L8n6

L8n5

L8n7

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L8n6 at Knotilus!
	L8n6 is $8_{10}^{3}$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L8n6's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X11,16,12,13 X7,14,8,15 X13,8,14,9 X15,12,16,5 X2536 X4,9,1,10
Gauss code	{1, -7, 2, -8}, {-5, 4, -6, 3}, {7, -1, -4, 5, 8, -2, -3, 6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(t(3)-1)(t(3)+1)(t(1)t(2)+t(3))}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)^{3/2}}}$ (db)
Jones polynomial	$q^{-2}+q^{-6}+q^{-7}+q^{-9}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{10}z^{-2}-2a^{8}z^{-2}-2a^{8}+a^{6}z^{-2}+z^{4}a^{4}+4z^{2}a^{4}+2a^{4}$ (db)
Kauffman polynomial	$a^{10}z^{6}-6a^{10}z^{4}+10a^{10}z^{2}+a^{10}z^{-2}-6a^{10}+a^{9}z^{5}-6a^{9}z^{3}+8a^{9}z-2a^{9}z^{-1}+a^{8}z^{6}-7a^{8}z^{4}+14a^{8}z^{2}+2a^{8}z^{-2}-9a^{8}+a^{7}z^{5}-6a^{7}z^{3}+8a^{7}z-2a^{7}z^{-1}+a^{6}z^{-2}-2a^{6}+a^{4}z^{4}-4a^{4}z^{2}+2a^{4}$ (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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 χ -3                 1 1 -5                 1 1 -7           1 1     0 -9                   0 -11       1 3 1       1 -13         2         2 -15     1             1 -17 1                 1 -19 1                 1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-5$ $i=-3$ $i=-1$ $r=-8$ ${\mathbb {Z} }$ ${\mathbb {Z} }$ $r=-7$ $r=-6$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{2}$ ${\mathbb {Z} }^{3}$ $r=-3$ ${\mathbb {Z} }$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-1$ $r=0$ ${\mathbb {Z} }$ ${\mathbb {Z} }$

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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