# L11n362

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n362 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{(t(1)-1) (t(3)-1)^2 \left(t(3) t(2)^2+t(3)^2 t(2)-t(3) t(2)+t(2)+t(3)\right)}{\sqrt{t(1)} t(2) t(3)^2}$ (db) Jones polynomial $3 q^7-5 q^6+10 q^5-12 q^4+14 q^3-13 q^2- q^{-2} +11 q+4 q^{-1} -7$ (db) Signature 2 (db) HOMFLY-PT polynomial $z^6 a^{-2} +z^6 a^{-4} +2 z^4 a^{-2} +3 z^4 a^{-4} -z^4 a^{-6} -z^4+4 z^2 a^{-4} -3 z^2 a^{-6} -z^2- a^{-2} +3 a^{-4} -4 a^{-6} + a^{-8} +1+ a^{-4} z^{-2} -2 a^{-6} z^{-2} + a^{-8} z^{-2}$ (db) Kauffman polynomial $6 z^4 a^{-8} -14 z^2 a^{-8} - a^{-8} z^{-2} +8 a^{-8} +3 z^7 a^{-7} -6 z^5 a^{-7} +10 z^3 a^{-7} -10 z a^{-7} +2 a^{-7} z^{-1} +5 z^8 a^{-6} -16 z^6 a^{-6} +33 z^4 a^{-6} -33 z^2 a^{-6} -2 a^{-6} z^{-2} +15 a^{-6} +2 z^9 a^{-5} +3 z^7 a^{-5} -13 z^5 a^{-5} +20 z^3 a^{-5} -12 z a^{-5} +2 a^{-5} z^{-1} +10 z^8 a^{-4} -25 z^6 a^{-4} +33 z^4 a^{-4} -24 z^2 a^{-4} - a^{-4} z^{-2} +9 a^{-4} +2 z^9 a^{-3} +6 z^7 a^{-3} -18 z^5 a^{-3} +14 z^3 a^{-3} -3 z a^{-3} +5 z^8 a^{-2} -5 z^6 a^{-2} -z^4 a^{-2} -3 z^2 a^{-2} +2 a^{-2} +6 z^7 a^{-1} +a z^5-10 z^5 a^{-1} -a z^3+3 z^3 a^{-1} -z a^{-1} +4 z^6-7 z^4+2 z^2+1$ (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 \
-3-2-10123456χ
15         33
13        42-2
11       61 5
9      64  -2
7     86   2
5    56    1
3   68     -2
1  37      4
-1 14       -3
-3 3        3
-51         -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=1$ $i=3$ $r=-3$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=0$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{6}$ $r=1$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=2$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=3$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=4$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=5$ ${\mathbb Z}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=6$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{3}$

### Computer Talk

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