ANNOUNCEMENT:     TRIANGLE GEOMETRY-TOPOLOGY SEMINAR

 

DATE:             Friday, April 4, 2003        

 

LOCATION:         Hempfield High School, Room 219

                  followed by dinner at Gatsby's on Main in Landisville

 

4:30 TALK:        James R. Hughes, Elizabethtown College

                  "Racks and Link Homotopy"

 

                              ABSTRACT: Link homotopy is the study of "links modulo knots; that is, modulo the equivalence relation that allows ambient isotopy as well as same-component        crossing changes.  In 1990, Habegger and Lin gave a "classification" of links up to link homotopy, consisting of a nice family of semidirect product decompositions of the group of string links mod link homotopy, and a "Markov-type" theorem describing how two string links with link homotopic closures are related.  Specifically, they proved that two such string links are related by a sequence of conjugations and "partial conjugations."  Partial conjugation is conjugation in the normal factor of any of the aforementioned family of semidirect product decompositions. In a forthcoming paper, I revisit the work of Habegger and Lin, using nonassociative algebraic objects called racks. Racks have previously been used successfully by Joyce, Kauffman, Fenn, Rourke, and others in the study of links up to ambient isotopy. In my talk, I will give a brief introduction to racks and link homotopy, followed by a summary of my results.  If time allows, I will give a prove a conjecture that has been in the back of my mind since Habegger and Lin's        classification first appeared: namely, that conjugation is extraneous, in the sense that any change to a string link resulting from conjugation can be produced by an appropriately-chosen sequence of partial conjugations.

 

 

5:00 TALK:        Ron Umble, Millersville University

                  "Canonical Cochain Operations on Permutahedra."

 

                              ABSTRACT: Let B be a module, graded ot ungraded. Uprooted planar trees represent compositions of derivations of the free tensor algebra TB, whereas downrooted planar trees represent compositions of coderivations of the free tensor coalgebra TB.  In the special case B=TA, corresponding tree representations become fibered trees with levels and there is a natural correspondence with faces of permutahedra.  Consequently, the tensor product and composition of maps in End(TTA) give rise to canonical operations on the module of cellular cochains of permutahedra C^*(P;End(TTA)).

 

 

EVERYONE WELCOME.