L8n1

L8a21

L8n2

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L8n1 at Knotilus!
	L8n1 is $8_{16}^{2}$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L8n1's Link Presentations]

Planar diagram presentation	X6172 X14,7,15,8 X4,15,1,16 X9,12,10,13 X3849 X5,11,6,10 X11,5,12,16 X13,2,14,3
Gauss code	{1, 8, -5, -3}, {-6, -1, 2, 5, -4, 6, -7, 4, -8, -2, 3, 7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {uv^{3}-2uv^{2}-2v+1}{{\sqrt {u}}v^{3/2}}}$ (db)
Jones polynomial	${\frac {2}{q^{9/2}}}-{\frac {2}{q^{7/2}}}+{\frac {2}{q^{5/2}}}-{\frac {2}{q^{3/2}}}-{\frac {2}{q^{11/2}}}-{\sqrt {q}}+{\frac {1}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{7}z^{-1}-a^{5}z^{3}-3a^{5}z-2a^{5}z^{-1}+a^{3}z^{5}+4a^{3}z^{3}+4a^{3}z+2a^{3}z^{-1}-az^{3}-3az-az^{-1}$ (db)
Kauffman polynomial	$3a^{7}z-a^{7}z^{-1}+a^{6}z^{4}+2a^{5}z^{5}-6a^{5}z^{3}+7a^{5}z-2a^{5}z^{-1}+a^{4}z^{6}-2a^{4}z^{4}+a^{4}z^{2}-a^{4}+3a^{3}z^{5}-10a^{3}z^{3}+8a^{3}z-2a^{3}z^{-1}+a^{2}z^{6}-3a^{2}z^{4}+a^{2}z^{2}+az^{5}-4az^{3}+4az-az^{-1}$ (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 χ 2             1 1 0               0 -2         2 1   1 -4       1 1     0 -6     1 1       0 -8   1 1         0 -10 1 1           0 -12 2             2

Integral Khovanov Homology (db, data source)

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

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.

L8a21 L8n2