Oriented Knot Complement Conjecture
Let be a closed oriented -manifold, and let and be knots in whose complements are homeomorphic by an orientation preserving homeomorphism and are not homeomorphic to . Oriented Knot Complement Conjecture. The knots and 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 -manifolds are determined by their complements, while counterexamples with orientation reversing homeomorphic complements are known.
Equivalent formulations 1Other wordings
Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.
The oriented knot complement conjecture
Let and be knots in a closed, oriented 3-manifold , and suppose that their complements are homeomorphic by an orientation-preserving homeomorphism. Oriented knot complement conjecture. There is an orientation-preserving homeomorphism of taking to . 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).
References
Primary source
Marc Kegel, “The Legendrian knot complement problem”, arXiv:1604.05196 (2018).
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