Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

On simple loops on a solid torus of general genus


Author: Takeshi Kaneto
Journal: Proc. Amer. Math. Soc. 86 (1982), 551-552
MSC: Primary 57N10; Secondary 57M05
DOI: https://doi.org/10.1090/S0002-9939-1982-0671234-5
MathSciNet review: 671234
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ l$ be a simple loop on the boundary of a solid torus $ T$ of genus $ g$. Let $ {\mathbf{m}}$ be a complete system of oriented meridian disks of $ T$. Let $ W(l,{\mathbf{m}})$ be a word obtained by reading the intersections $ l \cap {\mathbf{m}}$ along $ l$. We shall give a natural method for realizing the cyclic reductions of $ W(l,{\mathbf{m}})$ geometrically. This yields a simple proof of Whitehead-Zieschang's theorem related to the minimality of the intersections of two systems $ \{ {l_1}, \ldots ,{l_n}\} $ and $ {\mathbf{m}}$.


References [Enhancements On Off] (What's this?)

  • [1] J. S. Birman, Heegarrd splittings, diagrams and sewings for closed, orientable $ 3$-manifolds, Lecture notes for CBMS conference at Blacksburg, Va., Oct. 8-12, 1977.
  • [2] J. S. Birman and J. M. Montesinos, On minimal Heegaard splitings, Michigan Math. J. 27 (1980), 49-57. MR 555836 (81b:57007)
  • [3] T. Kaneto, On genus 2 Heegaard diagrams for the $ 3$-sphere, preprint.
  • [4] E. S. Rapaport, On free groups and their automorphisms, Acta Math. 99 (1958), 139-163. MR 0131452 (24:A1302)
  • [5] J. H. C. Whitehead, On equivalent sets of elements in a free group, Ann. of Math. (2) 37 (1963), 782-800. MR 1503309
  • [6] H. Zieschang, On simple systems of paths on complete pretzels, Amer. Math. Soc. Transl. (2) 92 (1970), 127-137.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 57N10, 57M05

Retrieve articles in all journals with MSC: 57N10, 57M05


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1982-0671234-5
Keywords: Solid torus of genus $ g$, complete system of meridian disks, simple loops on the boundary, cyclic reductions of words, minimal intersections, pseudominimal Heegaard diagrams
Article copyright: © Copyright 1982 American Mathematical Society

American Mathematical Society