L8a9

L8a8

L8a10

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L8a9 at Knotilus!
	L8a9 is $8_{8}^{2}$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L8a9's Link Presentations]

Planar diagram presentation	X8192 X10,4,11,3 X16,10,7,9 X2738 X4,16,5,15 X12,5,13,6 X14,11,15,12 X6,13,1,14
Gauss code	{1, -4, 2, -5, 6, -8}, {4, -1, 3, -2, 7, -6, 8, -7, 5, -3}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {t(2)^{2}t(1)^{2}-2t(2)t(1)^{2}+t(1)^{2}-2t(2)^{2}t(1)+5t(2)t(1)-2t(1)+t(2)^{2}-2t(2)+1}{t(1)t(2)}}$ (db)
Jones polynomial	$-{\frac {3}{q^{9/2}}}+{\frac {4}{q^{7/2}}}+q^{5/2}-{\frac {6}{q^{5/2}}}-3q^{3/2}+{\frac {6}{q^{3/2}}}+{\frac {1}{q^{11/2}}}+4{\sqrt {q}}-{\frac {6}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{5}(-z)+2a^{3}z^{3}+3a^{3}z+a^{3}z^{-1}-az^{5}-3az^{3}+z^{3}a^{-1}-4az-az^{-1}+za^{-1}$ (db)
Kauffman polynomial	$a^{6}z^{4}-a^{6}z^{2}+3a^{5}z^{5}-5a^{5}z^{3}+2a^{5}z+3a^{4}z^{6}-3a^{4}z^{4}+a^{3}z^{7}+6a^{3}z^{5}-13a^{3}z^{3}+7a^{3}z-a^{3}z^{-1}+6a^{2}z^{6}-8a^{2}z^{4}+z^{4}a^{-2}+2a^{2}z^{2}-z^{2}a^{-2}+a^{2}+az^{7}+6az^{5}+3z^{5}a^{-1}-13az^{3}-5z^{3}a^{-1}+7az+2za^{-1}-az^{-1}+3z^{6}-3z^{4}$ (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   \ -5 -4 -3 -2 -1 0 1 2 3 χ 6                 1 -1 4               2   2 2             2 1   -1 0           4 2     2 -2         3 3       0 -4       3 3         0 -6     2 4           2 -8   1 2             -1 -10   2               2 -12 1                 -1

Integral Khovanov Homology (db, data source)

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

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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