# L11a546

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a546 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{u v w^2 x-u v w^2+u v w x^2-3 u v w x+3 u v w-u v x^2+3 u v x-3 u v+u w^2 x^2-2 u w^2 x+u w^2-2 u w x^2+4 u w x-2 u w+u x^2-2 u x+u+v w^2 x^2-2 v w^2 x+v w^2-2 v w x^2+4 v w x-2 v w+v x^2-2 v x+v-3 w^2 x^2+3 w^2 x-w^2+3 w x^2-3 w x+w-x^2+x}{\sqrt{u} \sqrt{v} w x}$ (db) Jones polynomial $-\frac{18}{q^{9/2}}+\frac{19}{q^{7/2}}+q^{5/2}-\frac{22}{q^{5/2}}-4 q^{3/2}+\frac{18}{q^{3/2}}-\frac{1}{q^{17/2}}+\frac{2}{q^{15/2}}-\frac{8}{q^{13/2}}+\frac{11}{q^{11/2}}+9 \sqrt{q}-\frac{15}{\sqrt{q}}$ (db) Signature -1 (db) HOMFLY-PT polynomial $a^9 z^{-3} +a^9 z^{-1} -3 a^7 z^{-3} -4 a^7 z-6 a^7 z^{-1} +6 a^5 z^3+3 a^5 z^{-3} +12 a^5 z+9 a^5 z^{-1} -3 a^3 z^5-8 a^3 z^3-a^3 z^{-3} -10 a^3 z-4 a^3 z^{-1} -a z^5+a z^3+z^3 a^{-1} +2 a z$ (db) Kauffman polynomial $a^9 z^7-5 a^9 z^5+10 a^9 z^3-a^9 z^{-3} -10 a^9 z+5 a^9 z^{-1} +2 a^8 z^8-5 a^8 z^6+10 a^8 z^2+3 a^8 z^{-2} -10 a^8+3 a^7 z^9-5 a^7 z^7-7 a^7 z^5+22 a^7 z^3-3 a^7 z^{-3} -23 a^7 z+12 a^7 z^{-1} +a^6 z^{10}+11 a^6 z^8-40 a^6 z^6+31 a^6 z^4+10 a^6 z^2+6 a^6 z^{-2} -19 a^6+9 a^5 z^9-8 a^5 z^7-34 a^5 z^5+49 a^5 z^3-3 a^5 z^{-3} -27 a^5 z+12 a^5 z^{-1} +a^4 z^{10}+22 a^4 z^8-61 a^4 z^6+41 a^4 z^4+3 a^4 z^{-2} -10 a^4+6 a^3 z^9+12 a^3 z^7-57 a^3 z^5+53 a^3 z^3-a^3 z^{-3} -20 a^3 z+5 a^3 z^{-1} +13 a^2 z^8-17 a^2 z^6+2 a^2 z^4+z^4 a^{-2} +2 a^2 z^2+14 a z^7-21 a z^5+4 z^5 a^{-1} +15 a z^3-z^3 a^{-1} -6 a z+9 z^6-7 z^4+2 z^2$ (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 \
-8-7-6-5-4-3-2-10123χ
6           1-1
4          3 3
2         61 -5
0        93  6
-2       107   -3
-4      128    4
-6     1114     3
-8    78      -1
-10   613       7
-12  25        -3
-14  6         6
-1612          -1
-181           1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $r=-8$ ${\mathbb Z}$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}^{2}$ $r=-6$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-5$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=-4$ ${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{7}$ $r=-3$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{11}$ ${\mathbb Z}^{11}$ $r=-2$ ${\mathbb Z}^{14}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{12}$ $r=-1$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{10}$ $r=0$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{9}$ $r=1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=2$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=3$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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