L8a19

L8a18

L8a20

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L8a19 at Knotilus!
	L8a19 is $8_{6}^{3}$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L8a19's Link Presentations]

Planar diagram presentation	X6172 X12,4,13,3 X14,5,15,6 X8,12,9,11 X16,8,11,7 X10,13,5,14 X2,9,3,10 X4,16,1,15
Gauss code	{1, -7, 2, -8}, {3, -1, 5, -4, 7, -6}, {4, -2, 6, -3, 8, -5}

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(t(1)-1)(t(3)t(2)-t(2)+1)(t(2)t(3)-t(3)+1)}{{\sqrt {t(1)}}t(2)t(3)}}$ (db)
Jones polynomial	$q^{4}+q^{-4}-3q^{3}-3q^{-3}+5q^{2}+5q^{-2}-5q-5q^{-1}+8$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$-z^{6}+a^{2}z^{4}+z^{4}a^{-2}-4z^{4}+2a^{2}z^{2}+2z^{2}a^{-2}-5z^{2}+a^{2}+a^{-2}-2+a^{2}z^{-2}+a^{-2}z^{-2}-2z^{-2}$ (db)
Kauffman polynomial	$2az^{7}+2z^{7}a^{-1}+4a^{2}z^{6}+4z^{6}a^{-2}+8z^{6}+3a^{3}z^{5}+az^{5}+z^{5}a^{-1}+3z^{5}a^{-3}+a^{4}z^{4}-7a^{2}z^{4}-7z^{4}a^{-2}+z^{4}a^{-4}-16z^{4}-4a^{3}z^{3}-4az^{3}-4z^{3}a^{-1}-4z^{3}a^{-3}-a^{4}z^{2}+5a^{2}z^{2}+5z^{2}a^{-2}-z^{2}a^{-4}+12z^{2}+2az+2za^{-1}-2a^{2}-2a^{-2}-3-2az^{-1}-2a^{-1}z^{-1}+a^{2}z^{-2}+a^{-2}z^{-2}+2z^{-2}$ (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   \ -4 -3 -2 -1 0 1 2 3 4 χ 9                 1 1 7               2   -2 5             3 1   2 3           3 3     0 1         5 2       3 -1       2 5         3 -3     3 3           0 -5   1 3             2 -7   2               -2 -9 1                 1

Integral Khovanov Homology (db, data source)

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

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.

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