# L11n345

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n345 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{t(1) t(3)^3 t(2)^3-t(3)^3 t(2)^3+t(3)^2 t(2)^3-t(1) t(3)^3 t(2)^2+t(1) t(3) t(2)^2-t(3)^2 t(2)+t(2)+t(1)-t(1) t(3)-1}{\sqrt{t(1)} t(2)^{3/2} t(3)^{3/2}}$ (db) Jones polynomial $q^{10}-q^8+q^7-q^6+3 q^5-2 q^4+3 q^3-q^2+q$ (db) Signature 6 (db) HOMFLY-PT polynomial $-z^8 a^{-6} +z^6 a^{-4} -7 z^6 a^{-6} +z^6 a^{-8} +6 z^4 a^{-4} -17 z^4 a^{-6} +6 z^4 a^{-8} +11 z^2 a^{-4} -20 z^2 a^{-6} +9 z^2 a^{-8} -z^2 a^{-10} +7 a^{-4} -11 a^{-6} +4 a^{-8} + a^{-4} z^{-2} -2 a^{-6} z^{-2} + a^{-8} z^{-2}$ (db) Kauffman polynomial $z^4 a^{-12} -4 z^2 a^{-12} + a^{-12} -z^3 a^{-11} -z^4 a^{-10} -z^2 a^{-10} +2 z^7 a^{-9} -11 z^5 a^{-9} +11 z^3 a^{-9} +3 z^8 a^{-8} -18 z^6 a^{-8} +29 z^4 a^{-8} -16 z^2 a^{-8} - a^{-8} z^{-2} +5 a^{-8} +z^9 a^{-7} -3 z^7 a^{-7} -7 z^5 a^{-7} +20 z^3 a^{-7} -11 z a^{-7} +2 a^{-7} z^{-1} +4 z^8 a^{-6} -25 z^6 a^{-6} +48 z^4 a^{-6} -37 z^2 a^{-6} -2 a^{-6} z^{-2} +13 a^{-6} +z^9 a^{-5} -5 z^7 a^{-5} +4 z^5 a^{-5} +8 z^3 a^{-5} -11 z a^{-5} +2 a^{-5} z^{-1} +z^8 a^{-4} -7 z^6 a^{-4} +17 z^4 a^{-4} -18 z^2 a^{-4} - a^{-4} z^{-2} +8 a^{-4}$ (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 \
-2-1012345678χ
21          11
19          11
17       21  -1
15      11   0
13     231   0
11    112    2
9   12      1
7  21       1
5 13        2
3           0
11          1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=3$ $i=5$ $i=7$ $r=-2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=0$ ${\mathbb Z}^{3}$ ${\mathbb Z}^{2}$ $r=1$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=3$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=4$ ${\mathbb Z}^{2}$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=5$ ${\mathbb Z}$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=6$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=7$ $r=8$ ${\mathbb Z}$ ${\mathbb Z}$

### Computer Talk

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