2000/01/28 by Kazuo Habiro · 197 citations
Mathematics · #Equivalence (formal languages) #Equivalence relation #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Integer (computer science) #Mathematical Dynamics and Fractals #Relation (database) #Section (typography) #Type (biology) #math.GT #math.QA #msc:18D10 #msc:57M05 #msc:57M25
paper · pdf · doi:10.2140/gt.2000.4.1
published in Geometry & Topology 4(1), 1-83 (Mathematical Sciences Publishers) · 83 pages. Published copy, also available at http://www.maths.warwick.ac.uk/gt/GTVol4/paper1.abs.html
arxiv created 2000/01/28 · openalex publication_date 2000/01/28 · arxiv updated 2014/11/11 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
We introduce the concept of "claspers," which are surfaces in 3-manifolds with some additional structure on which surgery operations can be performed. Using claspers we define for each positive integer k an equivalence relation on links called "C k -equivalence," which is generated by surgery operations of a certain kind called "C k -moves". We prove that two knots in the 3-sphere are C k+1 -equivalent if and only if they have equal values of Vassiliev-Goussarov invariants of type k with values in any abelian groups. This result gives a characterization in terms of surgery operations of the informations that can be carried by Vassiliev-Goussarov invariants. In the last section we also describe outlines of some applications of claspers to other fields in 3-dimensional topology.