New techniques for computing geometric index

Kathryn B. Andrist (Author), Dennis Garity (Author), Dušan Repovš (Author), David Wright (Author)

Abstract

We introduce new general techniques for computing the geometric index of a link ▫$L$▫ in the interior of a solid torus ▫$T$▫. These techniques simplify and unify previous ad hoc methods used to compute the geometric index in specific examples and allow a simple computation of geometric index for new examples where the index was not previously known. The geometric index measures the minimum number of times any meridional disc of ▫$T$▫ must intersect ▫$L$▫. It is related to the algebraic index in the sense that adding up signed intersections of an interior simple closed curve ▫$C$▫ in ▫$T$▫ with a meridional disc gives ▫$\pm$▫ the algebraic index of ▫$C$▫ in ▫$T$▫. One key idea is introducing the notion of geometric index for solid chambers of the form ▫$B^2 \times I$▫ in ▫$T$▫. We prove that if a solid torus can be divided into solid chambers by meridional discs in a specific (and often easy to obtain) way, then the geometric index can be easily computed.

Keywords

algebraic index;geometric index;Whitehead link;Bing link;McMillan link;Gabai link;Antoine link;

Data

Language:	English
Year of publishing:	2017
Typology:	1.01 - Original Scientific Article
Organization:	UL FMF - Faculty of Mathematics and Physics
UDC:	515.124
COBISS:	18167129
ISSN:	1660-5446
Views:	395
Downloads:	296
Average score:	0 (0 votes)
Metadata:

Other data

Type (COBISS):	Article
Pages:	art. 237 (15 str.)
Volume:	ǂVol. ǂ14
Issue:	ǂiss. ǂ6
Chronology:	2017
DOI:	10.1007/s00009-017-1036-1
ID:	11216345