# L11a346

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a346 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{(t(1)-1) (t(2)-1) \left(t(2)^2 t(1)^2-4 t(2) t(1)^2+t(1)^2+t(2) t(1)+t(2)^2-4 t(2)+1\right)}{t(1)^{3/2} t(2)^{3/2}}$ (db) Jones polynomial $15 q^{9/2}-17 q^{7/2}+16 q^{5/2}-\frac{1}{q^{5/2}}-14 q^{3/2}+\frac{2}{q^{3/2}}+q^{17/2}-3 q^{15/2}+7 q^{13/2}-12 q^{11/2}+10 \sqrt{q}-\frac{6}{\sqrt{q}}$ (db) Signature 3 (db) HOMFLY-PT polynomial $z^3 a^{-7} +2 z a^{-7} + a^{-7} z^{-1} -2 z^5 a^{-5} -6 z^3 a^{-5} -7 z a^{-5} -4 a^{-5} z^{-1} +z^7 a^{-3} +4 z^5 a^{-3} +8 z^3 a^{-3} +11 z a^{-3} +6 a^{-3} z^{-1} -2 z^5 a^{-1} +a z^3-7 z^3 a^{-1} +3 a z-9 z a^{-1} +2 a z^{-1} -5 a^{-1} z^{-1}$ (db) Kauffman polynomial $-z^{10} a^{-2} -z^{10} a^{-4} -2 z^9 a^{-1} -6 z^9 a^{-3} -4 z^9 a^{-5} -5 z^8 a^{-2} -11 z^8 a^{-4} -8 z^8 a^{-6} -2 z^8-a z^7+z^7 a^{-1} +4 z^7 a^{-3} -7 z^7 a^{-5} -9 z^7 a^{-7} +20 z^6 a^{-2} +28 z^6 a^{-4} +9 z^6 a^{-6} -6 z^6 a^{-8} +7 z^6+5 a z^5+17 z^5 a^{-1} +31 z^5 a^{-3} +37 z^5 a^{-5} +15 z^5 a^{-7} -3 z^5 a^{-9} -13 z^4 a^{-2} -10 z^4 a^{-4} +3 z^4 a^{-6} +6 z^4 a^{-8} -z^4 a^{-10} -7 z^4-9 a z^3-33 z^3 a^{-1} -48 z^3 a^{-3} -41 z^3 a^{-5} -15 z^3 a^{-7} +2 z^3 a^{-9} -z^2 a^{-2} -6 z^2 a^{-4} -9 z^2 a^{-6} -4 z^2 a^{-8} +z^2 a^{-10} +z^2+7 a z+22 z a^{-1} +29 z a^{-3} +20 z a^{-5} +6 z a^{-7} + a^{-2} +3 a^{-4} +3 a^{-6} + a^{-8} +1-2 a z^{-1} -5 a^{-1} z^{-1} -6 a^{-3} z^{-1} -4 a^{-5} z^{-1} - a^{-7} z^{-1}$ (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 \
-4-3-2-101234567χ
18           1-1
16          2 2
14         51 -4
12        72  5
10       85   -3
8      97    2
6     78     1
4    79      -2
2   59       4
0  15        -4
-2 15         4
-4 1          -1
-61           1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=2$ $i=4$ $r=-4$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=0$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{7}$ $r=1$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=2$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=3$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=4$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=5$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=6$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=7$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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