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.

Beginning with the Jones polynomial, infinitely many new invariants of knots, links, and 3-manifolds were found during the 1980s.

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.

A clasper, like a framed link, is an embedded topological object in a 3-manifold on which one can perform surgery.

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.

There are four types of constituents: leaves, disk-leaves, nodes, and boxes.

Clasper surgery is most easily defined (after elimination of nodes, boxes, and disk-leaves as described below) as surgery along a link associated to the clasper by replacing each leaf with its core, and replacing each edge by a right Hopf link.

The following are the graphical conventions used when drawing claspers (and may be viewed as a definition for boxes, nodes, and disk-leaves): Habiro found 12 moves which relate claspers along which surgery gives the same result.

These moves form the core of clasper calculus, and give considerable power to the theory as a theorem-proving tool.

-moves, which are the local moves induced by surgeries on a simple tree claspers without boxes or disk-leaves and with

, the following conditions are equivalent: The corresponding statement is false for links.