# L11a397

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a397 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{-3 t(2) t(1)+3 t(2) t(3) t(1)-3 t(3) t(1)+4 t(1)+3 t(2)-4 t(2) t(3)+3 t(3)-3}{\sqrt{t(1)} \sqrt{t(2)} \sqrt{t(3)}}$ (db) Jones polynomial $-q^7+3 q^6-4 q^5+5 q^4-7 q^3+8 q^2-7 q+7-4 q^{-1} +4 q^{-2} - q^{-3} + q^{-4}$ (db) Signature 0 (db) HOMFLY-PT polynomial $-z^2 a^{-6} +z^4 a^{-4} +a^4 z^{-2} +z^2 a^{-4} +a^4+z^4 a^{-2} -2 a^2 z^2-2 a^2 z^{-2} -3 a^2+z^4+z^2+ z^{-2} +2$ (db) Kauffman polynomial $z^7 a^{-7} -4 z^5 a^{-7} +3 z^3 a^{-7} +3 z^8 a^{-6} -14 z^6 a^{-6} +16 z^4 a^{-6} -3 z^2 a^{-6} +3 z^9 a^{-5} -14 z^7 a^{-5} +18 z^5 a^{-5} -7 z^3 a^{-5} +z^{10} a^{-4} -z^8 a^{-4} -8 z^6 a^{-4} +a^4 z^4+10 z^4 a^{-4} -3 a^4 z^2-2 z^2 a^{-4} -a^4 z^{-2} +3 a^4+4 z^9 a^{-3} -17 z^7 a^{-3} +a^3 z^5+22 z^5 a^{-3} -10 z^3 a^{-3} -3 a^3 z+2 a^3 z^{-1} +z^{10} a^{-2} -3 z^8 a^{-2} +a^2 z^6+5 z^6 a^{-2} +2 a^2 z^4-6 z^4 a^{-2} -6 a^2 z^2+z^2 a^{-2} -2 a^2 z^{-2} +5 a^2+z^9 a^{-1} +a z^7-z^7 a^{-1} +a z^5-3 a z+2 a z^{-1} +z^8+z^4-3 z^2- z^{-2} +3$ (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
15           1-1
13          2 2
11         21 -1
9        32  1
7       42   -2
5      43    1
3     34     1
1    44      0
-1   36       3
-3  11        0
-5  3         3
-711          0
-91           1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-1$ $i=1$ $r=-4$ ${\mathbb Z}$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=0$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{4}$ $r=1$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=4$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=5$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=6$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=7$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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