# L11a334

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a334 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{t(2)^3 t(1)^3-3 t(2)^2 t(1)^3+2 t(2) t(1)^3-4 t(2)^3 t(1)^2+11 t(2)^2 t(1)^2-11 t(2) t(1)^2+3 t(1)^2+3 t(2)^3 t(1)-11 t(2)^2 t(1)+11 t(2) t(1)-4 t(1)+2 t(2)^2-3 t(2)+1}{t(1)^{3/2} t(2)^{3/2}}$ (db) Jones polynomial $q^{9/2}-\frac{8}{q^{9/2}}-4 q^{7/2}+\frac{14}{q^{7/2}}+9 q^{5/2}-\frac{20}{q^{5/2}}-15 q^{3/2}+\frac{22}{q^{3/2}}-\frac{1}{q^{13/2}}+\frac{3}{q^{11/2}}+20 \sqrt{q}-\frac{23}{\sqrt{q}}$ (db) Signature -1 (db) HOMFLY-PT polynomial $a z^7-2 a^3 z^5+3 a z^5-2 z^5 a^{-1} +a^5 z^3-5 a^3 z^3+4 a z^3-4 z^3 a^{-1} +z^3 a^{-3} +2 a^5 z-5 a^3 z+2 a z-2 z a^{-1} +z a^{-3} +a^5 z^{-1} -a^3 z^{-1}$ (db) Kauffman polynomial $-2 a^2 z^{10}-2 z^{10}-6 a^3 z^9-12 a z^9-6 z^9 a^{-1} -8 a^4 z^8-14 a^2 z^8-7 z^8 a^{-2} -13 z^8-6 a^5 z^7+15 a z^7+5 z^7 a^{-1} -4 z^7 a^{-3} -3 a^6 z^6+12 a^4 z^6+35 a^2 z^6+15 z^6 a^{-2} -z^6 a^{-4} +36 z^6-a^7 z^5+9 a^5 z^5+16 a^3 z^5+7 a z^5+10 z^5 a^{-1} +9 z^5 a^{-3} +4 a^6 z^4-10 a^4 z^4-27 a^2 z^4-9 z^4 a^{-2} +2 z^4 a^{-4} -24 z^4+2 a^7 z^3-8 a^5 z^3-21 a^3 z^3-13 a z^3-8 z^3 a^{-1} -6 z^3 a^{-3} -a^6 z^2+2 a^4 z^2+6 a^2 z^2+2 z^2 a^{-2} -z^2 a^{-4} +6 z^2-a^7 z+5 a^5 z+9 a^3 z+4 a z+2 z a^{-1} +z a^{-3} +a^4-a^5 z^{-1} -a^3 z^{-1}$ (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 \
-6-5-4-3-2-1012345χ
10           1-1
8          3 3
6         61 -5
4        93  6
2       116   -5
0      129    3
-2     1112     1
-4    911      -2
-6   511       6
-8  39        -6
-10 16         5
-12 2          -2
-141           1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $r=-6$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}$ $r=-3$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-2$ ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=-1$ ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{11}$ ${\mathbb Z}^{11}$ $r=0$ ${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{11}$ ${\mathbb Z}^{12}$ $r=1$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{11}$ ${\mathbb Z}^{11}$ $r=2$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=4$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=5$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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