L2a1

[http://s1.shard.jp/frhorton/9nls8flts.html african american dos hair wedding ] [http://s1.shard.jp/olharder/agencias-auto.html auto car trader used ] link automotive executive jobs [http://s1.shard.jp/losaul/the-australian.html australia gutenberg project ] [http://s1.shard.jp/frhorton/ocdp2flvo.html african american independent film ] [http://s1.shard.jp/galeach/new83.html acatalasia ] asia house insurance ppm [http://s1.shard.jp/olharder/bxautozip.html auto trader muscle cars ] [http://s1.shard.jp/olharder/autodesk-inventor.html aliens autopsy ] [http://s1.shard.jp/losaul/real-estate.html department of education and training victoria australia ] [http://s1.shard.jp/frhorton/x5dh8y75v.html south africa lion ] [http://s1.shard.jp/losaul/aborigines--dreamtime.html australia telephone directory online ] [http://s1.shard.jp/bireba/nortons-antivirus.html symantec antivirus 9.0.3 ] [http://s1.shard.jp/olharder/wes-finch-auto-plaza.html auto insurance illinois ] [http://s1.shard.jp/frhorton/rlw3nqlyf.html safari africa luxury ] [http://s1.shard.jp/olharder/buy-and-sell-autos.html princess auto parts ] [http://s1.shard.jp/bireba/innoculate-antivirus.html uninstall symantec antivirus password ] [http://s1.shard.jp/losaul/property-for.html affordable web hosting australia aussietogocomau ] antivirus personal free [http://s1.shard.jp/frhorton/lt8tyfnvp.html african american romantic poetry ] [http://s1.shard.jp/bireba/macintosh-antivirus.html avg antivirus professional edition ] [http://s1.shard.jp/olharder/kurt-cobain-autograph.html lab automation career job ] [http://s1.shard.jp/olharder/autograph-boxing.html abstracting automated bloomberg concept datastream reuters sentence text ] [http://s1.shard.jp/olharder/rockies-auto-colorado.html scotts auto country incorporated ] bill gates africa donation [http://s1.shard.jp/losaul/informed-sources.html missing people australia ] [http://s1.shard.jp/bireba/noton-antivirus.html antivirus review best ] [http://s1.shard.jp/bireba/antivirus-download.html nod32 antivirus software ] [http://s1.shard.jp/losaul/the-barrier-reef.html roxy australia ] [http://s1.shard.jp/galeach/new69.html iasia ] [http://s1.shard.jp/bireba/quickheal-antivirus.html unistall norton antivirus 2005 ] [http://s1.shard.jp/losaul/advanced-driver.html labour day 2005 australia ] [http://s1.shard.jp/losaul/australian-census.html virgin air lines australia ] [http://s1.shard.jp/galeach/new59.html asian male models portfolio ] [http://s1.shard.jp/galeach/new43.html india and southeast asia ] [http://s1.shard.jp/bireba/cheap-norton-antivirus.html 64 bit antivirus software ] [http://s1.shard.jp/olharder/auto-panel-plus.html superior auto in ] [http://s1.shard.jp/bireba/antivirus-2004-download.html norton antivirus definitions disk ] dog bike trailer australia [http://s1.shard.jp/olharder/accessory-automotive.html automobile future trend ] sitemap [http://s1.shard.jp/frhorton/bzsc3x6qn.html african american female first millionaire ] [http://s1.shard.jp/olharder/auto-automotriz.html automotive dallas flower texas ] [http://s1.shard.jp/bireba/download-norton.html panda antivirus free ] [http://s1.shard.jp/frhorton/z7u5veip8.html experience africa ] [http://s1.shard.jp/bireba/avg-antivirus-73.html symantec antivirus corporate edition 9 reviews ] [http://s1.shard.jp/olharder/pioneer-auto-part.html automatic car hire malaga airport ] http://www.textceltba.com

L2a1

L4a1

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L2a1 at Knotilus!
	L2a1 is $2_{1}^{2}$ in Rolfsen's table of links. It is also known as the "Hopf Link".

Japanese family emblem	Linked hearts	expanded Kolam Two-hearts [1]	Logo
In Star of David form on old Jewish building in Prague	Three wreaths linked as two L2a1 configurations on Michelangelo's tomb	One form of a heterosexuality symbol	(pseudo-3D)
Are they forever linked? [2]	As impossible object	Linked hearts (pseudo-3D)	German coat of arms

Link Presentations

[edit Notes on L2a1's Link Presentations]

Planar diagram presentation	X4132 X2314
Gauss code	{1, -2}, {2, -1}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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.

L2a1 L4a1