In the mathematical field of

low-dimensional topology In mathematics, low-dimensional topology is the branch of topology that studies manifolds, or more generally topological spaces, of four or fewer dimensions. Representative topics are the structure theory of 3-manifolds and 4-manifolds, knot the ...

, a clasper is a

surface A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is t ...

(with extra structure) in a

3-manifold In mathematics, a 3-manifold is a space that locally looks like Euclidean 3-dimensional space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane to a small enough observer, all 3-manifolds ...

on which surgery can be performed.

Motivation

Beginning with the

Jones polynomial In the mathematical field of knot theory, the Jones polynomial is a knot polynomial discovered by Vaughan Jones in 1984. Specifically, it is an invariant of an oriented knot or link which assigns to each oriented knot or link a Laurent polyno ...

, infinitely many new invariants of knots, links, and

s 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 In mathematics, a formal sum, formal series, or formal linear combination may be: *In group theory, an element of a free abelian group, a sum of finitely many elements from a given basis set multiplied by integer coefficients. *In linear algebra, an ...

of Jacobi diagrams (which carry a Lie algebra structure), and a representation of a ribbon Hopf algebra such as a

quantum group In mathematics and theoretical physics, the term quantum group denotes one of a few different kinds of noncommutative algebras with additional structure. These include Drinfeld–Jimbo type quantum groups (which are quasitriangular Hopf algebra ...

. 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 In mathematics, a knot is an embedding of the circle into three-dimensional Euclidean space, (also known as ). Often two knots are considered equivalent if they are ambient isotopic, that is, if there exists a continuous deformation of ...

, is an

embedded Embedded or embedding (alternatively imbedded or imbedding) may refer to: Science * Embedding, in mathematics, one instance of some mathematical object contained within another instance ** Graph embedding * Embedded generation, a distributed ge ...

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 In mathematics, the Kirby calculus in geometric topology, named after Robion Kirby, is a method for modifying framed links in the 3-sphere using a finite set of moves, the Kirby moves. Using four-dimensional Cerf theory, he proved that if ''M'' ...

on which only certain specific types of framed links are allowed. Claspers may also be interpreted algebraically, as a

diagram calculus A diagram is a symbolic representation of information using visualization techniques. Diagrams have been used since prehistoric times on walls of caves, but became more prevalent during the Enlightenment. Sometimes, the technique uses a three ...

for the braided strict monoidal category Cob of

oriented In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclockwise". A space is ...

connected surfaces with connected boundary. Additionally, most crucially, claspers may be roughly viewed as a topological realization of Jacobi diagrams, which are purely

combinatorial Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many ap ...

objects. This explains the

Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi iden ...

structure of the

graded vector space In mathematics, a graded vector space is a vector space that has the extra structure of a '' grading'' or a ''gradation'', which is a decomposition of the vector space into a direct sum of vector subspaces. Integer gradation Let \mathbb be ...

of Jacobi diagrams in terms of the Hopf algebra structure of Cob.

Definition

A clasper

G=\mathbf\cup\mathbf

is a compact surface embedded in the interior of a 3-manifold

M

equipped with a decomposition into two subsurfaces

\mathbf

and

\mathbf

, whose connected components are called the constituents and the edges of

G

correspondingly. Each edge of

G

is a band joining two constituents to one another, or joining one constituent to itself. 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.

Clasper calculus

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.

''C_n''-equivalence

Two knots, links, or 3-manifolds are said to be

C_n

-equivalent if they are related by

C_n

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

n

leaves. For a link

L\subset M

, a

C_1

-move is a crossing change. A

C_2

-move is a

Delta move Delta commonly refers to: * Delta (letter) (Δ or δ), a letter of the Greek alphabet * River delta, at a river mouth * D ( NATO phonetic alphabet: "Delta") * Delta Air Lines, US * Delta variant of SARS-CoV-2 that causes COVID-19 Delta may als ...

. Most applications of claspers use only

C_n

-moves.

Main results

For two knots

K

and

K^\prime

and a non-negative integer

k

, the following conditions are equivalent: #

K

and

K^\prime

are not distinguished by any invariant of type

k

. #

K

and

K^\prime

are

C_k

-equivalent. The corresponding statement is false for links.

Motivation

Definition

Clasper calculus

''Cn''-equivalence

Main results

Further reading

''C_n''-equivalence