T. J. Peters

www.cse.uconn.edu/~tpeters

Computational Topology :A Personal Overview

My Topological Emphasis:

General Topology (Point-Set Topology)

Mappings and Equivalences

Vertex, Edge, Face: Connectivity

Euler Operations

Thesis: M. Mantyla; “Computational Topology …”, 1983.

Contemporary Influences

• Grimm: Manifolds, charts, blending functions

• Blackmore: differential sweeps

• Kopperman, Herman: Digital topology

• Edelsbrunner, Zomordian, Carlsson : Algebraic

KnotPlot !

Comparing Knots

• Reduced two to simplest forms

• Need for equivalence

• Approximation as operation in geometric design

Unknot

BadApproximation!

Self-intersect?

Why Bad?

No Intersections!

Changes Knot Type

Now has 4Crossings

Good Approximation!

Respects Embedding

Via

Curvature (local)

Separation (global)

But recognizing unknot in NP (Hass, L, P, 1998)!!

NSF Workshop 1999 for Design

• Organized by D. R. Ferguson & R. Farouki

• SIAM News: Danger of self-intersections

• Crossings not detected by algorithms

• Would appear as intersections in projections

• Strong criterion for ‘lights-out’ manufacturing

Summary – Key Ideas

• Space Curves: intersection versus crossing

• Local and global arguments

• Knot equivalence via isotopy

• Extensions to surfaces

UMass, RasMol

Proof: 1. Local argument with curvature.

2. Global argument for separation.

(Similar to flow on normal field.)

Theorem: If an approximation of F has a unique intersection with each

normalof F, then it is ambient isotopic to F.

Good Approximation!

Respects Embedding

Via

Curvature (local)

Separation (global)

But recognizing unknot in NP (Hass, L, P, 1998)!!

Global separation

Mathematical Generalizations

• Equivalence classes: – Knot theory: isotopies & knots– General topology: homeomorphisms & spaces– Algebra: homorphisms & groups

• Manifolds (without boundary or with boundary)

Overview References

• Computation Topology Workshop, Summer Topology Conference, July 14, ‘05, Denison,

planning with Applied General Topology

• NSF, Emerging Trends in Computational Topology, 1999, xxx.lanl.gov/abs/cs/9909001

• Open Problems in Topology 2 (problems!!)

• I-TANGO,Regular Closed Sets (Top Atlas)

Credits• ROTATING IMMORTALITY

– www.bangor.ac.uk/cpm/sculmath/movimm.htm

• KnotPlot– www.knotplot.com

  

Credits• IBM Molecule

– http://domino.research.ibm.com/comm/pr.nsf/pages/rscd.bluegene-picaa.html

• Protein – Enzyme Complex– http://160.114.99.91/astrojan/protein/pictures/

parvalb.jpg

  

Acknowledgements, NSF

• I-TANGO: Intersections --- Topology, Accuracy and Numerics for Geometric Objects (in Computer Aided Design), May 1, 2002, #DMS-0138098.

• SGER: Computational Topology for Surface Reconstruction, NSF, October 1, 2002, #CCR - 0226504.

• Computational Topology for Surface Approximation, September 15, 2004,

#FMM -0429477.