L8n3

L8n2

L8n4

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L8n3 at Knotilus!
	L8n3 is $8_{7}^{3}$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L8n3's Link Presentations]

Planar diagram presentation	X6172 X5,12,6,13 X3849 X13,2,14,3 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	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {uv^{2}w^{2}-1}{{\sqrt {u}}vw}}$ (db)
Jones polynomial	$q^{-3}+q^{-5}+q^{-7}+q^{-9}$ (db)
Signature	-6 (db)
HOMFLY-PT polynomial	$a^{10}z^{-2}+a^{10}-z^{4}a^{8}-5z^{2}a^{8}-2a^{8}z^{-2}-6a^{8}+z^{6}a^{6}+6z^{4}a^{6}+10z^{2}a^{6}+a^{6}z^{-2}+5a^{6}$ (db)
Kauffman polynomial	$a^{12}+a^{10}z^{2}+a^{10}z^{-2}-3a^{10}+a^{9}z^{5}-5a^{9}z^{3}+6a^{9}z-2a^{9}z^{-1}+a^{8}z^{6}-6a^{8}z^{4}+11a^{8}z^{2}+2a^{8}z^{-2}-8a^{8}+a^{7}z^{5}-5a^{7}z^{3}+6a^{7}z-2a^{7}z^{-1}+a^{6}z^{6}-6a^{6}z^{4}+10a^{6}z^{2}+a^{6}z^{-2}-5a^{6}$ (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 -5             1 1 -7             1 1 -9         1     1 -11     1         1 -13     2 1       1 -15 1             1 -17 2 1           1 -19 1             1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-7$ $i=-5$ $i=-3$ $r=-6$ ${\mathbb {Z} }$ ${\mathbb {Z} }^{2}$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }^{2}$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-1$ $r=0$ ${\mathbb {Z} }$ ${\mathbb {Z} }$

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.

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