# L11n422

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n422 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{u^2 v^2 w^2-2 u^2 v^2 w-u^2 v w^2+2 u^2 v w-u^2 v-u v^2 w^2+u v^2 w+3 u v w^2-3 u v w-u w^2+u w+v w^3-2 v w^2+v w+2 w^2-w}{u v w^{3/2}}$ (db) Jones polynomial $q^9-2 q^8+5 q^7-6 q^6+8 q^5-8 q^4+8 q^3-5 q^2+4 q-1$ (db) Signature 2 (db) HOMFLY-PT polynomial $z^6 a^{-4} -z^4 a^{-2} +4 z^4 a^{-4} -2 z^4 a^{-6} -z^2 a^{-2} +6 z^2 a^{-4} -6 z^2 a^{-6} +z^2 a^{-8} + a^{-2} +3 a^{-4} -6 a^{-6} +2 a^{-8} + a^{-4} z^{-2} -2 a^{-6} z^{-2} + a^{-8} z^{-2}$ (db) Kauffman polynomial $z^9 a^{-5} +z^9 a^{-7} +2 z^8 a^{-4} +5 z^8 a^{-6} +3 z^8 a^{-8} +z^7 a^{-3} +z^7 a^{-5} +2 z^7 a^{-7} +2 z^7 a^{-9} -5 z^6 a^{-4} -17 z^6 a^{-6} -11 z^6 a^{-8} +z^6 a^{-10} +z^5 a^{-3} -8 z^5 a^{-5} -15 z^5 a^{-7} -6 z^5 a^{-9} +4 z^4 a^{-2} +11 z^4 a^{-4} +25 z^4 a^{-6} +14 z^4 a^{-8} -4 z^4 a^{-10} +z^3 a^{-1} +14 z^3 a^{-5} +18 z^3 a^{-7} +3 z^3 a^{-9} -3 z^2 a^{-2} -10 z^2 a^{-4} -23 z^2 a^{-6} -12 z^2 a^{-8} +4 z^2 a^{-10} -9 z a^{-5} -9 z a^{-7} - a^{-2} +5 a^{-4} +11 a^{-6} +5 a^{-8} - a^{-10} +2 a^{-5} z^{-1} +2 a^{-7} z^{-1} - a^{-4} z^{-2} -2 a^{-6} z^{-2} - a^{-8} z^{-2}$ (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 \
-1012345678χ
19         11
17        21-1
15       3  3
13      32  -1
11     53   2
9    33    0
7   55     0
5  25      3
3 23       -1
1 3        3
-11         -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=1$ $i=3$ $r=-1$ ${\mathbb Z}$ $r=0$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=2$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{5}$ $r=3$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=5$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=6$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=7$ ${\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=8$ ${\mathbb Z}$ ${\mathbb Z}$

### Computer Talk

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