Reidemeister Moves on 1-manifolds in the Real Line Cross the Disk.
by
Bobby Winters
Pittsburg State University

Let L₁ and L₂ be 1-manifolds that are proper and properly embedded in R ×D². Let J = [-1, 1] be the closed interval and let p:R ×D² --> R ×J be a projection map.

We say that L₁ and L₂ are equivalent if there is an isotopy h_t:X --> X with h₀ = 1_X and h₁(L₁) = L₂.

Let T = { ... < t_-1 < t₀ < t₁ < ... } subset R. For every i in Z, let D_i be a closed disk in (t_i, t_i+1) ×J subset R ×J. Suppose that C = { C_{i, j} | i in Z and 1 <= j <= n_i } is a set of disks such that C_{i, j} subset D_i - \partialD_i for 1 <= j <= n_i where n_i in N for every i in Z. Suppose that, for every i in Z,

p(L2) \cap ( [ti, ti+1] ×J )

is obtained from

p(L1) \cap ( [ti, ti+1] ×J )

by Reidemeister moves and each Reidemeister move is contained in C_{i, j} for some 1 <= i <= n_i. Then we say that L₂ is obtained from L₁ by a generalized countable Reidemeister move (GCR-move).

We say that L₁ is -equivalent to L₂ if p(L₂) is obtained from p(L₁) by a finite number of GCR-moves and isotopies of R ×J. Note that GCR-equivalence is an equivalence relation.

Theorem Let L₁ and L₂ be 1-manifolds that are proper in R ×D². Then L₁ is GCR-equivalent to L₂ iff L₁ is equivalent to L₂.

Date received: February 11, 2000