# L11n360

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n360 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{(t(1)-1) (t(3)-1)^2}{\sqrt{t(1)} t(3)}$ (db) Jones polynomial $q^3-2 q^2+3 q-3+4 q^{-1} -2 q^{-2} +3 q^{-3} + q^{-6} - q^{-7}$ (db) Signature -2 (db) HOMFLY-PT polynomial $-z^2 a^6-2 a^6+z^4 a^4+5 z^2 a^4+a^4 z^{-2} +5 a^4-z^4 a^2-3 z^2 a^2-2 a^2 z^{-2} -4 a^2-z^4-2 z^2+ z^{-2} +z^2 a^{-2} + a^{-2}$ (db) Kauffman polynomial $a^7 z^7-6 a^7 z^5+9 a^7 z^3-3 a^7 z+a^6 z^8-7 a^6 z^6+14 a^6 z^4-12 a^6 z^2+5 a^6+a^5 z^7-9 a^5 z^5+18 a^5 z^3-10 a^5 z+2 a^4 z^8-16 a^4 z^6+39 a^4 z^4-38 a^4 z^2-a^4 z^{-2} +15 a^4+a^3 z^9-5 a^3 z^7+2 a^3 z^5+13 a^3 z^3-13 a^3 z+2 a^3 z^{-1} +3 a^2 z^8-18 a^2 z^6+z^6 a^{-2} +35 a^2 z^4-4 z^4 a^{-2} -31 a^2 z^2+3 z^2 a^{-2} -2 a^2 z^{-2} +13 a^2- a^{-2} +a z^9-3 a z^7+2 z^7 a^{-1} -3 a z^5-8 z^5 a^{-1} +10 a z^3+6 z^3 a^{-1} -7 a z-z a^{-1} +2 a z^{-1} +2 z^8-8 z^6+6 z^4-2 z^2- z^{-2} +3$ (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 \
-7-6-5-4-3-2-101234χ
7           11
5          1 -1
3         21 1
1       121  0
-1      142   1
-3     123    2
-5    142     1
-7   113      3
-9   11       0
-11 111        1
-13            0
-151           -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-3$ $i=-1$ $i=1$ $r=-7$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{4}$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=0$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{4}$ ${\mathbb Z}$ $r=1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=2$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=3$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=4$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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