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Author Spotlight: Universal Molecular Retention with 11-Fold Expansion Microscopy
Published on: October 6, 2023
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Rank-expanding satellites, Whitehead doubles, and Heegaard Floer homology
Irving Dai1, Matthew Hedden2, Abhishek Mallick3
1Department of Mathematics The University of Texas at Austin Austin Texas USA.
Journal of Topology
|December 2, 2024
Summary
This study introduces rank-expanding satellite operators, mapping rank-one subgroups to infinite sets. It provides a Floer-theoretic condition for infinite-rank images under these operators, affirming a conjecture.
Area of Science:
- Knot theory
- Geometric topology
- Algebraic topology
Background:
- Satellite operators in knot theory are crucial for understanding knot complexity.
- Previous studies have explored specific cases of satellite operators, but a systematic analysis was lacking.
- A conjecture regarding the rank of satellite operators remained partially unproven.
Purpose of the Study:
- To systematically study rank-expanding satellite operators.
- To establish a Floer-theoretic condition for infinite-rank images under satellite operators.
- To affirm a conjecture concerning the properties of satellite operators.
Main Methods:
- Analysis of satellite operators acting on the concordance group.
- Development of a Floer-theoretic condition for companion knots.
- Application of methods suitable for patterns acting trivially in topological concordance.
Main Results:
- A large class of satellite operators are shown to be rank-expanding.
- A Floer-theoretic condition is established for companion knots to have infinite-rank images under satellites.
- An infinite linearly independent family of Whitehead doubles with negative -invariant companion knots is presented.
Conclusions:
- The study provides the first systematic analysis of rank-expanding satellite operators.
- The established Floer-theoretic condition offers a powerful tool for studying satellite operators.
- The results extend and recover existing theorems in knot theory, particularly those using instanton Floer homology.
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