# L11a300

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a300 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{t(2)^3 t(1)^3-3 t(2)^2 t(1)^3+2 t(2) t(1)^3-3 t(2)^3 t(1)^2+9 t(2)^2 t(1)^2-7 t(2) t(1)^2+2 t(1)^2+2 t(2)^3 t(1)-7 t(2)^2 t(1)+9 t(2) t(1)-3 t(1)+2 t(2)^2-3 t(2)+1}{t(1)^{3/2} t(2)^{3/2}}$ (db) Jones polynomial $15 q^{9/2}-18 q^{7/2}+16 q^{5/2}-\frac{1}{q^{5/2}}-14 q^{3/2}+\frac{3}{q^{3/2}}+q^{17/2}-4 q^{15/2}+8 q^{13/2}-12 q^{11/2}+10 \sqrt{q}-\frac{6}{\sqrt{q}}$ (db) Signature 3 (db) HOMFLY-PT polynomial $z^7 a^{-3} -2 z^5 a^{-1} +4 z^5 a^{-3} -2 z^5 a^{-5} +a z^3-6 z^3 a^{-1} +7 z^3 a^{-3} -5 z^3 a^{-5} +z^3 a^{-7} +2 a z-4 z a^{-1} +6 z a^{-3} -3 z a^{-5} +z a^{-7} + a^{-3} z^{-1} - a^{-5} z^{-1}$ (db) Kauffman polynomial $z^4 a^{-10} +4 z^5 a^{-9} -2 z^3 a^{-9} +8 z^6 a^{-8} -8 z^4 a^{-8} +2 z^2 a^{-8} +10 z^7 a^{-7} -13 z^5 a^{-7} +7 z^3 a^{-7} -2 z a^{-7} +8 z^8 a^{-6} -6 z^6 a^{-6} -5 z^4 a^{-6} +2 z^2 a^{-6} +4 z^9 a^{-5} +7 z^7 a^{-5} -30 z^5 a^{-5} +23 z^3 a^{-5} -7 z a^{-5} + a^{-5} z^{-1} +z^{10} a^{-4} +12 z^8 a^{-4} -33 z^6 a^{-4} +21 z^4 a^{-4} -3 z^2 a^{-4} - a^{-4} +7 z^9 a^{-3} -11 z^7 a^{-3} -11 z^5 a^{-3} +19 z^3 a^{-3} -6 z a^{-3} + a^{-3} z^{-1} +z^{10} a^{-2} +7 z^8 a^{-2} -31 z^6 a^{-2} +32 z^4 a^{-2} -9 z^2 a^{-2} +3 z^9 a^{-1} +a z^7-7 z^7 a^{-1} -4 a z^5-2 z^5 a^{-1} +5 a z^3+10 z^3 a^{-1} -2 a z-3 z a^{-1} +3 z^8-12 z^6+15 z^4-6 z^2$ (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 \
-4-3-2-101234567χ
18           1-1
16          3 3
14         51 -4
12        73  4
10       96   -3
8      96    3
6     79     2
4    79      -2
2   48       4
0  26        -4
-2 14         3
-4 2          -2
-61           1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=2$ $i=4$ $r=-4$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-1$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=0$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{7}$ $r=1$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=2$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=3$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=4$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{7}$ $r=5$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=6$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=7$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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