# L11n388

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n388 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{(u-1) (v-1) (w-1)^3}{\sqrt{u} \sqrt{v} w^{3/2}}$ (db) Jones polynomial $-3 q^{-6} +6 q^{-5} -9 q^{-4} +12 q^{-3} -q^2-10 q^{-2} +5 q+11 q^{-1} -7$ (db) Signature -2 (db) HOMFLY-PT polynomial $-a^6 z^{-2} -2 a^6-a^4 z^4+2 a^4 z^2+4 a^4 z^{-2} +7 a^4+a^2 z^6+2 a^2 z^4-2 a^2 z^2-5 a^2 z^{-2} -8 a^2-z^4+2 z^{-2} +3$ (db) Kauffman polynomial $3 a^4 z^8+3 a^2 z^8+6 a^5 z^7+13 a^3 z^7+7 a z^7+3 a^6 z^6+3 a^4 z^6+5 a^2 z^6+5 z^6-13 a^5 z^5-27 a^3 z^5-13 a z^5+z^5 a^{-1} -8 a^4 z^4-16 a^2 z^4-8 z^4+6 a^7 z^3+24 a^5 z^3+24 a^3 z^3+6 a z^3-3 a^4 z^2-3 a^2 z^2-6 a^7 z-19 a^5 z-21 a^3 z-8 a z+3 a^6+10 a^4+11 a^2+5+a^7 z^{-1} +5 a^5 z^{-1} +9 a^3 z^{-1} +5 a z^{-1} -a^6 z^{-2} -4 a^4 z^{-2} -5 a^2 z^{-2} -2 z^{-2}$ (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
5        1-1
3       4 4
1      31 -2
-1     84  4
-3    67   1
-5   64    2
-7  36     3
-9 36      -3
-11 3       3
-133        -3
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-3$ $i=-1$ $r=-5$ ${\mathbb Z}^{3}$ $r=-4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-3$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-2$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=-1$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=0$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{8}$ $r=1$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=2$ ${\mathbb Z}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=3$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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