# L11a81

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a81 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{5 u v^4-11 u v^3+12 u v^2-7 u v+2 u+2 v^5-7 v^4+12 v^3-11 v^2+5 v}{\sqrt{u} v^{5/2}}$ (db) Jones polynomial $-\frac{11}{q^{9/2}}+\frac{4}{q^{7/2}}-\frac{1}{q^{5/2}}+\frac{1}{q^{27/2}}-\frac{3}{q^{25/2}}+\frac{8}{q^{23/2}}-\frac{14}{q^{21/2}}+\frac{20}{q^{19/2}}-\frac{24}{q^{17/2}}+\frac{24}{q^{15/2}}-\frac{22}{q^{13/2}}+\frac{16}{q^{11/2}}$ (db) Signature -5 (db) HOMFLY-PT polynomial $a^{13} (-z)-a^{13} z^{-1} +3 a^{11} z^3+4 a^{11} z+a^{11} z^{-1} -2 a^9 z^5-a^9 z^3+4 a^9 z+2 a^9 z^{-1} -4 a^7 z^5-11 a^7 z^3-9 a^7 z-2 a^7 z^{-1} -a^5 z^5-a^5 z^3$ (db) Kauffman polynomial $-z^6 a^{16}+3 z^4 a^{16}-3 z^2 a^{16}+a^{16}-3 z^7 a^{15}+7 z^5 a^{15}-5 z^3 a^{15}+z a^{15}-5 z^8 a^{14}+9 z^6 a^{14}-4 z^4 a^{14}-5 z^9 a^{13}+4 z^7 a^{13}+5 z^5 a^{13}-4 z^3 a^{13}-z a^{13}+a^{13} z^{-1} -2 z^{10} a^{12}-12 z^8 a^{12}+34 z^6 a^{12}-31 z^4 a^{12}+15 z^2 a^{12}-3 a^{12}-13 z^9 a^{11}+18 z^7 a^{11}-3 z^5 a^{11}+2 z^3 a^{11}-4 z a^{11}+a^{11} z^{-1} -2 z^{10} a^{10}-19 z^8 a^{10}+45 z^6 a^{10}-31 z^4 a^{10}+7 z^2 a^{10}-8 z^9 a^9+z^7 a^9+17 z^5 a^9-15 z^3 a^9+7 z a^9-2 a^9 z^{-1} -12 z^8 a^8+17 z^6 a^8-4 z^4 a^8-5 z^2 a^8+3 a^8-10 z^7 a^7+17 z^5 a^7-15 z^3 a^7+9 z a^7-2 a^7 z^{-1} -4 z^6 a^6+3 z^4 a^6-z^5 a^5+z^3 a^5$ (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 \
-11-10-9-8-7-6-5-4-3-2-10χ
-4           11
-6          41-3
-8         7  7
-10        94  -5
-12       137   6
-14      1210    -2
-16     1212     0
-18    812      4
-20   612       -6
-22  28        6
-24 16         -5
-26 2          2
-281           -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-6$ $i=-4$ $r=-11$ ${\mathbb Z}$ $r=-10$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-9$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-8$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=-7$ ${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=-6$ ${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{12}$ ${\mathbb Z}^{12}$ $r=-5$ ${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{12}$ ${\mathbb Z}^{12}$ $r=-4$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{12}$ ${\mathbb Z}^{13}$ $r=-3$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=-2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=-1$ ${\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}$

### Computer Talk

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