# L11n386

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n386 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{(u-1) (v-1) (w-1) \left(w^2-w+1\right)}{\sqrt{u} \sqrt{v} w^{3/2}}$ (db) Jones polynomial $- q^{-8} +2 q^{-7} -4 q^{-6} +6 q^{-5} -8 q^{-4} +9 q^{-3} -6 q^{-2} +2 q+7 q^{-1} -3$ (db) Signature -2 (db) HOMFLY-PT polynomial $-z^4 a^6-3 z^2 a^6-a^6 z^{-2} -3 a^6+z^6 a^4+5 z^4 a^4+11 z^2 a^4+4 a^4 z^{-2} +10 a^4-3 z^4 a^2-10 z^2 a^2-5 a^2 z^{-2} -11 a^2+2 z^2+2 z^{-2} +4$ (db) Kauffman polynomial $a^9 z^5-3 a^9 z^3+a^9 z+2 a^8 z^6-5 a^8 z^4+a^8 z^2+3 a^7 z^7-9 a^7 z^5+9 a^7 z^3-5 a^7 z+a^7 z^{-1} +3 a^6 z^8-10 a^6 z^6+14 a^6 z^4-9 a^6 z^2-a^6 z^{-2} +5 a^6+a^5 z^9+2 a^5 z^7-17 a^5 z^5+35 a^5 z^3-22 a^5 z+5 a^5 z^{-1} +5 a^4 z^8-20 a^4 z^6+39 a^4 z^4-34 a^4 z^2-4 a^4 z^{-2} +16 a^4+a^3 z^9-8 a^3 z^5+26 a^3 z^3-26 a^3 z+9 a^3 z^{-1} +2 a^2 z^8-8 a^2 z^6+23 a^2 z^4-32 a^2 z^2-5 a^2 z^{-2} +17 a^2+a z^7-a z^5+3 a z^3-10 a z+5 a z^{-1} +3 z^4-8 z^2-2 z^{-2} +7$ (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 \
-7-6-5-4-3-2-1012χ
3         22
1        1 -1
-1       62 4
-3      45  1
-5     52   3
-7    34    1
-9   35     -2
-11  13      2
-13 13       -2
-15 1        1
-171         -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-3$ $i=-1$ $r=-7$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-3$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=0$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{6}$ $r=1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=2$ ${\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$

### Computer Talk

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