L8n8

L8n7

L9a1

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L8n8 at Knotilus!
	L8n8 is $8_{3}^{4}$ in the Rolfsen table of links.

Detail from an 18th century royal decree, Vietnam.

Link Presentations

[edit Notes on L8n8's Link Presentations]

Planar diagram presentation	X6172 X2536 X16,11,13,12 X3,11,4,10 X9,1,10,4 X7,15,8,14 X13,5,14,8 X12,15,9,16
Gauss code	{1, -2, -4, 5}, {2, -1, -6, 7}, {-5, 4, 3, -8}, {-7, 6, 8, -3}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(t(2)-t(3))(t(1)-t(4))}{{\sqrt {t(1)}}{\sqrt {t(2)}}{\sqrt {t(3)}}{\sqrt {t(4)}}}}$ (db)
Jones polynomial	$-q^{7/2}-q^{3/2}-2{\sqrt {q}}-{\frac {2}{\sqrt {q}}}-{\frac {1}{q^{3/2}}}-{\frac {1}{q^{7/2}}}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a^{3}z^{-3}-a^{-3}z^{-3}+a^{3}z+2a^{3}z^{-1}-za^{-3}-2a^{-3}z^{-1}-az^{3}-3az^{-3}+z^{3}a^{-1}+3a^{-1}z^{-3}-5az-6az^{-1}+5za^{-1}+6a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-a^{3}z^{5}-az^{5}-z^{5}a^{-1}-z^{5}a^{-3}-a^{2}z^{4}-z^{4}a^{-2}-2z^{4}+5a^{3}z^{3}+7az^{3}+7z^{3}a^{-1}+5z^{3}a^{-3}+6a^{2}z^{2}+6z^{2}a^{-2}+12z^{2}-6a^{3}z-14az-14za^{-1}-6za^{-3}-8a^{2}-8a^{-2}-15+4a^{3}z^{-1}+9az^{-1}+9a^{-1}z^{-1}+4a^{-3}z^{-1}+3a^{2}z^{-2}+3a^{-2}z^{-2}+6z^{-2}-a^{3}z^{-3}-3az^{-3}-3a^{-1}z^{-3}-a^{-3}z^{-3}$ (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 8                 1 1 6                 1 1 4             1     1 2         3         3 0       1 6 1       4 -2         3         3 -4     1             1 -6 1                 1 -8 1                 1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-2$ $i=0$ $i=2$ $r=-4$ ${\mathbb {Z} }$ ${\mathbb {Z} }$ $r=-3$ $r=-2$ ${\mathbb {Z} }$ $r=-1$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=0$ ${\mathbb {Z} }^{3}$ ${\mathbb {Z} }^{6}$ ${\mathbb {Z} }^{3}$ $r=1$ ${\mathbb {Z} }$ $r=2$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=3$ $r=4$ ${\mathbb {Z} }$ ${\mathbb {Z} }$

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.

L8n7 L9a1