L8a4

L8a3

L8a5

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L8a4 at Knotilus!
	L8a4 is $8_{12}^{2}$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L8a4's Link Presentations]

Planar diagram presentation	X6172 X10,4,11,3 X12,10,13,9 X16,13,5,14 X14,7,15,8 X8,15,9,16 X2536 X4,12,1,11
Gauss code	{1, -7, 2, -8}, {7, -1, 5, -6, 3, -2, 8, -3, 4, -5, 6, -4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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