# L11a293

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a293 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{(u-1) (v-1) \left(u^2 v^3-u^2 v^2+u^2 v+u v^4-u v^3+u v^2-u v+u+v^3-v^2+v\right)}{u^{3/2} v^{5/2}}$ (db) Jones polynomial $q^{7/2}-3 q^{5/2}+5 q^{3/2}-9 \sqrt{q}+\frac{11}{\sqrt{q}}-\frac{14}{q^{3/2}}+\frac{13}{q^{5/2}}-\frac{12}{q^{7/2}}+\frac{10}{q^{9/2}}-\frac{6}{q^{11/2}}+\frac{3}{q^{13/2}}-\frac{1}{q^{15/2}}$ (db) Signature -3 (db) HOMFLY-PT polynomial $a^5 z^5+3 a^5 z^3+2 a^5 z-a^3 z^7-4 a^3 z^5-5 a^3 z^3-3 a^3 z-a z^7-4 a z^5+z^5 a^{-1} -4 a z^3+3 z^3 a^{-1} +z a^{-1} +a z^{-1} - a^{-1} z^{-1}$ (db) Kauffman polynomial $a^9 z^3+3 a^8 z^4+6 a^7 z^5-3 a^7 z^3+10 a^6 z^6-15 a^6 z^4+6 a^6 z^2+12 a^5 z^7-26 a^5 z^5+15 a^5 z^3-4 a^5 z+10 a^4 z^8-22 a^4 z^6+5 a^4 z^4+3 a^4 z^2+6 a^3 z^9-10 a^3 z^7-13 a^3 z^5+19 a^3 z^3-6 a^3 z+2 a^2 z^{10}+4 a^2 z^8+z^8 a^{-2} -34 a^2 z^6-5 z^6 a^{-2} +38 a^2 z^4+8 z^4 a^{-2} -10 a^2 z^2-4 z^2 a^{-2} +9 a z^9+3 z^9 a^{-1} -38 a z^7-16 z^7 a^{-1} +47 a z^5+28 z^5 a^{-1} -17 a z^3-17 z^3 a^{-1} +2 z a^{-1} +a z^{-1} + a^{-1} z^{-1} +2 z^{10}-5 z^8-7 z^6+23 z^4-11 z^2-1$ (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 \
-6-5-4-3-2-1012345χ
8           1-1
6          2 2
4         31 -2
2        62  4
0       53   -2
-2      96    3
-4     67     1
-6    67      -1
-8   46       2
-10  26        -4
-12 14         3
-14 2          -2
-161           1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-4$ $i=-2$ $r=-6$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-3$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-2$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=-1$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=0$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{9}$ $r=1$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=4$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=5$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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