marilyn's cross
DESCRIPTION
Marilyn's Cross - Wikipedia articleTRANSCRIPT
12/26/12 Marilyn's Cross - Wikipedia, the free encyclopedia
1/3en.wikipedia.org/wiki/Marilyn's_Cross
The "cross" variation of Marilyn's
Cross.
Marilyn's CrossFrom Wikipedia, the free encyclopedia
In the language of the mathematical theory of knots, Marilyn's Cross is a rendering of a certain three-component Brunnian link whose projection has twelve crossings.
Marilyn's Cross was discovered independently by David Swart in 2010[1] and by Rick Mabry andLaura McCormick of Louisiana State University in Shreveport (http://www.lsus.edu) in 2011. Thefigure was named "Marilyn's Cross" by Mabry and McCormick in honor of Mabry's then just-deceased
cousin, Marilyn Henry.[citation needed]
Swart communicated with the knot theorist Dror Bar-Natan of the University of Toronto for help inidentifying the link. Without a complete list of all 12-crossing links, it seemed unknown, and "It is what it
is" became the link's erstwhile moniker.[2]
In June 2012 it was noticed by Mabry and McCormick that the link turns out to be equivalent to a 10-
crossing alternating link previously described by Slavik V. Jablan.[3] David Swart then observed in
August 2012 that it is also L10a140 in The Knot Atlas.[4]
Contents
1 Description2 The "cross" variation3 Brunnian link4 Invariants5 Simplification6 References
7 External links
Description
Marilyn's Cross is a particular rendering of a certain three-component Brunnian link.
The "cross" variation
This is another configuration of Marilyn's Cross that is topologically equivalent to the "Star-of-Davidversion" seen above. While the name "Marilyn's Cross" refers to any equivalent configuration, some callthis arrangement Marilyn's Cross and refer to the other as Marilyn's Star.
One benefit of the star configuration is that is allows for ease of comparison between Marilyn's Crossand similar links that were previously known, such as the Brunnian-Not-Borromean link.
Brunnian link
A Brunnian link is a link of connected components that, when one component is removed, becomes
entirely unconnected.[5] The best known example of a Brunnian link is the three-component Borromeanrings. A 12-crossing, three component example of a Brunnian link that, at first glance, looks very muchlike Marilyn's Cross is the alternating "Brunnian-Not-Borromean" link.
12/26/12 Marilyn's Cross - Wikipedia, the free encyclopedia
2/3en.wikipedia.org/wiki/Marilyn's_Cross
"Paper-clipped" components
The Borromean Rings
"Brunnian-Not-Borromean"
The components of Marilyn's Cross are pairwise "paper-clipped": any pair of components has acrossing configuration of over-over-under-under. This is in contrast with many other Brunnian links inwhich components are pairwise "stacked", each lying completely atop another.
Invariants
The Alexander polynomial[6] for Marilyn's Cross is
the multivariable Alexander polynomial (http://katlas.org/wiki/The_Multivariable_Alexander_Polynomial) is
the Conway polynomial is
the Jones polynomial factors nicely as
where (Notice that is essentially the Jones polynomial for the Whitehead link.)
The HOMFLY polynomial is
and the Kauffman polynomial is
Simplification
12/26/12 Marilyn's Cross - Wikipedia, the free encyclopedia
3/3en.wikipedia.org/wiki/Marilyn's_Cross
More than a year after Marilyn's Cross was discovered, it was finally noticed that it simplifies to a link with 10 crossings. Specifically, it turns outto be topologically equivalent to the link named L10a140 in the Thistlewaite Link Table(http://katlas.math.toronto.edu/wiki/The_Thistlethwaite_Link_Table) . This link is one of an infinite sequence of 3-component Brunnian linksdescribed by Slavik V. Jablan.
References
1. ^ Swart, David. Math Horizons. "It is what it is". April 2011.
2. ^ Dror Bar-Natan. A Link from David Swart (Academic Pensieve 2010-08-14). http://drorbn.net/AcademicPensieve/2010-08/index.html
3. ^ Jablan, Slavik V., Are Borromean Links So Rare?, Forma 14 (1999), 269–277. Online at the electronic journal Vismath athttp://www.mi.sanu.ac.rs/vismath/bor (page 4)
4. ^ L10a140. The Knot Atlas. http://katlas.org/wiki/Image:L10a140.gif
5. ^ Adams, Colin C. The Knot Book. American Mathematical Society. 1994.
6. ^ Collins, Julia. "The Alexander polynomial: The woefully overlooked granddaddy of knot polynomials". May, 2007.http://www.maths.ed.ac.uk/~s0681349/GeomClub.pdf
External links
Multivariable Alexander Polynomial (http://katlas.org/wiki/The_Multivariable_Alexander_Polynomial)The Knot Atlas (http://katlas.org)"It Is What It Is" (http://www.flickr.com/photos/dmswart/4878510547/)
Retrieved from "http://en.wikipedia.org/w/index.php?title=Marilyn%27s_Cross&oldid=517189387"Categories: Knot theory Geometric topology
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