# L11n40

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n40 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{2 u v^5-3 u v^4+3 u v^3-2 u v^2+u v+v^4-2 v^3+3 v^2-3 v+2}{\sqrt{u} v^{5/2}}$ (db) Jones polynomial $\frac{1}{q^{9/2}}-\frac{1}{q^{7/2}}+\frac{2}{q^{25/2}}-\frac{4}{q^{23/2}}+\frac{6}{q^{21/2}}-\frac{7}{q^{19/2}}+\frac{7}{q^{17/2}}-\frac{7}{q^{15/2}}+\frac{5}{q^{13/2}}-\frac{4}{q^{11/2}}$ (db) Signature -7 (db) HOMFLY-PT polynomial $-z a^{13}-a^{13} z^{-1} +z^5 a^{11}+4 z^3 a^{11}+4 z a^{11}+a^{11} z^{-1} -z^7 a^9-4 z^5 a^9-2 z^3 a^9+4 z a^9+2 a^9 z^{-1} -z^7 a^7-6 z^5 a^7-12 z^3 a^7-9 z a^7-2 a^7 z^{-1}$ (db) Kauffman polynomial $-3 z^2 a^{16}+a^{16}-z^5 a^{15}-3 z^3 a^{15}+z a^{15}-3 z^6 a^{14}+3 z^4 a^{14}-2 z^2 a^{14}-4 z^7 a^{13}+8 z^5 a^{13}-4 z^3 a^{13}-z a^{13}+a^{13} z^{-1} -3 z^8 a^{12}+7 z^6 a^{12}-7 z^4 a^{12}+9 z^2 a^{12}-3 a^{12}-z^9 a^{11}-z^7 a^{11}+7 z^5 a^{11}+z^3 a^{11}-4 z a^{11}+a^{11} z^{-1} -4 z^8 a^{10}+13 z^6 a^{10}-9 z^4 a^{10}+z^2 a^{10}-z^9 a^9+2 z^7 a^9+4 z^5 a^9-10 z^3 a^9+7 z a^9-2 a^9 z^{-1} -z^8 a^8+3 z^6 a^8+z^4 a^8-7 z^2 a^8+3 a^8-z^7 a^7+6 z^5 a^7-12 z^3 a^7+9 z a^7-2 a^7 z^{-1}$ (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 \
-9-8-7-6-5-4-3-2-10χ
-6         11
-8        110
-10       3  3
-12      21  -1
-14     53   2
-16    33    0
-18   44     0
-20  23      1
-22 24       -2
-24 2        2
-262         -2
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-8$ $i=-6$ $r=-9$ ${\mathbb Z}^{2}$ $r=-8$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-7$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-6$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-5$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{5}$ $r=-3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-2$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-1$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}$

### Computer Talk

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