# L11n423

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n423 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{(w-1) \left(u^2 v-u^2+u v^2 w^2-u v w^2-u v w-u v+u-v^2 w^2+v w^2\right)}{u v w^{3/2}}$ (db) Jones polynomial $-q^4- q^{-4} +3 q^3+3 q^{-3} -3 q^2-3 q^{-2} +5 q+5 q^{-1} -4$ (db) Signature 2 (db) HOMFLY-PT polynomial $z^6-a^2 z^4-z^4 a^{-2} +4 z^4-2 a^2 z^2-2 z^2 a^{-2} +3 z^2+a^2+ a^{-2} -2+a^2 z^{-2} + a^{-2} z^{-2} -2 z^{-2}$ (db) Kauffman polynomial $a^3 z^7+z^7 a^{-3} -4 a^3 z^5-4 z^5 a^{-3} +3 a^3 z^3+3 z^3 a^{-3} +3 a^2 z^8+3 z^8 a^{-2} -15 a^2 z^6-15 z^6 a^{-2} +20 a^2 z^4+20 z^4 a^{-2} -7 a^2 z^2-7 z^2 a^{-2} +a^2 z^{-2} + a^{-2} z^{-2} -2 a^2-2 a^{-2} +2 a z^9+2 z^9 a^{-1} -8 a z^7-8 z^7 a^{-1} +5 a z^5+5 z^5 a^{-1} +a z^3+z^3 a^{-1} +2 a z+2 z a^{-1} -2 a z^{-1} -2 a^{-1} z^{-1} +6 z^8-30 z^6+40 z^4-14 z^2+2 z^{-2} -3$ (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 \
-5-4-3-2-10123χ
9        1-1
7       2 2
5      22 0
3     321 2
1    34   1
-1   221   1
-3  24     2
-5 11      0
-7 2       2
-91        -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-1$ $i=1$ $i=3$ $r=-5$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=-1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}$ $r=1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=2$ ${\mathbb Z}$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=3$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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