# L11n236

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n236 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{\left(2 u^2 v-u^2-v+2\right) \left(u v^2+1\right)}{u^{3/2} v^{3/2}}$ (db) Jones polynomial $-\frac{1}{q^{7/2}}+\frac{1}{q^{9/2}}-\frac{2}{q^{11/2}}+\frac{1}{q^{19/2}}-\frac{1}{q^{21/2}}+\frac{1}{q^{23/2}}-\frac{1}{q^{25/2}}$ (db) Signature -6 (db) HOMFLY-PT polynomial $a^{13} (-z)+a^{11} z^5+6 a^{11} z^3+6 a^{11} z-a^9 z^7-6 a^9 z^5-9 a^9 z^3-3 a^9 z+a^9 z^{-1} -a^7 z^7-6 a^7 z^5-10 a^7 z^3-6 a^7 z-a^7 z^{-1}$ (db) Kauffman polynomial $-z^5 a^{15}+4 z^3 a^{15}-3 z a^{15}-z^6 a^{14}+4 z^4 a^{14}-2 z^2 a^{14}-z^5 a^{13}+5 z^3 a^{13}-3 z a^{13}-z^8 a^{12}+7 z^6 a^{12}-14 z^4 a^{12}+10 z^2 a^{12}-z^9 a^{11}+7 z^7 a^{11}-15 z^5 a^{11}+15 z^3 a^{11}-7 z a^{11}-2 z^8 a^{10}+13 z^6 a^{10}-22 z^4 a^{10}+10 z^2 a^{10}-z^9 a^9+6 z^7 a^9-9 z^5 a^9+4 z^3 a^9-z a^9-a^9 z^{-1} -z^8 a^8+5 z^6 a^8-4 z^4 a^8-2 z^2 a^8+a^8-z^7 a^7+6 z^5 a^7-10 z^3 a^7+6 z a^7-a^7 z^{-1}$ (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
-6          11
-8         110
-10        1  1
-12      211  2
-14     121   0
-16    121    0
-18   122     -1
-20  11       0
-22 121       0
-24           0
-261          1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-8$ $i=-6$ $i=-4$ $r=-10$ ${\mathbb Z}$ $r=-9$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=-8$ ${\mathbb Z}^{2}$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}_2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{2}$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=-3$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}$

### Computer Talk

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