2 z^6-6 a^2 z^4+a^2 z^2+a z^5-2 a z^3+a z$ (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 \
-7-6-5-4-3-2-1012χ
2         11
0        2 -2
-2       41 3
-4      54  -1
-6     52   3
-8    45    1
-10   55     0
-12  14      3
-14 25       -3
-16 1        1
-182         -2
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-4$ $i=-2$ $r=-7$ ${\mathbb Z}^{2}$ $r=-6$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-5$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-3$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-2$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=0$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{4}$ $r=1$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=2$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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