06-1350/Class Notes for Tuesday October 10
Question 1. Can you embed a trinion (a.k.a. a sphere with three holes, a pair of pants, or a band theta graph) in so that each boundary component would be unknotted yet each pair of boundary components would be knotted? How about, so that at least one pair of boundary components would be knotted?
Dror's Speculation. Yes and yes.
Question 2. A trinion γ is embedded in so that its boundary is the trivial 3-component link. Does it follow that γ is trivial?
Dror's Speculation. No.
Question 3. Suppose two trinions γ1 and γ2 are knotted so that the pushforwards and are equal for any link L which is "drawn" on the parameter space Γ of γ1 and γ2. Does it follow that γ1 and γ2 are equivalent?
Dror's Speculation. I'm clueless.
Question 4. A trinion γ is embedded in so that its "strapped boundary" is equivalent to the strapped boundary of the trivially embedded trinion. Does it follow that γ is trivial?
Dror's Speculation. If yes, it will have terrific consequences. If no, it will explain some of the misery we encounter when we deal with "associators". I would really like to understand this one.
Also see Some Questions About Trinions.