# L11n30

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n30 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{2 (u-1) (v-1)}{\sqrt{u} \sqrt{v}}$ (db) Jones polynomial $-\frac{3}{q^{9/2}}+\frac{1}{q^{7/2}}-\frac{2}{q^{5/2}}+\frac{1}{q^{3/2}}+\frac{1}{q^{19/2}}-\frac{1}{q^{17/2}}+\frac{2}{q^{15/2}}-\frac{2}{q^{13/2}}+\frac{2}{q^{11/2}}-\frac{1}{\sqrt{q}}$ (db) Signature -1 (db) HOMFLY-PT polynomial $-z a^9-a^9 z^{-1} +z^3 a^7+2 z a^7+a^7 z^{-1} +z^3 a^5+2 z a^5+2 a^5 z^{-1} -2 z a^3-2 a^3 z^{-1} -z a$ (db) Kauffman polynomial $a^{10} z^8-7 a^{10} z^6+16 a^{10} z^4-14 a^{10} z^2+4 a^{10}+a^9 z^9-6 a^9 z^7+10 a^9 z^5-4 a^9 z^3-a^9 z^{-1} +3 a^8 z^8-19 a^8 z^6+38 a^8 z^4-30 a^8 z^2+9 a^8+a^7 z^9-4 a^7 z^7+8 a^7 z^3-3 a^7 z-a^7 z^{-1} +2 a^6 z^8-11 a^6 z^6+18 a^6 z^4-13 a^6 z^2+4 a^6+2 a^5 z^7-10 a^5 z^5+14 a^5 z^3-9 a^5 z+2 a^5 z^{-1} +a^4 z^6-4 a^4 z^4+4 a^4 z^2-2 a^4+2 a^3 z^3-5 a^3 z+2 a^3 z^{-1} +a^2 z^2+a z$ (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 \
-9-8-7-6-5-4-3-2-10χ
0         11
-2        220
-4       1111
-6      23  1
-8     211  2
-10    131   1
-12   22     0
-14   11     0
-16 12       -1
-18          0
-201         -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-4$ $i=-2$ $i=0$ $r=-9$ ${\mathbb Z}$ $r=-8$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}^{2}$ $r=-6$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-5$ ${\mathbb Z}$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}_2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-3$ ${\mathbb Z}$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-2$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}_2$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}^{2}$ ${\mathbb Z}$

### Computer Talk

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