# L8a8

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L8a8 at Knotilus! L8a8 is $8^2_{7}$ in the Rolfsen table of links, and the "seized Carrick bend" of practical knot-tying.  The simplest Celtic or pseudo-Celtic linear decorative knot.  Decorative variant with big loops at ends  Coat of arms of Bressauc, Jura, Switzerland.

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{u^2 v^2-2 u^2 v+u^2-2 u v^2+3 u v-2 u+v^2-2 v+1}{u v}$ (db) Jones polynomial $-q^{9/2}+3 q^{7/2}-4 q^{5/2}+5 q^{3/2}-6 \sqrt{q}+\frac{4}{\sqrt{q}}-\frac{4}{q^{3/2}}+\frac{2}{q^{5/2}}-\frac{1}{q^{7/2}}$ (db) Signature 1 (db) HOMFLY-PT polynomial $-z^3 a^{-3} +a^3 z-z a^{-3} +a^3 z^{-1} +z^5 a^{-1} -2 a z^3+3 z^3 a^{-1} -4 a z+3 z a^{-1} -a z^{-1}$ (db) Kauffman polynomial $-a z^7-z^7 a^{-1} -2 a^2 z^6-3 z^6 a^{-2} -5 z^6-a^3 z^5-2 a z^5-5 z^5 a^{-1} -4 z^5 a^{-3} +5 a^2 z^4+z^4 a^{-2} -3 z^4 a^{-4} +9 z^4+3 a^3 z^3+10 a z^3+12 z^3 a^{-1} +4 z^3 a^{-3} -z^3 a^{-5} -2 a^2 z^2+2 z^2 a^{-2} +2 z^2 a^{-4} -2 z^2-3 a^3 z-7 a z-6 z a^{-1} -2 z a^{-3} -a^2+a^3 z^{-1} +a z^{-1}$ (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 \
-4-3-2-101234χ
10        11
8       2 -2
6      21 1
4     32  -1
2    32   1
0   24    2
-2  22     0
-4 13      2
-6 1       -1
-81        1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=0$ $i=2$ $r=-4$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=-1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=0$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}$ $r=1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=3$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=4$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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