L8n5

L8n4

L8n6

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L8n5 at Knotilus!
	L8n5 is is $8_{9}^{3}$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L8n5's Link Presentations]

Planar diagram presentation	X6172 X5,12,6,13 X8493 X2,14,3,13 X14,7,15,8 X9,16,10,11 X11,10,12,5 X4,15,1,16
Gauss code	{1, -4, 3, -8}, {-2, -1, 5, -3, -6, 7}, {-7, 2, 4, -5, 8, 6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(t(1)-1)(t(2)-1)(t(3)-1)}{{\sqrt {t(1)}}{\sqrt {t(2)}}{\sqrt {t(3)}}}}$ (db)
Jones polynomial	$q^{-7}-2q^{-6}+3q^{-5}-2q^{-4}+4q^{-3}-2q^{-2}+2q^{-1}$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$a^{6}z^{2}+a^{6}z^{-2}+a^{6}-a^{4}z^{4}-3a^{4}z^{2}-2a^{4}z^{-2}-4a^{4}+2a^{2}z^{2}+a^{2}z^{-2}+3a^{2}$ (db)
Kauffman polynomial	$a^{8}z^{4}-2a^{8}z^{2}+a^{8}+2a^{7}z^{5}-4a^{7}z^{3}+a^{6}z^{6}-2a^{6}z^{2}+a^{6}z^{-2}-a^{6}+3a^{5}z^{5}-5a^{5}z^{3}+4a^{5}z-2a^{5}z^{-1}+a^{4}z^{6}-a^{4}z^{4}+3a^{4}z^{2}+2a^{4}z^{-2}-4a^{4}+a^{3}z^{5}-a^{3}z^{3}+4a^{3}z-2a^{3}z^{-1}+3a^{2}z^{2}+a^{2}z^{-2}-3a^{2}$ (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   \ -6 -5 -4 -3 -2 -1 0 χ -1             2 2 -3           2 2 0 -5         2     2 -7         2     2 -9     3 2       1 -11   1 2         1 -13   1           -1 -15 1             1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-3$ $i=-1$ $r=-6$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }^{3}$ $r=-3$ ${\mathbb {Z} }^{2}$ $r=-2$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-1$ ${\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=0$ ${\mathbb {Z} }^{2}$ ${\mathbb {Z} }^{2}$

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.

L8n4 L8n6