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Oriented Knot Complement Conjecture

Let MM be a closed oriented 33-manifold, and let K1K_1 and K2K_2 be knots in MM whose complements are homeomorphic by an orientation preserving homeomorphism and are not homeomorphic to S1×D2S^1\times D^2. Oriented Knot Complement Conjecture. The knots K1K_1 and K2K_2 are orientation preserving equivalent. This conjecture is stated as still open in general manifolds. It is motivated by the fact that knots in several classes of 33-manifolds are determined by their complements, while counterexamples with orientation reversing homeomorphic complements are known.

Equivalent formulations 1

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

  1. The oriented knot complement conjecture

    Let K1K_1 and K2K_2 be knots in a closed, oriented 3-manifold MM, and suppose that their complements are homeomorphic by an orientation-preserving homeomorphism. Oriented knot complement conjecture. There is an orientation-preserving homeomorphism of MM taking K1K_1 to K2K_2. The source presents this as an equivalent form of the cosmetic surgery conjecture and gives no resolution.

    source: Steven A. Bleiler, Craig D. Hodgson and Jeffrey R. Weeks, “Cosmetic surgery on knots”, arXiv:math/9911247 (1999).

Sources & referencesView supporting material

Primary source

Marc Kegel, “The Legendrian knot complement problem”, arXiv:1604.05196 (2018).

3-manifolds3-manifold-topologyknot-complementsknot-theory

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