L8n4

L8n3

L8n5

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L8n4 at Knotilus!
	L8n4 is $8_{8}^{3}$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L8n4's Link Presentations]

Planar diagram presentation	X6172 X5,12,6,13 X3849 X2,14,3,13 X14,7,15,8 X9,16,10,11 X11,10,12,5 X15,1,16,4
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 $u$, $v$, $w$, ...)	${\frac {t(3)t(2)^{2}-t(1)t(3)^{2}t(2)+t(1)t(3)t(2)-t(3)t(2)+t(2)-t(1)t(3)}{{\sqrt {t(1)}}t(2)t(3)}}$ (db)
Jones polynomial	$q^{-7}-q^{-6}+2q^{-5}-2q^{-4}+3q^{-3}-q^{-2}+2q^{-1}$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$a^{6}z^{2}+a^{6}z^{-2}+2a^{6}-a^{4}z^{4}-4a^{4}z^{2}-2a^{4}z^{-2}-6a^{4}+2a^{2}z^{2}+a^{2}z^{-2}+4a^{2}$ (db)
Kauffman polynomial	$z^{4}a^{8}-3z^{2}a^{8}+a^{8}+z^{5}a^{7}-2z^{3}a^{7}+z^{6}a^{6}-3z^{4}a^{6}+4z^{2}a^{6}+a^{6}z^{-2}-3a^{6}+2z^{5}a^{5}-5z^{3}a^{5}+6za^{5}-2a^{5}z^{-1}+z^{6}a^{4}-4z^{4}a^{4}+10z^{2}a^{4}+2a^{4}z^{-2}-8a^{4}+z^{5}a^{3}-3z^{3}a^{3}+6za^{3}-2a^{3}z^{-1}+3z^{2}a^{2}+a^{2}z^{-2}-5a^{2}$ (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           1 2 1 -5         2     2 -7       1 2     1 -9     1 1       0 -11     1         1 -13 1 1           0 -15 1             1

Integral Khovanov Homology (db, data source)

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

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.

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