Convergence and Isotopy Type for Graphs of Finite Total Curvature
نویسندگان
چکیده
Generalizing Milnor’s result that an FTC (finite total curvature) knot has an isotopic inscribed polygon, we show that any two nearby knotted FTC graphs are isotopic by a small isotopy. We also show how to obtain sharper constants when the starting curve is smooth. We apply our main theorem to prove a limiting result for essential subarcs of a knot.
منابع مشابه
Total Curvature of Graphs in Space
The Fáry-Milnor Theorem says that any embedding of the circle S1 into R3 of total curvature less than 4π is unknotted. More generally, a (finite) graph consists of a finite number of edges and vertices. Given a topological type of graphs Γ, what limitations on the isotopy class of Γ are implied by a bound on total curvature? Especially: what does “total curvature” mean for a graph? I shall disc...
متن کاملIsotopy Convergence Theorem
When approximating a space curve, it is natural to consider whether the knot type of the original curve is preserved in the approximant. This preservation is of strong contemporary interest in computer graphics and visualization. We establish a criterion to preserve knot type under approximation that relies upon convergence in both distance and total curvature.
متن کاملHigh Order Compact Finite Difference Schemes for Solving Bratu-Type Equations
In the present study, high order compact finite difference methods is used to solve one-dimensional Bratu-type equations numerically. The convergence analysis of the methods is discussed and it is shown that the theoretical order of the method is consistent with its numerical rate of convergence. The maximum absolute errors in the solution at grid points are calculated and it is shown that the ...
متن کاملMean curvature flows and isotopy problems
In this note, we discuss the mean curvature flow of graphs of maps between Riemannian manifolds. Special emphasis will be placed on estimates of the flow as a non-linear parabolic system of differential equations. Several global existence theorems and applications to isotopy problems in geometry and topology will be presented. The results are based on joint works of the author with his collabor...
متن کاملLocal Moves on Spatial Graphs and Finite Type Invariants
We define Ak-moves for embeddings of a finite graph into the 3-sphere for each natural number k. Let Ak-equivalence denote an equivalence relation generated by Ak-moves and ambient isotopy. Ak-equivalence implies Ak−1-equivalence. Let F be an Ak−1-equivalence class of the embeddings of a finite graph into the 3-sphere. Let G be the quotient set of F under Ak-equivalence. We show that the set G ...
متن کامل