# L11a327

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a327 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{-u^3 v^5+u^3 v^4-u^3 v^3+u^3 v^2+u^2 v^5-4 u^2 v^4+5 u^2 v^3-5 u^2 v^2+2 u^2 v+2 u v^4-5 u v^3+5 u v^2-4 u v+u+v^3-v^2+v-1}{u^{3/2} v^{5/2}}$ (db) Jones polynomial $q^{9/2}-2 q^{7/2}+5 q^{5/2}-9 q^{3/2}+11 \sqrt{q}-\frac{14}{\sqrt{q}}+\frac{13}{q^{3/2}}-\frac{12}{q^{5/2}}+\frac{9}{q^{7/2}}-\frac{5}{q^{9/2}}+\frac{2}{q^{11/2}}-\frac{1}{q^{13/2}}$ (db) Signature -1 (db) HOMFLY-PT polynomial $a^3 z^7+6 a^3 z^5+13 a^3 z^3+10 a^3 z+2 a^3 z^{-1} -a z^9-8 a z^7+z^7 a^{-1} -25 a z^5+6 z^5 a^{-1} -36 a z^3+13 z^3 a^{-1} -22 a z+10 z a^{-1} -3 a z^{-1} + a^{-1} z^{-1}$ (db) Kauffman polynomial $a^7 z^5-3 a^7 z^3+2 a^7 z+2 a^6 z^6-4 a^6 z^4+a^6 z^2+3 a^5 z^7-5 a^5 z^5+2 a^5 z^3-a^5 z+4 a^4 z^8-10 a^4 z^6+z^6 a^{-4} +14 a^4 z^4-4 z^4 a^{-4} -7 a^4 z^2+4 z^2 a^{-4} +3 a^3 z^9-6 a^3 z^7+2 z^7 a^{-3} +10 a^3 z^5-6 z^5 a^{-3} -10 a^3 z^3+4 z^3 a^{-3} +9 a^3 z-2 a^3 z^{-1} +a^2 z^{10}+4 a^2 z^8+3 z^8 a^{-2} -17 a^2 z^6-9 z^6 a^{-2} +30 a^2 z^4+10 z^4 a^{-2} -18 a^2 z^2-7 z^2 a^{-2} +3 a^2+ a^{-2} +6 a z^9+3 z^9 a^{-1} -21 a z^7-10 z^7 a^{-1} +43 a z^5+21 z^5 a^{-1} -47 a z^3-28 z^3 a^{-1} +24 a z+12 z a^{-1} -3 a z^{-1} - a^{-1} z^{-1} +z^{10}+3 z^8-15 z^6+26 z^4-21 z^2+3$ (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 \
-6-5-4-3-2-1012345χ
10           1-1
8          1 1
6         41 -3
4        51  4
2       64   -2
0      85    3
-2     67     1
-4    67      -1
-6   36       3
-8  26        -4
-10 14         3
-12 1          -1
-141           1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $r=-6$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=-3$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-2$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=-1$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=0$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{8}$ $r=1$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=3$ ${\mathbb Z}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=4$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=5$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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