# L11a1

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a1 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{(u-1) (v-1) \left(v^4-5 v^3+7 v^2-5 v+1\right)}{\sqrt{u} v^{5/2}}$ (db) Jones polynomial $-q^{13/2}+4 q^{11/2}-9 q^{9/2}+15 q^{7/2}-21 q^{5/2}+24 q^{3/2}-25 \sqrt{q}+\frac{21}{\sqrt{q}}-\frac{16}{q^{3/2}}+\frac{10}{q^{5/2}}-\frac{5}{q^{7/2}}+\frac{1}{q^{9/2}}$ (db) Signature 1 (db) HOMFLY-PT polynomial $-z^7 a^{-1} +2 a z^5-3 z^5 a^{-1} +2 z^5 a^{-3} -a^3 z^3+3 a z^3-4 z^3 a^{-1} +4 z^3 a^{-3} -z^3 a^{-5} -a z+2 z a^{-3} -z a^{-5} +a^3 z^{-1} -2 a z^{-1} +2 a^{-1} z^{-1} - a^{-3} z^{-1}$ (db) Kauffman polynomial $-2 z^{10} a^{-2} -2 z^{10}-7 a z^9-13 z^9 a^{-1} -6 z^9 a^{-3} -9 a^2 z^8-17 z^8 a^{-2} -9 z^8 a^{-4} -17 z^8-5 a^3 z^7+5 a z^7+12 z^7 a^{-1} -6 z^7 a^{-3} -8 z^7 a^{-5} -a^4 z^6+20 a^2 z^6+36 z^6 a^{-2} +9 z^6 a^{-4} -4 z^6 a^{-6} +44 z^6+10 a^3 z^5+14 a z^5+16 z^5 a^{-1} +25 z^5 a^{-3} +12 z^5 a^{-5} -z^5 a^{-7} +a^4 z^4-11 a^2 z^4-19 z^4 a^{-2} -z^4 a^{-4} +5 z^4 a^{-6} -25 z^4-4 a^3 z^3-9 a z^3-14 z^3 a^{-1} -18 z^3 a^{-3} -8 z^3 a^{-5} +z^3 a^{-7} +2 z^2 a^{-2} -z^2 a^{-4} -2 z^2 a^{-6} +z^2-a^3 z-4 a z-2 z a^{-1} +3 z a^{-3} +2 z a^{-5} +1+a^3 z^{-1} +2 a z^{-1} +2 a^{-1} z^{-1} + a^{-3} z^{-1}$ (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 \
-5-4-3-2-10123456χ
14           11
12          3 -3
10         61 5
8        93  -6
6       126   6
4      129    -3
2     1312     1
0    1014      4
-2   611       -5
-4  410        6
-6 16         -5
-8 4          4
-101           -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=0$ $i=2$ $r=-5$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-2$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=-1$ ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{10}$ $r=0$ ${\mathbb Z}^{14}\oplus{\mathbb Z}_2^{11}$ ${\mathbb Z}^{13}$ $r=1$ ${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{12}$ ${\mathbb Z}^{12}$ $r=2$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{12}$ ${\mathbb Z}^{12}$ $r=3$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=5$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=6$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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