Illinois Journal of Mathematics

Minimal genus of links and fibering of canonical surfaces

A. Stoimenow

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Abstract

This paper contains some further applications of the study of knot diagrams by genus. Introducing a procedure of regularization for knot generators, and using invariants derived from the Jones polynomial (degrees, congruences, and the Fiedler–Polyak–Viro Gauß diagram formulas for its Vassiliev invariants), we examine the existence of genus-minimizing diagrams for almost alternating and almost positive knots. In particular, we examine the existence of such knots such that either all or none of their almost alternating/positive diagrams have the minimal genus property. We prove that the genus of almost positive non-split links is determined by the Alexander polynomial.

Article information

Source
Illinois J. Math., Volume 59, Number 2 (2015), 399-448.

Dates
Received: 9 August 2015
Revised: 19 January 2016
First available in Project Euclid: 5 May 2016

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1462450708

Digital Object Identifier
doi:10.1215/ijm/1462450708

Mathematical Reviews number (MathSciNet)
MR3499519

Zentralblatt MATH identifier
1342.57011

Subjects
Primary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45}
Secondary: 57N10: Topology of general 3-manifolds [See also 57Mxx] 53D10: Contact manifolds, general 57M15: Relations with graph theory [See also 05Cxx]

Citation

Stoimenow, A. Minimal genus of links and fibering of canonical surfaces. Illinois J. Math. 59 (2015), no. 2, 399--448. doi:10.1215/ijm/1462450708. https://projecteuclid.org/euclid.ijm/1462450708


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