# L11n222

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n222 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{(u v+1) \left(u^2 v^4-u v^3+2 u v^2-u v+1\right)}{u^{3/2} v^{5/2}}$ (db) Jones polynomial $-\frac{1}{q^{9/2}}-\frac{1}{q^{27/2}}+\frac{2}{q^{25/2}}-\frac{2}{q^{23/2}}+\frac{2}{q^{21/2}}-\frac{2}{q^{19/2}}+\frac{1}{q^{17/2}}-\frac{1}{q^{13/2}}$ (db) Signature -7 (db) HOMFLY-PT polynomial $a^{13} \left(-z^3\right)-2 a^{13} z-a^{13} z^{-1} +a^{11} z^7+8 a^{11} z^5+19 a^{11} z^3+15 a^{11} z+3 a^{11} z^{-1} -a^9 z^9-9 a^9 z^7-28 a^9 z^5-36 a^9 z^3-17 a^9 z-2 a^9 z^{-1}$ (db) Kauffman polynomial $-z^3 a^{17}+z a^{17}-2 z^4 a^{16}+3 z^2 a^{16}-z^5 a^{15}+z a^{15}-2 z^4 a^{14}+2 z^2 a^{14}-a^{14}-z^3 a^{13}-z a^{13}+a^{13} z^{-1} -z^8 a^{12}+8 z^6 a^{12}-19 z^4 a^{12}+14 z^2 a^{12}-3 a^{12}-z^9 a^{11}+9 z^7 a^{11}-27 z^5 a^{11}+34 z^3 a^{11}-18 z a^{11}+3 a^{11} z^{-1} -z^8 a^{10}+8 z^6 a^{10}-19 z^4 a^{10}+15 z^2 a^{10}-3 a^{10}-z^9 a^9+9 z^7 a^9-28 z^5 a^9+36 z^3 a^9-17 z a^9+2 a^9 z^{-1}$ (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 \
-10-9-8-7-6-5-4-3-2-10χ
-8          11
-10          11
-12        1  1
-14      1    1
-16     111   -1
-18    21     1
-20    11     0
-22  22       0
-24 11        0
-26 1         -1
-281          1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-10$ $i=-8$ $i=-6$ $r=-10$ ${\mathbb Z}$ $r=-9$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-8$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=-7$ ${\mathbb Z}^{2}$ $r=-6$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-5$ ${\mathbb Z}$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}$

### Computer Talk

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