Knot invariant
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In the
A knot invariant is a quantity defined on the set of all knots, which takes the same value for any two equivalent knots. For example, a knot group is a knot invariant.[5]
Typically a knot invariant is a
combinatorialquantity defined on knot diagrams. Thus if two knot diagrams differ with respect to some knot invariant, they must represent different knots. However, as is generally the case with topological invariants, if two knot diagrams share the same values with respect to a [single] knot invariant, then we still cannot conclude that the knots are the same.[6]
From the modern perspective, it is natural to define a knot invariant from a
Other invariants can be defined by considering some integer-valued function of knot diagrams and taking its minimum value over all possible diagrams of a given knot. This category includes the crossing number, which is the minimum number of crossings for any diagram of the knot, and the bridge number, which is the minimum number of bridges for any diagram of the knot.
Historically, many of the early knot invariants are not defined by first selecting a diagram but defined intrinsically, which can make computing some of these invariants a challenge. For example,
The
By
In recent years, there has been much interest in
There is also growing interest from both knot theorists and scientists in understanding "physical" or geometric properties of knots and relating it to topological invariants and knot type. An old result in this direction is the Fáry–Milnor theorem states that if the total curvature of a knot K in satisfies
where κ(p) is the
An example of a "physical" invariant is ropelength, which is the length of unit-diameter rope needed to realize a particular knot type.
Other invariants
- Linking number – Numerical invariant that describes the linking of two closed curves in three-dimensional space
- Finite type invariant (or Vassiliev or Vassiliev–Goussarov invariant)
- Stick number – Smallest number of edges of an equivalent polygonal path for a knot
Sources
- ISBN 9781470410209
- ^ ISBN 9789401004466.
- ^ ISBN 9781470454999"A knot invariant is a function from the set of knots to some other set whose value depends only on the equivalence class of the knot."
- ^ ISBN 9781470447816"A knot invariant is a mathematical property or quantity associated with a knot that does not change as we perform triangular moves on the knot.
- ISBN 9781447121589. "Likewise," with knot invariants, "a quantity inv(L) = inv(L') for any two equivalent links L and L'."
- ISBN 9781421424071.
- PMID 27868114.
- ^ Skerritt, Matt (June 27, 2003). "An Introduction to Knot Theory" (PDF). carmamaths.org. p. 22. Archived (PDF) from the original on November 19, 2022. Retrieved November 19, 2022.
- ^ Hodorog, Mădălina (February 2, 2010). "Basic Knot Theory" (PDF). www.dk-compmath.jku.at/people/mhodorog/. p. 47. Archived (PDF) from the original on November 19, 2022. Retrieved November 19, 2022.
- JSTOR 1970594.
Further reading
- Rolfsen, Dale (2003). Knots and Links. Providence, RI: AMS. ISBN 0-8218-3436-3.
- Adams, Colin Conrad (2004). The Knot Book: an Elementary Introduction to the Mathematical Theory of Knots (Repr., with corr ed.). Providence, RI: AMS. ISBN 0-8218-3678-1.
- Burde, Gerhard; Zieschang, Heiner (2002). Knots (2nd rev. and extended ed.). New York: De Gruyter. ISBN 3-11-017005-1.
External links
- Cha, Jae Choon; Livingston, Charles. "KnotInfo: Table of Knot Invariants". Indiana.edu. Retrieved 17 August 2021.
- "Invariants", The Knot Atlas.