Tsuyoshi KOBAYASHI Home Page

2009 Nara Topology Seminar

‚VŒŽ‚Q‚W“úi‰ÎjŒßŒã‚SF‚Q‚O`‚T:‚T‚O

êŠF“ޗǏ—Žq‘åŠw—Šw•”‚b“‚SŠK‚S‚R‚Pi‰‰KŽºj

u‰‰ŽÒFProf. Kenneth J. Shackletonr (Univ. of Tokyo IPMU)

u‰‰‘è–ځF On the coarse geometry of Weil-Petersson's metric on Teichmueller space

Abstract: We discuss the synthetic geometry of the pants
graph in comparison with the Weil-Petersson metric, whose
geometry the pants graph coarsely models following work
of Brock's. We also restrict our attention to the 5-holed
sphere and the 2-holed torus, finding that the boundary is
visible - any two points in the bordification may connected
by an (in)finite geodesic - and that pseudo-Anosov mapping
classes, when raised to a high power, have an invariant
geodesic axis.

‚VŒŽ‚P‚S“úi‰ÎjŒßŒã‚SF‚Q‚O`‚T:‚T‚O

êŠF“ޗǏ—Žq‘åŠw—Šw•”‚b“‚SŠK‚S‚R‚Pi‰‰KŽºj

u‰‰ŽÒFProf. Joseph Maher (Oklahoma State University)

u‰‰‘è–ځF A Comparison of 3-Manifold Widths

Abstract: Scharlemann and Thompson define the width of a 3-manifold $M$ as the
minimum, over all thin position handle decompositions of $M$, of the
maximum number of 1-handles at each level of the decomposition.  Gromov
defines a width of a 3-manifold more geometrically, so that the width of
3-manifold is the minimum, over all projections of $M$ onto a graph, of
the maximum area of the preimage of a point on the graph under the
projection.

We will show that the Scharlemann-Thompson width and the Gromov width are
comparable; in other words, that there is a constant $K$ such that for
any hyperbolic 3-manifold $M$, the Scharlemann-Thompson width is no more
than $K$ times the Gromov width of $M$, and the Gromov width of $M$ is no
more than $K$ times the square of the Scharlemann-Thompson width.

This is work in progress with Diane Hoffoss.

6ŒŽ5“úi‰ÎjŒßŒã‚SF‚Q‚O`‚T:‚T‚O

êŠF“ޗǏ—Žq‘åŠw—Šw•”‚b“‚SŠK‚S‚R‚Pi‰‰KŽºj

u‰‰ŽÒFProf. Chaim Goodman-Strauss (Univ. Arkansas)

u‰‰‘è–ځF Undecidable Games and Puzzles

Abstract: TSimple games and puzzles quickly lead us to one of the
hallmark achievements of twentieth century mathematics, the
recognition that there are true but formally unprovable mathematical
statements. We will discuss undecidable problems in what might
be viewed as a branch of recreational mathematics,tilings of the
plane. Puzzles and toys will be provided.

6ŒŽ2“úi‰ÎjŒßŒã3F00`5:00

êŠF“ޗǏ—Žq‘åŠw—Šw•”‚b“‚SŠKC434i¬u‹`Žºj

u‰‰ŽÒFˆÀ•”“NÆ (‘åãŽs—§‘åŠw‘åŠw‰@—ŠwŒ¤‹†‰È)

u‰‰‘è–ځF The band-unknotting number of a knot
(joint work with Ryuji Higa)

Abstract: This is a joint work with Ryuji Higa. A band-move is a local move
of a link diagram which is performed by adding a band. We define the
band-unknotting number of a knot K to be the minimum number of
band-moves needed to transform a diagram of K into that of the trivial
knot. Note that, in the definition of the band-unknotting number of a
knot K, we may use Reidemeister moves after applying a band-move and
the sequence from a diagram of K to that of the trivial knot may
contain a diagram of a link.

In this talk, we show that the band-unknotting number of a knot K is
less than or equal to half the crossing number of K and the equality
holds if and only if K is the trivial knot or the figure-eight knot.
To prove this, we give a characterization of the figure-eight knot. We
also determine the band-unknotting number of a twist knot.