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 mutation is an
operation
Operation or Operations may refer to:
Arts, entertainment and media
* ''Operation'' (game), a battery-operated board game that challenges dexterity
* Operation (music), a term used in musical set theory
* ''Operations'' (magazine), Multi-Ma ...
on a knot that can produce different knots. Suppose ''K'' is a knot given in the form of a
knot diagram. Consider a disc ''D'' in the projection plane of the diagram whose boundary circle intersects ''K'' exactly four times. We may suppose that (after planar isotopy) the disc is geometrically round and the four points of intersection on its boundary with ''K'' are equally spaced. The part of the knot inside the disc is a
tangle
Tangle may refer to: Science, Technology, Engineering & Mathematics
*''The Tangle'' is the name of the ledger, a directed acyclic graph, used for the cryptocurrency IOTA
*Tangle (mathematics), a topological object
Natural sciences & medicine ...
. There are two reflections that switch pairs of endpoints of the tangle. There is also a rotation that results from composition of the reflections. A mutation replaces the original tangle by a tangle given by any of these operations. The result will always be a knot and is called a mutant of ''K''.
Mutants can be difficult to distinguish as they have a number of the same invariants. They have the same
hyperbolic volume
In the mathematical field of knot theory, the hyperbolic volume of a hyperbolic link is the volume of the link's complement with respect to its complete hyperbolic metric. The volume is necessarily a finite real number, and is a topological inv ...
(by a result of Ruberman), and have the same
HOMFLY polynomial
In the mathematical field of knot theory, the HOMFLY polynomial or HOMFLYPT polynomial, sometimes called the generalized Jones polynomial, is a 2-variable knot polynomial, i.e. a knot invariant in the form of a polynomial of variables ''m'' and ' ...
s.
Examples
*
Conway and Kinoshita-Terasaka mutant pair, distinguished as
knot genus 3 and 2, respectively.
References
Further reading
*Colin Adams, ''The Knot Book'', American Mathematical Society,
External links
A list of pairs of mutant nodes
Knot operations
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