# L11n303

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n303 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{-t(2)^2 t(3)^4+t(1) t(2)^2 t(3)^3-t(1) t(2) t(3)^3+t(2) t(3)^3-t(1) t(2)^2 t(3)^2+2 t(1) t(2) t(3)^2-2 t(2) t(3)^2+t(3)^2-t(1) t(2) t(3)+t(2) t(3)-t(3)+t(1)}{\sqrt{t(1)} t(2) t(3)^2}$ (db) Jones polynomial $-q^8+2 q^7-3 q^6+4 q^5-4 q^4+4 q^3-2 q^2+2 q+1+ q^{-2}$ (db) Signature 2 (db) HOMFLY-PT polynomial $-z^6 a^{-2} +z^6 a^{-4} -8 z^4 a^{-2} +6 z^4 a^{-4} -z^4 a^{-6} +z^4-19 z^2 a^{-2} +14 z^2 a^{-4} -3 z^2 a^{-6} +5 z^2-17 a^{-2} +13 a^{-4} -3 a^{-6} +7-5 a^{-2} z^{-2} +4 a^{-4} z^{-2} - a^{-6} z^{-2} +2 z^{-2}$ (db) Kauffman polynomial $z^8 a^{-2} +z^8 a^{-4} +z^8 a^{-6} +z^8+z^7 a^{-1} +2 z^7 a^{-3} +3 z^7 a^{-5} +2 z^7 a^{-7} -10 z^6 a^{-2} -6 z^6 a^{-4} -2 z^6 a^{-6} +2 z^6 a^{-8} -8 z^6-9 z^5 a^{-1} -16 z^5 a^{-3} -14 z^5 a^{-5} -6 z^5 a^{-7} +z^5 a^{-9} +31 z^4 a^{-2} +15 z^4 a^{-4} -z^4 a^{-6} -6 z^4 a^{-8} +21 z^4+22 z^3 a^{-1} +41 z^3 a^{-3} +26 z^3 a^{-5} +4 z^3 a^{-7} -3 z^3 a^{-9} -39 z^2 a^{-2} -16 z^2 a^{-4} +3 z^2 a^{-6} +3 z^2 a^{-8} -23 z^2-19 z a^{-1} -35 z a^{-3} -19 z a^{-5} -2 z a^{-7} +z a^{-9} +22 a^{-2} +13 a^{-4} - a^{-8} +11+5 a^{-1} z^{-1} +9 a^{-3} z^{-1} +5 a^{-5} z^{-1} + a^{-7} z^{-1} -5 a^{-2} z^{-2} -4 a^{-4} z^{-2} - a^{-6} z^{-2} -2 z^{-2}$ (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 \
-4-3-2-101234567χ
17           1-1
15          1 1
13         21 -1
11        21  1
9       22   0
7     132    0
5     24     2
3   121      0
1    3       3
-1  1         1
-31           1
-51           1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=1$ $i=3$ $i=5$ $r=-4$ ${\mathbb Z}$ ${\mathbb Z}$ $r=-3$ $r=-2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=0$ ${\mathbb Z}^{3}$ ${\mathbb Z}^{2}$ $r=1$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ ${\mathbb Z}$ $r=2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{3}$ $r=3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=4$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=5$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=6$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=7$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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