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Almost alternating diagrams and fibered links in the 3-sphere.Geometry/Topology
|Speaker:||Hiroshi Goda, UCD and Kobe University|
|Start time:||Wed, May 19 1999, 4:10PM|
Let L be an oriented link with an alternating diagram D. It is known that L is a fibered link if and only if the surface R obtained by applying Seifert's algorithm to D is a Hopf plumbing. Here, we call R a Hopf plumbing if R is obtained by successively plumbing finite number of Hopf bands to a disk.
In this talk, we discuss its extension, and we show the following theorem: Let R be a Seifert surface obtained by applying Seifert's algorithm to an almost alternating diagrams. Then R is a fiber surface if and only if R is a Hopf plumbing.
We also show that the above theorem can not be extended to 2-almost alternating diagrams, that is, we give examples of 2-almost alternating diagrams for knots whose Seifert surface obtained by Seifert's algorithm is a fiber surfaces that is not a Hopf plumbing. This is shown by using a criterion of Melvin-Morton.
A diagram D is called almost alternating (resp. 2-almost alternating) if D becomes an alternating diagram after one crossing change (resp. two crossing changes).
This is joint work with Mikami Hirasawa and Ryosuke Yamamoto.