Reidemeister/Roseman-type Moves to Embedded Foams in 4-dimensional Space

Abstract

The dual to a tetrahedron consists of a single vertex at which four edges and six faces are incident. Along each edge, three faces converge. A 2-foam is a compact topological space such that each point has a neighborhood homeomorphic to a neighborhood of that complex. Knotted foams in 4-dimensional space are to knotted surfaces, as knotted trivalent graphs are to classical knots. The diagram of a knotted foam consists of a generic projection into 4-space with crossing information indicated via a broken surface. In this paper, a finite set of moves to foams are presented that are analogous to the Reidemeister-type moves for knotted graphs. These moves include the Roseman moves for knotted surfaces. Given a pair of diagrams of isotopic knotted foams there is a finite sequence of moves taken from this set that, when applied to one diagram sequentially, produces the other diagram.Comment: 18 pages, 29 figures, Be aware: the figure on page 3 takes some time to load. A higher resolution version is found at http://www.southalabama.edu/mathstat/personal_pages/carter/Moves2Foams.pdf . If you want to use to any drawings, please contact m