Covering moves

Abstract

Abstract.In this paper we give a positive answer to a long standing question posed by Montesinos, by introducing new covering moves, in order to relate any two colored diagrams representing the same 3-manifold as simple branched 3-covering of S .

Coverings, correspondences, and noncommutative geometry

“…A subshift of finite type naturally determines a noncommutative space in the form of associated Cuntz-Krieger algebras. The covering moves (or colored Reidemeister moves) of [31] will determine correspondences between these noncommutative spaces.…”

On 4–fold covering moves

“…Then κ fixes the front edges and sends each back edge to its opposite edge. It is customary to identify, see [8] for example, the three transpositions of S 3 with three colors Red, Green and Blue. The above exact sequence then suggests the identification of the transpositions of S 4 with three colors that come in light and dark shades.…”

“…Each ball lifts to a handlebody and the braid lifts to a homeomorphism between the boundaries of the handlebodies, and so we get a Heegaard splitting of the covering manifold. (See Figure 5, for more details consult the references, for example [8]). This move and its inverse will also be referred to as an addition or deletion of a trivial sheet.…”

“…The plan of the proofs is the one introduced by Piergallini in [8] and [9]: Given L 1 and L 2 , two colored link representations of the same 3-manifold M , first isotope the links so that they are in plat form. Each plat gives a Heegaard splitting of M whose gluing homeomorphism is determined by lifting the braid (viewed as a homeomorphism of the disk).…”

“…An excellent background source is [2] and the references contained therein. We also assume the reader is familiar with the contents of [4] as well as with Piergallini's papers regarding covering moves [8], [9]. Finally we mention that the author's thesis [1] contains extensive background material.…”