An estimate of the triple point numbers of surface-knots by quandle cocycle invariants
type: Item
instance of: scholarly article
title: An estimate of the triple point numbers of surface-knots by quandle cocycle invariants
zbMATH Open document ID: 1052.57031
author: Eri Hatakenaka
published in: Topology and its Applications
publication date: 2004-08-19T00:00:00Z
review text: A surface knot is a closed 2 manifold embedded in \(\mathbb{R}^4\), considered up to ambient isotopy; surface knots are represented by surface knot diagrams, i.e. immersed surfaces in \(\mathbb{R}^3\) with singular set enhanced to show crossing information. This singular set consists of a union of simple closed curves and arcs which may intersect in a number of triple points. The triple point number of a surface knot \(K\) is the minimal number of triple points over all diagrams of \(K\); it is analagous to the crossing number for ordinary knots. In this paper, the author establishes a lower bound on the triple point number of oriented surface knots which have nontrival values of quandle 3-cocyle invariants over the dihedral quandle \(R_5\).
reviewed by: Sam Nelson
Mathematics Subject Classification ID: 57Q45
Mathematics Subject Classification ID: 57Q35
zbMATH DE Number: 2092794
zbMATH Keywords: surface knots
zbMATH Keywords: quandle cocyle invariants
zbMATH Keywords: triple point number
MaRDI profile type: Publication
full work available at URL: j.topol.2003.09.006
OpenAlex ID: W1986694022
cites work: Q4375394
cites work: State-sum invariants of knotted curves and surfaces from quandle cohomology
cites work: Alexander numbering of knotted surface diagrams
cites work: Some calculations of cohomology groups of finite Alexander quandles
cites work: The 3-cocycles of the Alexander quandles \(\mathbb F_q[T]/(T-\omega)\)
cites work: On non-orientable surfaces in 4-space which are projected with at most one triple point
cites work: AN INFINITE FAMILY OF NON-INVERTIBLE SURFACES IN 4-SPACE
cites work: The 2-twist-spun trefoil has the triple point number four
cites work: On simply knotted spheres in \(R^ 4\)
DOI: 10.1016/J.TOPOL.2003.09.006
Recommended article: Q5462529
Recommended article: Triple point numbers of surface-links and symmetric quandle cocycle invariants
Recommended article: Triple point cancelling numbers of surface links and quandle cocycle invariants
Recommended article: Computations of quandle cocycle invariants of knotted curves and surfaces
Recommended article: Q3678318
Recommended article: Quandle homotopy invariants of knotted surfaces
Recommended article: On the triple point number of surface-links in Yoshikawa’s table
Recommended article: Non-additivity for triple point numbers on the connected sum of surface-knots
Recommended article: Triple point invariants of non-orientable surface-links
Recommended article: Quandle cohomology and state-sum invariants of knotted curves and surfaces
description: scientific article; zbMATH DE number 2092794