# L11a216

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a216 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{4 u^2 v^3-6 u^2 v^2+4 u^2 v-u^2+2 u v^4-8 u v^3+13 u v^2-8 u v+2 u-v^4+4 v^3-6 v^2+4 v}{u v^2}$ (db) Jones polynomial $-\frac{8}{q^{9/2}}+\frac{3}{q^{7/2}}-\frac{1}{q^{5/2}}+\frac{1}{q^{27/2}}-\frac{4}{q^{25/2}}+\frac{8}{q^{23/2}}-\frac{13}{q^{21/2}}+\frac{18}{q^{19/2}}-\frac{20}{q^{17/2}}+\frac{20}{q^{15/2}}-\frac{18}{q^{13/2}}+\frac{12}{q^{11/2}}$ (db) Signature -5 (db) HOMFLY-PT polynomial $a^{13} (-z)+3 a^{11} z^3+3 a^{11} z-a^{11} z^{-1} -2 a^9 z^5-2 a^9 z^3+3 a^9 z+3 a^9 z^{-1} -3 a^7 z^5-8 a^7 z^3-7 a^7 z-2 a^7 z^{-1} -a^5 z^5-2 a^5 z^3-a^5 z$ (db) Kauffman polynomial $a^{16} z^6-2 a^{16} z^4+a^{16} z^2+4 a^{15} z^7-10 a^{15} z^5+7 a^{15} z^3-a^{15} z+6 a^{14} z^8-13 a^{14} z^6+7 a^{14} z^4-a^{14} z^2+4 a^{13} z^9+2 a^{13} z^7-22 a^{13} z^5+18 a^{13} z^3-3 a^{13} z+a^{12} z^{10}+14 a^{12} z^8-36 a^{12} z^6+27 a^{12} z^4-8 a^{12} z^2-a^{12}+8 a^{11} z^9-3 a^{11} z^7-19 a^{11} z^5+16 a^{11} z^3-3 a^{11} z+a^{11} z^{-1} +a^{10} z^{10}+14 a^{10} z^8-29 a^{10} z^6+17 a^{10} z^4+a^{10} z^2-3 a^{10}+4 a^9 z^9+5 a^9 z^7-19 a^9 z^5+19 a^9 z^3-10 a^9 z+3 a^9 z^{-1} +6 a^8 z^8-4 a^8 z^6-5 a^8 z^4+8 a^8 z^2-3 a^8+6 a^7 z^7-11 a^7 z^5+12 a^7 z^3-8 a^7 z+2 a^7 z^{-1} +3 a^6 z^6-4 a^6 z^4+a^6 z^2+a^5 z^5-2 a^5 z^3+a^5 z$ (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          31-2
-8         5  5
-10        73  -4
-12       115   6
-14      97    -2
-16     1111     0
-18    810      2
-20   510       -5
-22  38        5
-24 15         -4
-26 3          3
-281           -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-6$ $i=-4$ $r=-11$ ${\mathbb Z}$ $r=-10$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-9$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-8$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-7$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=-6$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{11}$ $r=-5$ ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=-4$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{11}$ ${\mathbb Z}^{11}$ $r=-3$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=-2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-1$ ${\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}$

### Computer Talk

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