# L11a358

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a358 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{t(2)^3 t(1)^4-t(2)^2 t(1)^4+t(2)^4 t(1)^3-3 t(2)^3 t(1)^3+3 t(2)^2 t(1)^3-t(2) t(1)^3-t(2)^4 t(1)^2+3 t(2)^3 t(1)^2-3 t(2)^2 t(1)^2+3 t(2) t(1)^2-t(1)^2-t(2)^3 t(1)+3 t(2)^2 t(1)-3 t(2) t(1)+t(1)-t(2)^2+t(2)}{t(1)^2 t(2)^2}$ (db) Jones polynomial $-q^{15/2}+3 q^{13/2}-5 q^{11/2}+7 q^{9/2}-9 q^{7/2}+9 q^{5/2}-9 q^{3/2}+7 \sqrt{q}-\frac{6}{\sqrt{q}}+\frac{3}{q^{3/2}}-\frac{2}{q^{5/2}}+\frac{1}{q^{7/2}}$ (db) Signature 3 (db) HOMFLY-PT polynomial $z^7 a^{-1} +z^7 a^{-3} -a z^5+5 z^5 a^{-1} +4 z^5 a^{-3} -z^5 a^{-5} -4 a z^3+8 z^3 a^{-1} +3 z^3 a^{-3} -3 z^3 a^{-5} -3 a z+6 z a^{-1} -z a^{-3} -z a^{-5} + a^{-1} z^{-1} - a^{-3} z^{-1}$ (db) Kauffman polynomial $-z^{10} a^{-2} -z^{10}-2 a z^9-5 z^9 a^{-1} -3 z^9 a^{-3} -a^2 z^8-2 z^8 a^{-2} -5 z^8 a^{-4} +2 z^8+12 a z^7+24 z^7 a^{-1} +6 z^7 a^{-3} -6 z^7 a^{-5} +6 a^2 z^6+19 z^6 a^{-2} +11 z^6 a^{-4} -6 z^6 a^{-6} +8 z^6-24 a z^5-37 z^5 a^{-1} +2 z^5 a^{-3} +10 z^5 a^{-5} -5 z^5 a^{-7} -11 a^2 z^4-23 z^4 a^{-2} -4 z^4 a^{-4} +7 z^4 a^{-6} -3 z^4 a^{-8} -20 z^4+19 a z^3+25 z^3 a^{-1} -z^3 a^{-3} -2 z^3 a^{-5} +4 z^3 a^{-7} -z^3 a^{-9} +6 a^2 z^2+7 z^2 a^{-2} +z^2 a^{-4} -z^2 a^{-6} +z^2 a^{-8} +10 z^2-5 a z-9 z a^{-1} -3 z a^{-3} -z a^{-7} - a^{-2} + a^{-1} z^{-1} + a^{-3} z^{-1}$ (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
16           11
14          2 -2
12         31 2
10        42  -2
8       53   2
6      55    0
4     44     0
2    46      2
0   23       -1
-2  14        3
-4 12         -1
-6 1          1
-81           -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=2$ $i=4$ $r=-5$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=0$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{4}$ $r=1$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=2$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{5}$ $r=3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=4$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=5$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=6$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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