# L11n73

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n73 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{2 u v^3-5 u v^2+4 u v-2 u-2 v^3+4 v^2-5 v+2}{\sqrt{u} v^{3/2}}$ (db) Jones polynomial $\frac{9}{q^{9/2}}-\frac{9}{q^{7/2}}+\frac{7}{q^{5/2}}-\frac{6}{q^{3/2}}+\frac{1}{q^{17/2}}-\frac{3}{q^{15/2}}+\frac{5}{q^{13/2}}-\frac{8}{q^{11/2}}-\sqrt{q}+\frac{3}{\sqrt{q}}$ (db) Signature -3 (db) HOMFLY-PT polynomial $a^7 \left(-z^3\right)-a^7 z+a^5 z^5+2 a^5 z^3+2 a^5 z+a^5 z^{-1} +a^3 z^5+a^3 z^3-2 a^3 z-a^3 z^{-1} -a z^3-a z$ (db) Kauffman polynomial $a^{10} z^4-2 a^{10} z^2+3 a^9 z^5-5 a^9 z^3+a^9 z+a^8 z^8-2 a^8 z^6+5 a^8 z^4-3 a^8 z^2+a^7 z^9-2 a^7 z^7+5 a^7 z^5-2 a^7 z^3+a^7 z+4 a^6 z^8-10 a^6 z^6+12 a^6 z^4-2 a^6 z^2+a^5 z^9+2 a^5 z^7-7 a^5 z^5+8 a^5 z^3-4 a^5 z+a^5 z^{-1} +3 a^4 z^8-5 a^4 z^6+2 a^4 z^4-a^4+4 a^3 z^7-8 a^3 z^5+3 a^3 z^3-3 a^3 z+a^3 z^{-1} +3 a^2 z^6-6 a^2 z^4+a^2 z^2+a z^5-2 a z^3+a z$ (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      43  -1
-6     53   2
-8    44    0
-10   45     -1
-12  25      3
-14 13       -2
-16 2        2
-181         -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-4$ $i=-2$ $r=-7$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-4$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{4}$ $r=-3$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=0$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{4}$ $r=1$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=2$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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