# L11n342

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n342 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{(v-1) (w-1) \left(u v w-u w^2-u+v^2 w^2+v^2-v w\right)}{\sqrt{u} v^{3/2} w^{3/2}}$ (db) Jones polynomial $-q^5+ q^{-5} +q^4- q^{-4} +4 q^{-3} -q^2-3 q^{-2} +2 q+4 q^{-1} -2$ (db) Signature 2 (db) HOMFLY-PT polynomial $z^6-2 a^2 z^4+5 z^4+a^4 z^2-8 a^2 z^2-z^2 a^{-4} +8 z^2+3 a^4-10 a^2- a^{-4} +8+2 a^4 z^{-2} -5 a^2 z^{-2} - a^{-2} z^{-2} +4 z^{-2}$ (db) Kauffman polynomial $a^3 z^9+a z^9+a^4 z^8+5 a^2 z^8+4 z^8-4 a^3 z^7+4 z^7 a^{-1} -7 a^4 z^6-30 a^2 z^6+z^6 a^{-4} -24 z^6-a^3 z^5-24 a z^5-25 z^5 a^{-1} -z^5 a^{-3} +z^5 a^{-5} +18 a^4 z^4+58 a^2 z^4-3 z^4 a^{-2} -5 z^4 a^{-4} +42 z^4+17 a^3 z^3+52 a z^3+40 z^3 a^{-1} +z^3 a^{-3} -4 z^3 a^{-5} -21 a^4 z^2-51 a^2 z^2+z^2 a^{-2} +4 z^2 a^{-4} -33 z^2-18 a^3 z-37 a z-23 z a^{-1} -2 z a^{-3} +2 z a^{-5} +11 a^4+24 a^2+2 a^{-2} - a^{-4} +17+5 a^3 z^{-1} +9 a z^{-1} +5 a^{-1} z^{-1} + a^{-3} z^{-1} -2 a^4 z^{-2} -5 a^2 z^{-2} - a^{-2} z^{-2} -4 z^{-2}$ (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 \
-6-5-4-3-2-1012345χ
11           1-1
9            0
7        111 1
5       21   -1
3      221   1
1     341    0
-1    123     2
-3   23       1
-5  21        1
-7 14         3
-9            0
-111           1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-1$ $i=1$ $i=3$ $r=-6$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{4}$ ${\mathbb Z}^{2}$ $r=-3$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=0$ ${\mathbb Z}^{3}$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=1$ ${\mathbb Z}$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=2$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=3$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=4$ ${\mathbb Z}$ $r=5$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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