Various Aspects of Classical Links II
∼ Decompositions by Various Surfaces ∼

December 1 (Mon), 2014

Place: C431, Nara Women's University
(Kitauoya Nishimachi, Nara 630-8506 JAPAN)

    10:30-12:00 Alexander Zupan (University of Texas at Austin)
    Bridge spectra of knots in the 3-sphere
    The bridge spectrum of a knot in the 3-sphere is a compilation of its genus g bridge numbers over all possible genera. I will discuss several recent results about this topic, including the classification of bridge spectra of most iterated torus knots and bounds on the bridge spectra of a class of sufficiently twisted torus knots. The second result is based on joint work with Sean Bowman and Scott Taylor. Both collections of knots are interesting in that their bridge spectra exhibit gaps; that is, the bridge number falls by more than one as genus increases. I will conclude with several open questions about bridge spectra.
    14:00-15:30 Ryan Blair (California State University, Long Beach)
    Knots with compressible thin levels
    Recent advancements in the study of high-distance surfaces in knot exteriors have made it possible to construct knots in a way that puts heavy restrictions on all essential and strongly irreducible surfaces in the knot exterior. We use these techniques to construct the first examples of knots in Gabai thin position which admit compressible thin levels. This is joint work with Alex Zupan.
    16:00-16:40 Toshio Saito (Joetsu University of Education)
    Essential tangle spheres of knots
    It is shown by Ozawa that a knot in the 3-sphere has a unique essential tangle decomposition if it admits an essential free 2-tangle decomposition. We show that Ozawa's result cannot be generalized even if a knot admits an essential 2-tangle decomposition such that one of the decomposed tangles is of tunnel number one. The same can be said in case that admits an essential free n-tangle decomposition for n>2.

Organaizers : Tsuyoshi Kobayashi, Yeonhee Jang (Nara Women's University)