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On the 1-bridge genus of small knots

✍ Scribed by Phoebe Hoidn

Publisher: Elsevier Science
Year: 2000
Tongue: English
Weight: 151 KB
Volume: 106
Category: Article
ISSN: 0166-8641
DOI: 10.1016/s0166-8641(99)00085-1

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✦ Synopsis

A Heegaard splitting for S 3 gives a 1-bridge presentation for a knot k ⊂ S 3 if the knot intersects each handlebody of the Heegaard splitting in an arc which forms the interior part of the boundary of a disk in the handlebody. The minimal g for which the knot has a 1-bridge presentation of genus g is called the 1-bridge genus g 1 (k). The object of the article is the behaviour of this invariant under the connected sum k 1 #k 2 . More precisely, for small knots k 1 and k 2 which are knots which do not have essential closed surfaces in their exteriors, our purpose is to show the following inequality:

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