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The sheet numbers of 2-knots with non-trivial fundamental quandles
by
Shin SATOH
Kobe University
An embedded 2-sphere in R4 is called a 2-knot, which is described by a diagram through a projection of R4 onto R3. Such a diagram is regarded as a disjoint union of compact connected surfaces divided by crossing information. The sheet number of a 2-knot is the minimal number of such surfaces for all possible diagrams of the 2-knot. We prove that if the fundamental quandle of a 2-knot is non-trivial (in particular, if the fundamental group of the complement of the 2-knot is non-trivial), then its sheet number is greater than or equal to four. This gives an alternative proof of our previous result that the sheet numbers of the 0- and 2-twist-spun trefoils are equal to four.
Date received: October 25, 2006
Copyright © 2006 by the author(s). The author(s) of this work and the organizers of the conference have granted their consent to include this abstract in Topology Atlas. Document # catp-02.