# L11n430

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n430 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{(v-1) (w-1) \left(u^2 w^2-u v w+u v-u w^2+u w-v\right)}{u v w^{3/2}}$ (db) Jones polynomial $-q^3+3 q^2-5 q+8-7 q^{-1} +9 q^{-2} -6 q^{-3} +5 q^{-4} -3 q^{-5} + q^{-6}$ (db) Signature 0 (db) HOMFLY-PT polynomial $a^4 z^4+2 a^4 z^2+a^4 z^{-2} +a^4-a^2 z^6-4 a^2 z^4-6 a^2 z^2-z^2 a^{-2} -2 a^2 z^{-2} -4 a^2- a^{-2} +2 z^4+5 z^2+ z^{-2} +4$ (db) Kauffman polynomial $a^6 z^6-3 a^6 z^4+a^6 z^2+3 a^5 z^7-10 a^5 z^5+6 a^5 z^3+4 a^4 z^8-15 a^4 z^6+16 a^4 z^4-9 a^4 z^2-a^4 z^{-2} +3 a^4+2 a^3 z^9-4 a^3 z^7-3 a^3 z^5+5 a^3 z^3+z^3 a^{-3} -2 a^3 z-z a^{-3} +2 a^3 z^{-1} +7 a^2 z^8-29 a^2 z^6+43 a^2 z^4+3 z^4 a^{-2} -27 a^2 z^2-4 z^2 a^{-2} -2 a^2 z^{-2} +7 a^2+2 a^{-2} +2 a z^9-6 a z^7+z^7 a^{-1} +6 a z^5-z^5 a^{-1} +2 a z^3+4 z^3 a^{-1} -4 a z-3 z a^{-1} +2 a z^{-1} +3 z^8-13 z^6+27 z^4-21 z^2- z^{-2} +7$ (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 \
-6-5-4-3-2-10123χ
7         1-1
5        2 2
3       31 -2
1      52  3
-1     45   1
-3    53    2
-5   36     3
-7  23      -1
-9 13       2
-11 2        -2
-131         1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-1$ $i=1$ $r=-6$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-2$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{5}$ $r=-1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=0$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{5}$ $r=1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=2$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=3$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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