Markov's theorem in 3--manifolds

2004/05/26 by Sofia Lambropoulou, Colin P. Rourke · 3 citations
Mathematics · #math.GT #msc:57M25

paper · pdf

published as Topology and its Applications, Vol. 78, pp. 95--122 (1997) · 29 pages, 41 figures, LaTex document

arxiv created 2004/05/26 · arxiv updated 2009/12/01

Abstract

In this paper we first give a one-move version of Markov's braid theorem for knot isotopy in S³ that sharpens the classical theorem. Then a relative version of Markov's theorem concerning a fixed braided portion in the knot. We also prove an analogue of Markov's theorem for knot isotopy in knot complements. Finally we extend this last result to prove a Markov theorem for links in an arbitrary orientable 3--manifold.

Cited by

• From annular to toroidal knotoids and their universal bracket polynomials
• Topology and Algebra of Bonded Knots and Braids
• The HOMFLYPT skein module of S¹ × S² via braids

Related

• Towards a version of Markov's theorem for ribbon torus-links in\n \ℝ⁴
• Algebraic Markov equivalence for links in 3-manifolds
• On Markov's Theorem
• The Markov Theorem for transverse knots
• Markov's Theorem