MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

A knot in S^3 is uniquely decomposed into a connected sum of prime knots. What is known for knots in other three-manifolds?

share|cite|improve this question
What definition of connect-sum do you want to use? I suppose the most natural one would be to take connect-sum of the ambient manifolds along common 3-balls that intersect the knots in unknotted arcs. Is that what you're interested in? – Ryan Budney Apr 7 '10 at 6:20
Perhaps that will work. But even better to say: this raises another question, namely what all possible definitions of connected sum(s) are! – John Vrem Apr 7 '10 at 6:46
@Ryan- I would ask also whether there is a little discs operad action in such a context. What would happen with your work with Fred Cohen for long knots in other 3-manifolds? – Daniel Moskovich Apr 7 '10 at 7:02
There is a 2-cubes/discs action for "knots" in a variety of 3-manifolds but they're maybe not what most people would like to call knots. Things like self-embeddings of $\mathbb{R}\times \Sigma$ in itself with support contained in $[-1,1]\times \Sigma$ where $\Sigma$ is a compact 2-manifold. Presumably there's more global algebraic structure lurking in the background for general knots in 3-manifolds but I haven't found a strong formalism for it. – Ryan Budney Apr 7 '10 at 19:38

This was addressed in Miyazaki, Conjugation and the prime decomposition of knots in closed, oriented 3-manifolds, using the definition of connected sum suggested by Ryan Budney in the comments.

The main results are that a prime decomposition of K exists iff a meridian of K is not null-homotopic in the complement of K, and if a prime decomposition of K does not contain a particular knot R in S1×S2 then it is the unique decomposition of K, whereas knots with this summand R can admit several prime decompositions.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.