In the
mathematical
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
field of
knot theory
In the mathematical field of topology, knot theory is the study of knot (mathematics), mathematical knots. While inspired by knots which appear in daily life, such as those in shoelaces and rope, a mathematical knot differs in that the ends are ...
, a split link is a
link that has a (topological) 2-sphere in its complement separating one or more link components from the others. A split link is said to be splittable, and a link that is not split is called a non-split link or not splittable. Whether a link is split or non-split corresponds to whether the
link complement
In mathematics, the knot complement of a tame knot ''K'' is the space where the knot is not. If a knot is embedded in the 3-sphere, then the complement is the 3-sphere minus the space near the knot. To make this precise, suppose that ''K'' is a ...
is reducible or irreducible as 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 lo ...
.
A link with an alternating diagram, i.e. an
alternating link
Alternating may refer to:
Mathematics
* Alternating algebra, an algebra in which odd-grade elements square to zero
* Alternating form, a function formula in algebra
* Alternating group, the group of even permutations of a finite set
* Alternati ...
, will be non-split if and only if this diagram is connected. This is a result of the work of
William Menasco William W. Menasco is a topologist and a professor at the University at Buffalo. He is best known for his work in knot theory.
Biography
Menasco received his B.A. from the University of California, Los Angeles in 1975, and his Ph.D. from the Univ ...
.
[.] A split link has many connected, non-alternating link diagrams.
References
Links (knot theory)
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