PATCH Day on October 26

On October 26th, the second PATCH seminar will be held at Bryn Mawr in the Park Science Center.  There will be two talks, starting at 4:10 PM:

Greg Schneider from the University at Buffalo will speak on

Title: Box-dot diagrams for “regular” rational tangles

Abstract: We introduce a new presentation for rational tangles which illustrates a geometric connection to the number theory of positive regular continued fractions.  This presentation also admits a suitable extension to the contact setting, allowing us to define a natural Legendrian embedding of a particular class of rational tangles into the standard contact Euclidean 3-space.  We will briefly discuss how these box-dot diagrams, along with an associated construction, can be used to determine when the Legendrian flyping operation yields tangles which are not Legendrian isotopic.

Charles Livingston from Indiana University will speak on

Title: The non-orientable 4-genus of knots

Abstract: Every knot in the 3-sphere bounds an embedded orientable surface in the 4-ball.  The minimum genus of such a surface defines the 4-genus of a knot.  This is a well-studied invariant, closely related to more general problems in 4-manifold theory.  The case of non-orientable surfaces has been much more challenging; the natural analogs of basic results in the orientable setting have yielded intractable questions in the non-orientable setting.  In this talk I will first review some of the techniques and results of the orientable case.  Then, I will discuss some of their extensions and progress in the non-orientable setting.  The talk will also touch on the distinction between the topological and smooth settings.

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