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	ABOUT: [Introduction][History][Related areas][Subfields] POINTERS: [Texts][Software][Web links][Selected topics here] 57M: Low-dimensional topology

Introduction

History

Applications and related fields

Subfields

• 57M05: Fundamental group, presentations, free differential calculus
• 57M07: Topological methods in group theory
• 57M10: Covering spaces
• 57M12: Special coverings, e.g. branched
• 57M15: Relations with graph theory, See also 05Cxx
• 57M20: Two-dimensional complexes
• 57M25: Knots and links in S^3, For higher dimensions, See 57Q45
• 57M27: Invariants of knots and 3-manifolds [new in 2000]
• 57M30: Wild knots and surfaces, etc., wild embeddings
• 57M35: Dehn's lemma, sphere theorem, loop theorem, asphericity
• 57M40: Characterizations of E^3 and S^3 (Poincaré conjecture), See also 57N12
• 57M50: Geometric structures on low-dimensional manifolds
• 57M60: Group actions in low dimensions
• 57M99: None of the above but in this section

Parent field: 57: Manifolds and Cell Complexes

Browse all (old) classifications for this area at the AMS.

Textbooks, reference works, and tutorials

A good survey paper is Raymond Lickorish, "Polynomials for Links", Bull. London Math. Soc. 20 (1988) 558-588. See also L. H. Kauffman's book "Knots and Physics".

Online Primer on Knot theory [Charilaos Aneziris]

Software and tables

Knot Plot, a program to visualize and manipulate knots in three and four dimensions

Other web sites with this focus

• Knot Square
• Atlas of oriented knots and links
• A physicist's view of Knots and Braids

Selected topics at this site

• What is knot theory
• Do knot complements determine knot type?
• Distinct links with homeomorphic complements in S^3
• Whitney's Theorem classifying immersions of circles into the plane
• Papakyriakopolous' Theorems (historical)
• Parameterizing a tubular neighborhood of a knot.
• Alexander's horned sphere and other wild embeddings of S^2 into S^3
• References to computational knot theory
• The computational complexity of knot and link problems
• How hard is it to distinguish among knots?
• Can one determine whether a knot is really knotted from its projection to the plane?
• Triangulating polygons in R^3 -- when is it even possible? (problems if knotted)
• Curves in R^4 are unknotted; generalize?
• The Smith conjecture: fixed points under periodic homeomorphisms of the sphere are unknotted.
• Connections between knot theory and statistical mechanics (the Jones polynomial)
• Literature highlights: Knot theory and Functional Analysis
• Applications of the Schönflies theorem
• The p-colorability of knots and the dihedral groups
• Conway's game of rational tangles (a two-strand braid).