L8a8

L8a7

L8a9

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L8a8 at Knotilus!
	L8a8 is $8_{7}^{2}$ 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.

Link Presentations

[edit Notes on L8a8's Link Presentations]

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}

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {u^{2}v^{2}-2u^{2}v+u^{2}-2uv^{2}+3uv-2u+v^{2}-2v+1}{uv}}$ (db)
Jones polynomial	$-q^{9/2}+3q^{7/2}-4q^{5/2}+5q^{3/2}-6{\sqrt {q}}+{\frac {4}{\sqrt {q}}}-{\frac {4}{q^{3/2}}}+{\frac {2}{q^{5/2}}}-{\frac {1}{q^{7/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{3}a^{-3}+a^{3}z-za^{-3}+a^{3}z^{-1}+z^{5}a^{-1}-2az^{3}+3z^{3}a^{-1}-4az+3za^{-1}-az^{-1}$ (db)
Kauffman polynomial	$-az^{7}-z^{7}a^{-1}-2a^{2}z^{6}-3z^{6}a^{-2}-5z^{6}-a^{3}z^{5}-2az^{5}-5z^{5}a^{-1}-4z^{5}a^{-3}+5a^{2}z^{4}+z^{4}a^{-2}-3z^{4}a^{-4}+9z^{4}+3a^{3}z^{3}+10az^{3}+12z^{3}a^{-1}+4z^{3}a^{-3}-z^{3}a^{-5}-2a^{2}z^{2}+2z^{2}a^{-2}+2z^{2}a^{-4}-2z^{2}-3a^{3}z-7az-6za^{-1}-2za^{-3}-a^{2}+a^{3}z^{-1}+az^{-1}$ (db)

Khovanov Homology

The coefficients of the monomials $t^{r}q^{j}$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$).

\   r    \    j   \ -4 -3 -2 -1 0 1 2 3 4 χ 10                 1 1 8               2   -2 6             2 1   1 4           3 2     -1 2         3 2       1 0       2 4         2 -2     2 2           0 -4   1 3             2 -6   1               -1 -8 1                 1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=0$ $i=2$ $r=-4$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }^{2}$ $r=-1$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=0$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{3}$ $r=1$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=2$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=3$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=4$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

Computer Talk

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

Modifying This Page

Read me first: Modifying Knot Pages See/edit the Link Page master template (intermediate). See/edit the Link_Splice_Base (expert). Back to the top.

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