# L11a313

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a313 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{(t(1)-1) (t(2)-1) (t(1)+t(2)) (t(1) t(2)+1) \left(t(2)^2-t(2)+1\right)}{t(1)^{3/2} t(2)^{5/2}}$ (db) Jones polynomial $-q^{11/2}+3 q^{9/2}-6 q^{7/2}+10 q^{5/2}-13 q^{3/2}+14 \sqrt{q}-\frac{16}{\sqrt{q}}+\frac{13}{q^{3/2}}-\frac{10}{q^{5/2}}+\frac{6}{q^{7/2}}-\frac{3}{q^{9/2}}+\frac{1}{q^{11/2}}$ (db) Signature -1 (db) HOMFLY-PT polynomial $-a^3 z^5-z^5 a^{-3} -3 a^3 z^3-3 z^3 a^{-3} -2 a^3 z-2 z a^{-3} +a z^7+z^7 a^{-1} +4 a z^5+4 z^5 a^{-1} +5 a z^3+5 z^3 a^{-1} +2 a z+2 z a^{-1} +a z^{-1} - a^{-1} z^{-1}$ (db) Kauffman polynomial $-2 z^{10} a^{-2} -2 z^{10}-5 a z^9-9 z^9 a^{-1} -4 z^9 a^{-3} -7 a^2 z^8+z^8 a^{-2} -3 z^8 a^{-4} -3 z^8-7 a^3 z^7+7 a z^7+30 z^7 a^{-1} +15 z^7 a^{-3} -z^7 a^{-5} -5 a^4 z^6+13 a^2 z^6+13 z^6 a^{-2} +12 z^6 a^{-4} +19 z^6-3 a^5 z^5+12 a^3 z^5+4 a z^5-33 z^5 a^{-1} -18 z^5 a^{-3} +4 z^5 a^{-5} -a^6 z^4+5 a^4 z^4-9 a^2 z^4-16 z^4 a^{-2} -14 z^4 a^{-4} -17 z^4+3 a^5 z^3-11 a^3 z^3-11 a z^3+19 z^3 a^{-1} +12 z^3 a^{-3} -4 z^3 a^{-5} +a^6 z^2-a^4 z^2+7 z^2 a^{-2} +5 z^2 a^{-4} +4 z^2+4 a^3 z+4 a z-4 z a^{-1} -4 z a^{-3} +1-a z^{-1} - a^{-1} z^{-1}$ (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
12           11
10          2 -2
8         41 3
6        62  -4
4       74   3
2      76    -1
0     97     2
-2    69      3
-4   47       -3
-6  26        4
-8 14         -3
-10 2          2
-121           -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $r=-5$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-2$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-1$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=0$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{9}$ $r=1$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=2$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=3$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=4$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=5$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=6$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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