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Clasper (mathematics)
In the mathematical field of low-dimensional topology, a clasper is a surface (with extra structure) in a 3-manifold on which surgery can be performed.
== Motivation ==
Beginning with the Jones polynomial, infinitely many new invariants of knots, links, and 3-manifolds were found during the 1980s. The study of these new `quantum' invariants expanded rapidly into a sub-discipline of low-dimensional topology called quantum topology. A quantum invariant is typically constructed from two ingredients: a formal sum of Jacobi diagrams (which carry a Lie algebra structure), and a representation of a ribbon Hopf algebra such as a quantum group. It is not clear a-priori why either of these ingredients should have anything to do with low-dimensional topology. Thus one of the main problems in quantum topology has been to interpret quantum invariants topologically.
The theory of claspers comes to provide such an interpretation. A clasper, like a framed link, is an embedded topological object in a 3-manifold on which one can perform surgery. In fact, clasper calculus can be thought of as a variant of Kirby calculus on which only certain specific types of framed links are allowed. Claspers may also be interpreted algebraically, as a diagram calculus for the braided strict monoidal category Cob of oriented connected surfaces with connected boundary. Additionally, most crucially, claspers may be roughly viewed as a topological realization of Jacobi diagrams, which are purely combinatorial objects. This explains the Lie algebra structure of the graded vector space of Jacobi diagrams in terms of the Hopf algebra structure of Cob.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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