06-1350/Class Notes for Tuesday October 10

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Plastic trinions

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 ${\mathbb {R} }^{3}$ 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 $\gamma$ is embedded in ${\mathbb {R} }^{3}$ so that its boundary is the trivial 3-component link. Does it follow that $\gamma$ is trivial?

Dror's Speculation. No.

Question 3. Suppose two trinions $\gamma _{1}$ and $\gamma _{2}$ are knotted so that the pushforwards $\gamma _{1\star }L$ and $\gamma _{2\star }L$ are equal for any link $L$ which is "drawn" on the parameter space $\Gamma$ of $\gamma _{1}$ and $\gamma _{2}$. Does it follow that $\gamma _{1}$ and $\gamma _{2}$ are equivalent?

Dror's Speculation. I'm clueless.

Question 4. A trinion $\gamma$ is embedded in ${\mathbb {R} }^{3}$ so that its "strapped boundary" is equivalent to the strapped boundary of the trivially embedded trinion. Does it follow that $\gamma$ 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.

The standardly embedded strapped trinion