# L11n204

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n204 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{\left(t(1) t(2)^2+1\right) \left(t(1)^2 t(2)^3+1\right)}{t(1)^{3/2} t(2)^{5/2}}$ (db) Jones polynomial $-\frac{1}{q^{9/2}}-\frac{1}{q^{13/2}}+\frac{1}{q^{21/2}}-\frac{1}{q^{23/2}}$ (db) Signature -8 (db) HOMFLY-PT polynomial $a^{13} \left(-z^3\right)-3 a^{13} z-a^{13} z^{-1} +a^{11} z^7+8 a^{11} z^5+20 a^{11} z^3+17 a^{11} z+3 a^{11} z^{-1} -a^9 z^9-9 a^9 z^7-28 a^9 z^5-36 a^9 z^3-18 a^9 z-2 a^9 z^{-1}$ (db) Kauffman polynomial $-z a^{15}-a^{14}+z^3 a^{13}-3 z a^{13}+a^{13} z^{-1} -z^8 a^{12}+8 z^6 a^{12}-20 z^4 a^{12}+17 z^2 a^{12}-3 a^{12}-z^9 a^{11}+9 z^7 a^{11}-28 z^5 a^{11}+37 z^3 a^{11}-20 z a^{11}+3 a^{11} z^{-1} -z^8 a^{10}+8 z^6 a^{10}-20 z^4 a^{10}+17 z^2 a^{10}-3 a^{10}-z^9 a^9+9 z^7 a^9-28 z^5 a^9+36 z^3 a^9-18 z a^9+2 a^9 z^{-1}$ (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 \
-8-7-6-5-4-3-2-10χ
-8        11
-10        11
-12      1  1
-14    1    1
-16    11   0
-18  11     0
-20   1     -1
-2211       0
-241        1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-10$ $i=-8$ $i=-6$ $r=-8$ ${\mathbb Z}$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}$

### Computer Talk

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