New techniques for computing geometric index

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

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.

• Language: English
• Keywords: algebraic index, geometric index, Whitehead link, Bing link, McMillan link, Gabai link, Antoine link
• Work type: Article
• Typology: 1.01 - Original Scientific Article
• Organization: PEF - Faculty of Education; FMF - Faculty of Mathematics and Physics
• Year: 2017
• Number of pages: Str. 1-15
• Numbering: art. 237, iss. 6
• PID: 20.500.12556/RUL-109537
• UDC: 515.124
• ISSN on article: 1660-5446
• DOI: 10.1007/s00009-017-1036-1
• COBISS.SI-ID: 18167129
• Publication date in RUL: 05.09.2019

Record is a part of a journal

• Title: Mediterranean journal of mathematics
• Shortened title: Mediterr. j. math.
• Publisher: Springer Nature, Birkhäuser
• ISSN: 1660-5446
• COBISS.SI-ID: 13561433