# L11n233

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n233 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{-u^3 v^5+u^2 v^4-3 u^2 v^3+2 u^2 v^2-u^2 v-u v^4+2 u v^3-3 u v^2+u v-1}{u^{3/2} v^{5/2}}$ (db) Jones polynomial $-\frac{1}{q^{9/2}}-\frac{1}{q^{13/2}}-\frac{1}{q^{15/2}}+\frac{3}{q^{17/2}}-\frac{4}{q^{19/2}}+\frac{5}{q^{21/2}}-\frac{5}{q^{23/2}}+\frac{4}{q^{25/2}}-\frac{3}{q^{27/2}}+\frac{1}{q^{29/2}}$ (db) Signature -7 (db) HOMFLY-PT polynomial $-2 a^{13} z^3-4 a^{13} z-a^{13} z^{-1} +a^{11} z^7+9 a^{11} z^5+23 a^{11} z^3+19 a^{11} z+3 a^{11} z^{-1} -a^9 z^9-9 a^9 z^7-28 a^9 z^5-37 a^9 z^3-19 a^9 z-2 a^9 z^{-1}$ (db) Kauffman polynomial $a^{18} z^4-a^{18} z^2+3 a^{17} z^5-5 a^{17} z^3+a^{17} z+3 a^{16} z^6-4 a^{16} z^4+a^{15} z^7+3 a^{15} z^5-7 a^{15} z^3+3 a^{15} z+3 a^{14} z^6-2 a^{14} z^4-a^{14} z^2+a^{14}+4 a^{13} z^5-6 a^{13} z^3+5 a^{13} z-a^{13} z^{-1} +a^{12} z^8-9 a^{12} z^6+26 a^{12} z^4-21 a^{12} z^2+3 a^{12}+a^{11} z^9-10 a^{11} z^7+32 a^{11} z^5-41 a^{11} z^3+22 a^{11} z-3 a^{11} z^{-1} +a^{10} z^8-9 a^{10} z^6+23 a^{10} z^4-19 a^{10} z^2+3 a^{10}+a^9 z^9-9 a^9 z^7+28 a^9 z^5-37 a^9 z^3+19 a^9 z-2 a^9 z^{-1}$ (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 \
-11-10-9-8-7-6-5-4-3-2-10χ
-8           11
-10           11
-12         1  1
-14       2    2
-16      211   -2
-18     32     1
-20    221     -1
-22   33       0
-24  23        1
-26 12         -1
-28 2          2
-301           -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-10$ $i=-8$ $i=-6$ $r=-11$ ${\mathbb Z}$ $r=-10$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-9$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-8$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}$ $r=-7$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-6$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-5$ ${\mathbb Z}$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-4$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=-3$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}$

### Computer Talk

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