# L11a204

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a204 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{2 u^2 v^4-4 u^2 v^3+4 u^2 v^2-u^2 v-u v^4+5 u v^3-7 u v^2+5 u v-u-v^3+4 v^2-4 v+2}{u v^2}$ (db) Jones polynomial $q^{3/2}-2 \sqrt{q}+\frac{4}{\sqrt{q}}-\frac{8}{q^{3/2}}+\frac{10}{q^{5/2}}-\frac{13}{q^{7/2}}+\frac{13}{q^{9/2}}-\frac{12}{q^{11/2}}+\frac{9}{q^{13/2}}-\frac{6}{q^{15/2}}+\frac{3}{q^{17/2}}-\frac{1}{q^{19/2}}$ (db) Signature -3 (db) HOMFLY-PT polynomial $z^5 a^7+3 z^3 a^7+2 z a^7-z^7 a^5-4 z^5 a^5-4 z^3 a^5+z a^5+2 a^5 z^{-1} -z^7 a^3-5 z^5 a^3-10 z^3 a^3-10 z a^3-3 a^3 z^{-1} +z^5 a+4 z^3 a+4 z a+a z^{-1}$ (db) Kauffman polynomial $-z^5 a^{11}+2 z^3 a^{11}-z a^{11}-3 z^6 a^{10}+6 z^4 a^{10}-2 z^2 a^{10}-4 z^7 a^9+6 z^5 a^9-4 z^8 a^8+6 z^6 a^8-3 z^4 a^8+2 z^2 a^8-3 z^9 a^7+5 z^7 a^7-7 z^5 a^7+6 z^3 a^7-3 z a^7-z^{10} a^6-3 z^8 a^6+11 z^6 a^6-16 z^4 a^6+6 z^2 a^6-5 z^9 a^5+14 z^7 a^5-17 z^5 a^5+3 z^3 a^5+4 z a^5-2 a^5 z^{-1} -z^{10} a^4-z^8 a^4+6 z^6 a^4-5 z^4 a^4-3 z^2 a^4+3 a^4-2 z^9 a^3+3 z^7 a^3+4 z^5 a^3-12 z^3 a^3+12 z a^3-3 a^3 z^{-1} -2 z^8 a^2+3 z^6 a^2+6 z^4 a^2-9 z^2 a^2+3 a^2-2 z^7 a+7 z^5 a-7 z^3 a+4 z a-a z^{-1} -z^6+4 z^4-4 z^2+1$ (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 \
-8-7-6-5-4-3-2-10123χ
4           1-1
2          1 1
0         31 -2
-2        51  4
-4       64   -2
-6      74    3
-8     66     0
-10    67      -1
-12   36       3
-14  36        -3
-16 14         3
-18 2          -2
-201           1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-4$ $i=-2$ $r=-8$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}$ $r=-5$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-4$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=-3$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=-2$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=-1$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=0$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{5}$ $r=1$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=2$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=3$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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