57Mxx Knots and 3-manifolds
This subtopic introduces the core ideas in knots and 3-manifolds, including foundational concepts, standard methods, and the main questions used to organize the area. Typical uses include building mathematical background, framing related research problems, and supporting applications in neighboring fields where these concepts provide useful structure.
Specific topics
57M05 Fundamental group, presentations, free differential calculus
Overview
Fundamental group, presentations, free differential calculus. This topic studies knots and 3-manifolds through diagrammatic, geometric, and algebraic invariants, including decomposition and surgery techniques.
Related Wikipedia Page
Wikipedia: Knot theory
Useful Links
Key Ideas
- knot and link invariants
- surgery and decomposition
- interplay between combinatorial and geometric methods
Typical Uses
Used to analyze embeddings in 3-space and classify manifolds via knot and link constructions.
Applications
- DNA and molecular topology models
- Quantum invariants
- 3-manifold classification
References
Recommended Textbooks
57M07 Topological methods in group theory
Overview
Topological methods in group theory. This topic studies knots and 3-manifolds through diagrammatic, geometric, and algebraic invariants, including decomposition and surgery techniques.
Related Wikipedia Page
Wikipedia: Knot theory
Useful Links
Key Ideas
- knot and link invariants
- surgery and decomposition
- interplay between combinatorial and geometric methods
Typical Uses
Used to analyze embeddings in 3-space and classify manifolds via knot and link constructions.
Applications
- DNA and molecular topology models
- Quantum invariants
- 3-manifold classification
References
Recommended Textbooks
57M10 Covering spaces and low-dimensional topology
Overview
Covering spaces and low-dimensional topology. This topic studies knots and 3-manifolds through diagrammatic, geometric, and algebraic invariants, including decomposition and surgery techniques.
Related Wikipedia Page
Wikipedia: Knot theory
Useful Links
Key Ideas
- knot and link invariants
- surgery and decomposition
- interplay between combinatorial and geometric methods
Typical Uses
Used to analyze embeddings in 3-space and classify manifolds via knot and link constructions.
Applications
- DNA and molecular topology models
- Quantum invariants
- 3-manifold classification
References
Recommended Textbooks
57M12 Special coverings
Overview
Special coverings. This topic studies knots and 3-manifolds through diagrammatic, geometric, and algebraic invariants, including decomposition and surgery techniques.
Related Wikipedia Page
Wikipedia: Knot theory
Useful Links
Key Ideas
- knot and link invariants
- surgery and decomposition
- interplay between combinatorial and geometric methods
Typical Uses
Used to analyze embeddings in 3-space and classify manifolds via knot and link constructions.
Applications
- DNA and molecular topology models
- Quantum invariants
- 3-manifold classification
References
Recommended Textbooks
57M15 Relations with graph theory
Overview
Relations with graph theory. This topic studies knots and 3-manifolds through diagrammatic, geometric, and algebraic invariants, including decomposition and surgery techniques.
Related Wikipedia Page
Wikipedia: Knot theory
Useful Links
Key Ideas
- knot and link invariants
- surgery and decomposition
- interplay between combinatorial and geometric methods
Typical Uses
Used to analyze embeddings in 3-space and classify manifolds via knot and link constructions.
Applications
- DNA and molecular topology models
- Quantum invariants
- 3-manifold classification
References
Recommended Textbooks
57M20 Two-dimensional complexes (manifolds) (MSC2000)
Overview
Two-dimensional complexes (manifolds) (MSC2000). This topic studies knots and 3-manifolds through diagrammatic, geometric, and algebraic invariants, including decomposition and surgery techniques.
Related Wikipedia Page
Wikipedia: Knot theory
Useful Links
Key Ideas
- knot and link invariants
- surgery and decomposition
- interplay between combinatorial and geometric methods
Typical Uses
Used to analyze embeddings in 3-space and classify manifolds via knot and link constructions.
Applications
- DNA and molecular topology models
- Quantum invariants
- 3-manifold classification
References
Recommended Textbooks
57M25 Knots and links in the $3$-sphere
Overview
Knots and links in the $3$-sphere. This topic studies knots and 3-manifolds through diagrammatic, geometric, and algebraic invariants, including decomposition and surgery techniques.
Related Wikipedia Page
Wikipedia: Knot theory
Useful Links
Key Ideas
- knot and link invariants
- surgery and decomposition
- interplay between combinatorial and geometric methods
Typical Uses
Used to analyze embeddings in 3-space and classify manifolds via knot and link constructions.
Applications
- DNA and molecular topology models
- Quantum invariants
- 3-manifold classification
References
Recommended Textbooks
57M27 Invariants of knots and $3$-manifolds
Overview
Invariants of knots and $3$-manifolds. This topic studies knots and 3-manifolds through diagrammatic, geometric, and algebraic invariants, including decomposition and surgery techniques.
Related Wikipedia Page
Wikipedia: Knot theory
Useful Links
Key Ideas
- knot and link invariants
- surgery and decomposition
- interplay between combinatorial and geometric methods
Typical Uses
Used to analyze embeddings in 3-space and classify manifolds via knot and link constructions.
Applications
- DNA and molecular topology models
- Quantum invariants
- 3-manifold classification
References
Recommended Textbooks
57M30 Wild knots and surfaces, etc., wild embeddings
Overview
Wild knots and surfaces, etc., wild embeddings. This topic studies knots and 3-manifolds through diagrammatic, geometric, and algebraic invariants, including decomposition and surgery techniques.
Related Wikipedia Page
Wikipedia: Knot theory
Useful Links
Key Ideas
- knot and link invariants
- surgery and decomposition
- interplay between combinatorial and geometric methods
Typical Uses
Used to analyze embeddings in 3-space and classify manifolds via knot and link constructions.
Applications
- DNA and molecular topology models
- Quantum invariants
- 3-manifold classification
References
Recommended Textbooks
57M35 Dehn's lemma, the loop theorem, the sphere theorem, and generalizations
Overview
Dehn's lemma, the loop theorem, the sphere theorem, and generalizations. This topic studies knots and 3-manifolds through diagrammatic, geometric, and algebraic invariants, including decomposition and surgery techniques.
Related Wikipedia Page
Wikipedia: Knot theory
Useful Links
Key Ideas
- knot and link invariants
- surgery and decomposition
- interplay between combinatorial and geometric methods
Typical Uses
Used to analyze embeddings in 3-space and classify manifolds via knot and link constructions.
Applications
- DNA and molecular topology models
- Quantum invariants
- 3-manifold classification
References
Recommended Textbooks
57M40 Characterizations of $E^3$ and $S^3$
Overview
Characterizations of $E^3$ and $S^3$. This topic studies knots and 3-manifolds through diagrammatic, geometric, and algebraic invariants, including decomposition and surgery techniques.
Related Wikipedia Page
Wikipedia: Knot theory
Useful Links
Key Ideas
- knot and link invariants
- surgery and decomposition
- interplay between combinatorial and geometric methods
Typical Uses
Used to analyze embeddings in 3-space and classify manifolds via knot and link constructions.
Applications
- DNA and molecular topology models
- Quantum invariants
- 3-manifold classification
References
Recommended Textbooks
57M50 General geometric and topological aspects of $3$-manifolds
Overview
General geometric and topological aspects of $3$-manifolds. This topic studies knots and 3-manifolds through diagrammatic, geometric, and algebraic invariants, including decomposition and surgery techniques.
Related Wikipedia Page
Wikipedia: Knot theory
Useful Links
Key Ideas
- knot and link invariants
- surgery and decomposition
- interplay between combinatorial and geometric methods
Typical Uses
Used to analyze embeddings in 3-space and classify manifolds via knot and link constructions.
Applications
- DNA and molecular topology models
- Quantum invariants
- 3-manifold classification
References
Recommended Textbooks
57M60 Group actions in low dimensions
Overview
Group actions in low dimensions. This topic studies knots and 3-manifolds through diagrammatic, geometric, and algebraic invariants, including decomposition and surgery techniques.
Related Wikipedia Page
Wikipedia: Knot theory
Useful Links
Key Ideas
- knot and link invariants
- surgery and decomposition
- interplay between combinatorial and geometric methods
Typical Uses
Used to analyze embeddings in 3-space and classify manifolds via knot and link constructions.
Applications
- DNA and molecular topology models
- Quantum invariants
- 3-manifold classification
References
Recommended Textbooks
57M99 None of the above
Overview
None of the above. This topic studies knots and 3-manifolds through diagrammatic, geometric, and algebraic invariants, including decomposition and surgery techniques.
Related Wikipedia Page
Wikipedia: Knot theory
Useful Links
Key Ideas
- knot and link invariants
- surgery and decomposition
- interplay between combinatorial and geometric methods
Typical Uses
Used to analyze embeddings in 3-space and classify manifolds via knot and link constructions.
Applications
- DNA and molecular topology models
- Quantum invariants
- 3-manifold classification
References
Recommended Textbooks