06-1350/Class Notes for Tuesday October 10

06-1350/Navigation Panel   # Week of... Links 1 Sep 11 About, Tue, Thu 2 Sep 18 Tue, Kurlin(P), Thu 3 Sep 25 Tue, Photo, Thu 4 Oct 2 HW1, Tue, Thu 5 Oct 9 Tue(P), Thu(P) 6 Oct 16 HW2, Tue(P), Thu(P) 7 Oct 23 Tue(P), Thu 8 Oct 30 HW3, Tue, Thu 9 Nov 6 Tue ([math]\displaystyle{ \nu }[/math]), Thu 10 Nov 13 Tue, Thu 11 Nov 20 HW4(P), Thu 12 Nov 27 Thu 13 Dec 4 Syzygies in Asymptote, Final Jan 8 Grades Note. (P) means "contains a problem that Dror cares about". Add your name / see who's in! On to 07-1352

Some Questions

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

Dror's Speculation. No.

Question 3. Suppose two trinions [math]\displaystyle{ \gamma_1 }[/math] and [math]\displaystyle{ \gamma_2 }[/math] are knotted so that the pushforwards [math]\displaystyle{ \gamma_{1\star}L }[/math] and [math]\displaystyle{ \gamma_{2\star}L }[/math] are equal for any link [math]\displaystyle{ L }[/math] which is "drawn" on the parameter space [math]\displaystyle{ \Gamma }[/math] of [math]\displaystyle{ \gamma_1 }[/math] and [math]\displaystyle{ \gamma_2 }[/math]. Does it follow that [math]\displaystyle{ \gamma_1 }[/math] and [math]\displaystyle{ \gamma_2 }[/math] are equivalent?

Dror's Speculation. I'm clueless.

The standardly embedded strapped trinion

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