Andrew Marshall's paper Linear configurations of complete graphs, K_4 and K_5 in R ³, and higher dimensional analogs has been published in the current issue of the Journal of Knot Theory and its Ramifications.Andrew Marshall's paper Linear configurations of complete graphs, K_4 and K_5 in R ³, and higher dimensional analogs has been published in the current issue of the Journal of Knot Theory and its Ramifications.
The paper is part of a larger project to understand configuration spaces of graphs sitting in 3 dimensional space, and other ambient spaces.
A configuration space is a space parametrizing all the possible positions of a system. A graph is an abstract network made up of nodes and edges. How a graph can sit within a space can tell us about the ambient space, as well as tell us something about the graph. The groups of symmetries of loops of positions of graphs generalize a century old algebraic system called a Braid Group. These are studied in disparate settings including protein folding, cryptology, and quantum mechanics.
This work was made possible by the guidance of Professor Allen Hatcher of Cornell University, and the support of Ithaca College's department of Mathematics.
For details, see here:
http://www.worldscientific.com/doi/abs/10.1142/S0218216517500286?src=recsys
https://www.ithaca.edu/intercom/article.php/20170402135350944