# L11a231

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a231 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{2 u^2 v^3-7 u^2 v^2+6 u^2 v-2 u^2+u v^4-8 u v^3+13 u v^2-8 u v+u-2 v^4+6 v^3-7 v^2+2 v}{u v^2}$ (db) Jones polynomial $\frac{21}{q^{9/2}}-\frac{19}{q^{7/2}}+\frac{14}{q^{5/2}}-\frac{9}{q^{3/2}}+\frac{1}{q^{21/2}}-\frac{3}{q^{19/2}}+\frac{7}{q^{17/2}}-\frac{13}{q^{15/2}}+\frac{17}{q^{13/2}}-\frac{21}{q^{11/2}}-\sqrt{q}+\frac{4}{\sqrt{q}}$ (db) Signature -3 (db) HOMFLY-PT polynomial $-a^{11} z^{-1} +4 a^9 z+3 a^9 z^{-1} -5 a^7 z^3-6 a^7 z-2 a^7 z^{-1} +2 a^5 z^5+2 a^5 z^3+a^5 z+a^3 z^5-a^3 z^3-2 a^3 z-a z^3$ (db) Kauffman polynomial $a^{12} z^6-3 a^{12} z^4+3 a^{12} z^2-a^{12}+3 a^{11} z^7-8 a^{11} z^5+7 a^{11} z^3-3 a^{11} z+a^{11} z^{-1} +4 a^{10} z^8-5 a^{10} z^6-6 a^{10} z^4+9 a^{10} z^2-3 a^{10}+4 a^9 z^9-2 a^9 z^7-11 a^9 z^5+15 a^9 z^3-12 a^9 z+3 a^9 z^{-1} +2 a^8 z^{10}+6 a^8 z^8-17 a^8 z^6+8 a^8 z^4+3 a^8 z^2-3 a^8+11 a^7 z^9-20 a^7 z^7+9 a^7 z^5+6 a^7 z^3-7 a^7 z+2 a^7 z^{-1} +2 a^6 z^{10}+12 a^6 z^8-34 a^6 z^6+31 a^6 z^4-8 a^6 z^2+7 a^5 z^9-7 a^5 z^7-2 a^5 z^5+5 a^5 z^3+10 a^4 z^8-19 a^4 z^6+15 a^4 z^4-5 a^4 z^2+8 a^3 z^7-13 a^3 z^5+6 a^3 z^3-2 a^3 z+4 a^2 z^6-5 a^2 z^4+a z^5-a z^3$ (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 \
-9-8-7-6-5-4-3-2-1012χ
2           11
0          3 -3
-2         61 5
-4        94  -5
-6       105   5
-8      119    -2
-10     1010     0
-12    711      4
-14   610       -4
-16  28        6
-18 15         -4
-20 2          2
-221           -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-4$ $i=-2$ $r=-9$ ${\mathbb Z}$ $r=-8$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-6$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{6}$ $r=-5$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=-4$ ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{10}$ $r=-3$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{11}$ ${\mathbb Z}^{11}$ $r=-2$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{10}$ $r=-1$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=0$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{6}$ $r=1$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=2$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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