Claspers and finite type invariants of links.

By Kazuo Habiro. 2000

343 citations recorded by the source corpus.

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 dene for each positive integer k an equivalence relation on links called \Ck {equivalence," which is generated by surgery operations of a certain kind called \Ck {moves". We prove that two knots in the 3{sphere are Ck+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 elds in 3{dimensional topology.

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/7e6de64d9b85ca85742d6f9197b62eb784f4f0b1

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In Applied Mathematics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Cohomological structure of the mapping class group and beyond (2005, background): “In his theory developed in [ 27 ], Habiro introduced the concept of homology cobordism of surfaces and proposed interesting problems concerning it. Goussarov [26] also studied the same thing in his theory.”
New perspectives on self-linking (2003, background): “[19] • Capped grope cobordism of class n.”
On Knots with trivial Alexander polynomial (2002, background): “For a reference on claspers1 and their associated surgery, we refer the reader to [14, 15, 16] and also [3, 9].”
A unified Witten–Reshetikhin–Turaev invariant for integral homology spheres (2006, background): “An interesting generalization of integral homology spheres (more precisely, integral homology 3-balls) are homology cylinders over a surface [12, 15] (see also [8, 14]).”
Knot concordance and von Neumann $\rho$-invariants (2004, methodology): “For those familiar with the language of claspers [ Hab ], J can also be described as the knot obtained by performing clasper surgery on the unknot along the height 2 clasper as shown in Figure 3.6 (we have used a convention wherein an edge corresponds to a left-handed Hopf clasp).”
Grope cobordism of classical knots (2000, methodology): “As an example, one can use the Vassiliev degree v( ) := (number of vertices of )=2 to obtain exactly the well-known Vassiliev ltration used in Theorem 2: The main theorem of Habiro [14] states that two knots represent the same element in K=Fk if and only if they share the same Vassiliev invariants o”
Finite-type invariants of w-knotted objects, I: w-knots and the Alexander polynomial (2014, methodology): “21 In the case of usual knots and usual chord diagrams, Jacobi diagrams have a topological interpretation using the Goussarov–Habiro calculus of claspers [41; 46].”
Noncommutative knot theory. (2002, background): “Habiro wherein the loop is arbitrary but the analogue of the infection parameter J is increasingly subtle (for example, in Stanford’s case, J must lie in the nth term of the lower central series of the pure braid group; and, in the claspers that Habiro associated to Vassiliev theory, the analogue of”
Calculus of clovers and finite type invariants of 3–manifolds (2000, methodology): “In particular, the basic claspers, which were Habiro’s main tool in the theory for links, are not allowed in this setting.”
Infinitesimal Morita homomorphisms and the tree-level of the LMO invariant (2008, background): “Fundamental in their works is the notion of Ydequivalence for 3-manifolds [12, 15, 11, 9].”
Problems on invariants of knots and 3-manifolds (2004, methodology): “Habiro [165], see also [153, \Theorem 5"]) Two mstrand string links L and L0 are Cd -equivalent if and only if v(L) = v(L0) for any A-valued nite type invariant v of degree < d for any abelian group A.”
Problems on Mapping Class Groups And Related Topics (2006, methodology): “3 below) of Ivanov [22] for g ≥ 2 and of the author [27] for g ≤ 1 that the restriction of φ to φn(G) is the restriction of an automorphism ψ of the mapping class group.”