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Computational Topology : A Personal Overview

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T. J. Peters

www.cse.uconn.edu/~tpeters

General Topology (Point-Set Topology)

Mappings and Equivalences

Vertex, Edge, Face:

Connectivity

Euler Operations

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

- Grimm: Manifolds, charts, blending functions
- Blackmore: differential sweeps
- Kopperman, Herman: Digital topology
- Edelsbrunner, Zomordian, Carlsson : Algebraic

- Reduced two to simplest forms
- Need for equivalence
- Approximation as operation in geometric design

Unknot

Bad

Approximation!

Self-intersect?

Why Bad?

No

Intersections!

Changes

Knot Type

Now has 4

Crossings

Good

Approximation!

Respects Embedding

Via

Curvature (local)

Separation (global)

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

- 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

- Space Curves: intersection versus crossing
- Local and global arguments
- Knot equivalence via isotopy
- Extensions to surfaces

UMass, RasMol

Theorem: If an approximation of F has a unique intersection with each normalof F, then it is ambient isotopic to F.

Proof:

1. Local argument with curvature.

2. Global argument for separation.

(Similar to flow on normal field.)

Good

Approximation!

Respects Embedding

Via

Curvature (local)

Separation (global)

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

Global separation

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

- Manifolds (without boundary or with boundary)

- 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)

- ROTATING IMMORTALITY
- www.bangor.ac.uk/cpm/sculmath/movimm.htm

- KnotPlot
- www.knotplot.com

- 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

- 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.