# L11a86

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a86 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{2 t(1) t(2)^3-4 t(2)^3-8 t(1) t(2)^2+9 t(2)^2+9 t(1) t(2)-8 t(2)-4 t(1)+2}{\sqrt{t(1)} t(2)^{3/2}}$ (db) Jones polynomial $-\sqrt{q}+\frac{3}{\sqrt{q}}-\frac{6}{q^{3/2}}+\frac{10}{q^{5/2}}-\frac{14}{q^{7/2}}+\frac{14}{q^{9/2}}-\frac{15}{q^{11/2}}+\frac{12}{q^{13/2}}-\frac{9}{q^{15/2}}+\frac{5}{q^{17/2}}-\frac{2}{q^{19/2}}+\frac{1}{q^{21/2}}$ (db) Signature -3 (db) HOMFLY-PT polynomial $-a^{11} z^{-1} +3 z a^9+2 a^9 z^{-1} -3 z^3 a^7-2 z a^7+z^5 a^5-z^3 a^5-3 z a^5-a^5 z^{-1} +z^5 a^3+z^3 a^3+z a^3-z^3 a-z a$ (db) Kauffman polynomial $-z^6 a^{12}+4 z^4 a^{12}-5 z^2 a^{12}+2 a^{12}-2 z^7 a^{11}+6 z^5 a^{11}-5 z^3 a^{11}+2 z a^{11}-a^{11} z^{-1} -2 z^8 a^{10}+z^6 a^{10}+10 z^4 a^{10}-13 z^2 a^{10}+5 a^{10}-2 z^9 a^9+z^7 a^9+5 z^5 a^9-6 z^3 a^9+5 z a^9-2 a^9 z^{-1} -z^{10} a^8-2 z^8 a^8+3 z^6 a^8+4 z^4 a^8-6 z^2 a^8+3 a^8-5 z^9 a^7+9 z^7 a^7-9 z^5 a^7+4 z^3 a^7-z^{10} a^6-4 z^8 a^6+7 z^6 a^6-5 z^4 a^6+2 z^2 a^6-a^6-3 z^9 a^5+2 z^7 a^5-2 z^5 a^5+3 z^3 a^5-2 z a^5+a^5 z^{-1} -4 z^8 a^4+3 z^6 a^4+3 z^4 a^4-3 z^2 a^4-4 z^7 a^3+5 z^5 a^3-3 z^6 a^2+6 z^4 a^2-3 z^2 a^2-z^5 a+2 z^3 a-z a$ (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          2 -2
-2         41 3
-4        73  -4
-6       73   4
-8      77    0
-10     87     1
-12    58      3
-14   47       -3
-16  15        4
-18 14         -3
-20 1          1
-221           -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-4$ $i=-2$ $r=-9$ ${\mathbb Z}$ $r=-8$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-5$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-4$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{8}$ $r=-3$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=-2$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=-1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=0$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{4}$ $r=1$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=2$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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