# L8a8

## Contents

 (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L8a8's page at Knotilus. Visit L8a8's page at the original Knot Atlas. 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. Alternate decorative variant Circular arcs only Decorative variant with big loops at ends Coat of arms of Bressauc, Jura, Switzerland.

 Planar diagram presentation X8192 X10,4,11,3 X16,10,7,9 X2738 X14,12,15,11 X12,5,13,6 X4,13,5,14 X6,16,1,15 Gauss code {1, -4, 2, -7, 6, -8}, {4, -1, 3, -2, 5, -6, 7, -5, 8, -3}

### 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 −z3a−3 + a3z−za−3 + a3z−1 + z5a−1−2az3 + 3z3a−1−4az + 3za−1−az−1 (db) Kauffman polynomial −az7−z7a−1−2a2z6−3z6a−2−5z6−a3z5−2az5−5z5a−1−4z5a−3 + 5a2z4 + z4a−2−3z4a−4 + 9z4 + 3a3z3 + 10az3 + 12z3a−1 + 4z3a−3−z3a−5−2a2z2 + 2z2a−2 + 2z2a−4−2z2−3a3z−7az−6za−1−2za−3−a2 + a3z−1 + az−1 (db)

### Khovanov Homology

 The coefficients of the monomials trqj are shown, along with their alternating sums χ (fixed j, alternation over r). The squares with yellow highlighting are those on the "critical diagonals", where j−2r = s + 1 or j−2r = s−1, where s = 1 is the signature of L8a8. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
\ 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.

Read me first: Modifying Knot Pages

See/edit the Link Page master template (intermediate).