# L11n199

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n199 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{u^3 v^4+u^2 v^5-u^2 v^4+u^2 v^2+u v^3-u v+u+v}{u^{3/2} v^{5/2}}$ (db) Jones polynomial $-\frac{1}{q^{7/2}}+\frac{1}{q^{9/2}}-\frac{2}{q^{11/2}}+\frac{1}{q^{13/2}}-\frac{1}{q^{15/2}}+\frac{1}{q^{17/2}}-\frac{1}{q^{19/2}}$ (db) Signature -7 (db) HOMFLY-PT polynomial $-z a^{13}-a^{13} z^{-1} +z^5 a^{11}+6 z^3 a^{11}+8 z a^{11}+3 a^{11} z^{-1} -z^7 a^9-6 z^5 a^9-10 z^3 a^9-6 z a^9-2 a^9 z^{-1} -z^7 a^7-6 z^5 a^7-10 z^3 a^7-5 z a^7$ (db) Kauffman polynomial $z^2 a^{14}-a^{14}+z^3 a^{13}-z a^{13}+a^{13} z^{-1} -z^8 a^{12}+7 z^6 a^{12}-15 z^4 a^{12}+13 z^2 a^{12}-3 a^{12}-z^9 a^{11}+7 z^7 a^{11}-17 z^5 a^{11}+22 z^3 a^{11}-14 z a^{11}+3 a^{11} z^{-1} -2 z^8 a^{10}+12 z^6 a^{10}-20 z^4 a^{10}+12 z^2 a^{10}-3 a^{10}-z^9 a^9+6 z^7 a^9-11 z^5 a^9+11 z^3 a^9-8 z a^9+2 a^9 z^{-1} -z^8 a^8+5 z^6 a^8-5 z^4 a^8-z^7 a^7+6 z^5 a^7-10 z^3 a^7+5 z a^7$ (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
-6        11
-8       110
-10      1  1
-12    111  1
-14   121   0
-16   11    0
-18  22     0
-201        1
-2211       0
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-8$ $i=-6$ $i=-4$ $r=-8$ ${\mathbb Z}$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=-5$ ${\mathbb Z}^{2}$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}$

### Computer Talk

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