# L11n349

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n349 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{u v^2 w^3-2 u v^2 w^2+2 u v^2 w-u v^2-u v w^3+2 u v w^2-3 u v w+2 u v+u w+v^3 \left(-w^2\right)-2 v^2 w^3+3 v^2 w^2-2 v^2 w+v^2+v w^3-2 v w^2+2 v w-v}{\sqrt{u} v^{3/2} w^{3/2}}$ (db) Jones polynomial $-q^6+5 q^5-7 q^4+10 q^3+ q^{-3} -10 q^2-2 q^{-2} +10 q+6 q^{-1} -8$ (db) Signature 2 (db) HOMFLY-PT polynomial $z^6 a^{-2} +2 z^4 a^{-2} -z^4 a^{-4} -2 z^4+a^2 z^2-2 z^2 a^{-2} -4 z^2+2 a^2-5 a^{-2} +3 a^{-4} -2 a^{-2} z^{-2} + a^{-4} z^{-2} + z^{-2}$ (db) Kauffman polynomial $z^3 a^{-7} +5 z^4 a^{-6} -z^2 a^{-6} -2 a^{-6} +2 z^7 a^{-5} +z^3 a^{-5} +z a^{-5} +3 z^8 a^{-4} -5 z^6 a^{-4} +8 z^4 a^{-4} -6 z^2 a^{-4} - a^{-4} z^{-2} +3 a^{-4} +z^9 a^{-3} +5 z^7 a^{-3} -15 z^5 a^{-3} +15 z^3 a^{-3} -8 z a^{-3} +2 a^{-3} z^{-1} +6 z^8 a^{-2} +a^2 z^6-14 z^6 a^{-2} -4 a^2 z^4+14 z^4 a^{-2} +5 a^2 z^2-16 z^2 a^{-2} -2 a^{-2} z^{-2} -2 a^2+9 a^{-2} +z^9 a^{-1} +2 a z^7+5 z^7 a^{-1} -5 a z^5-20 z^5 a^{-1} +2 a z^3+17 z^3 a^{-1} +a z-8 z a^{-1} +2 a^{-1} z^{-1} +3 z^8-8 z^6+7 z^4-6 z^2- z^{-2} +3$ (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 \
-4-3-2-1012345χ
13         1-1
11        4 4
9       53 -2
7      52  3
5     55   0
3    55    0
1   46     2
-1  24      -2
-3  4       4
-512        -1
-71         1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=1$ $i=3$ $r=-4$ ${\mathbb Z}$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{2}$ $r=-2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-1$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=0$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{5}$ $r=1$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=2$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{4}$ $r=5$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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