# L11n417

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n417 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{u^2 v w^3-u^2 w^3+u^2 w^2-u^2 w+u^2+u v^2 w^3-u v w^3+u v-u-v^2 w^3+v^2 w^2-v^2 w+v^2-v}{u v w^{3/2}}$ (db) Jones polynomial $-q^7+2 q^6-3 q^5+4 q^4-4 q^3+5 q^2-3 q+4- q^{-1} + q^{-2}$ (db) Signature 4 (db) HOMFLY-PT polynomial $-z^6 a^{-2} -z^6 a^{-4} -5 z^4 a^{-2} -4 z^4 a^{-4} +z^4 a^{-6} +z^4-9 z^2 a^{-2} -2 z^2 a^{-4} +4 z^2 a^{-6} +4 z^2-10 a^{-2} +4 a^{-4} +2 a^{-6} - a^{-8} +5-5 a^{-2} z^{-2} +4 a^{-4} z^{-2} - a^{-6} z^{-2} +2 z^{-2}$ (db) Kauffman polynomial $z a^{-9} +2 z^2 a^{-8} - a^{-8} +z^5 a^{-7} -z^3 a^{-7} -2 z a^{-7} + a^{-7} z^{-1} +3 z^6 a^{-6} -10 z^4 a^{-6} +6 z^2 a^{-6} - a^{-6} z^{-2} +5 z^7 a^{-5} -23 z^5 a^{-5} +31 z^3 a^{-5} -19 z a^{-5} +5 a^{-5} z^{-1} +4 z^8 a^{-4} -19 z^6 a^{-4} +27 z^4 a^{-4} -21 z^2 a^{-4} -4 a^{-4} z^{-2} +13 a^{-4} +z^9 a^{-3} +z^7 a^{-3} -25 z^5 a^{-3} +50 z^3 a^{-3} -35 z a^{-3} +9 a^{-3} z^{-1} +5 z^8 a^{-2} -29 z^6 a^{-2} +55 z^4 a^{-2} -46 z^2 a^{-2} -5 a^{-2} z^{-2} +22 a^{-2} +z^9 a^{-1} -4 z^7 a^{-1} -z^5 a^{-1} +18 z^3 a^{-1} -19 z a^{-1} +5 a^{-1} z^{-1} +z^8-7 z^6+18 z^4-21 z^2-2 z^{-2} +11$ (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
15         1-1
13        1 1
11       32 -1
9      221 1
7     33   0
5    221   1
3   24     2
1  21      1
-1 14       3
-3          0
-51         1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=1$ $i=3$ $i=5$ $r=-4$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{4}$ ${\mathbb Z}^{2}$ $r=-1$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=0$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=2$ ${\mathbb Z}$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}$ $r=4$ ${\mathbb Z}$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=5$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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