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Knot invariant
Function of a knot that takes the same value for equivalent knots

In mathematics, particularly in knot theory, a knot invariant is a quantity assigned to each knot that remains unchanged under ambient isotopy or homeomorphism, helping to distinguish equivalent knots. Invariants range from simple yes/no properties like tricolorability to complex algebraic constructions such as the Khovanov homology and knot Floer homology. Important invariants derived from diagrams include knot polynomials like the Jones polynomial, while geometric invariants include the crossing number and ropelength. The knot complement serves as a complete invariant by the Gordon–Luecke theorem, illustrating the deep connections between topology and geometry in knot classification.

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Other invariants

  • Linking number – Numerical invariant that describes the linking of two closed curves in three-dimensional space
  • Finite type invariant – Type of invariant in Knot theory (or Vassiliev or Vassiliev–Goussarov invariant)
  • Stick number – Smallest number of edges of an equivalent polygonal path for a knot
  • Arnold invariants – Mathematical invariants used to classify plane curves

Sources

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.

References

  1. Schultens, Jennifer (2014). Introduction to 3-manifolds, p.113. American Mathematical Society. ISBN 9781470410209 /wiki/ISBN_(identifier) ↩

  2. Ricca, Renzo L.; ed. (2012). An Introduction to the Geometry and Topology of Fluid Flows, p.67. Springer Netherlands. ISBN 9789401004466. /wiki/ISBN_(identifier) ↩

  3. Purcell, Jessica (2020). Hyperbolic Knot Theory, p.7. American Mathematical Society. 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." /wiki/ISBN_(identifier) ↩

  4. Purcell, Jessica (2020). Hyperbolic Knot Theory, p.7. American Mathematical Society. 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." /wiki/ISBN_(identifier) ↩

  5. Messer, Robert and Straffin, Philip D. (2018). Topology Now!, p.50. American Mathematical Society. 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. /wiki/ISBN_(identifier) ↩

  6. Ricca, Renzo L.; ed. (2012). An Introduction to the Geometry and Topology of Fluid Flows, p.67. Springer Netherlands. ISBN 9789401004466. /wiki/ISBN_(identifier) ↩

  7. Morishita, Masanori (2011). Knots and Primes: An Introduction to Arithmetic Topology, p.16. Springer London. ISBN 9781447121589. "Likewise," with knot invariants, "a quantity inv(L) = inv(L') for any two equivalent links L and L'." /wiki/ISBN_(identifier) ↩

  8. Ault, Shaun V. (2018). Understanding Topology: A Practical Introduction, p.245. Johns Hopkins University Press. ISBN 9781421424071. /wiki/ISBN_(identifier) ↩

  9. Messer, Robert and Straffin, Philip D. (2018). Topology Now!, p.50. American Mathematical Society. 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. /wiki/ISBN_(identifier) ↩

  10. Horner, Kate; Miller, Mark; Steedb, Jonathan; Sutcliffe, Paul (August 20, 2016). "Knot theory in modern chemistry". Chemical Society Reviews. 45 (23). Royal Society of Chemistry: 6409–6658. doi:10.1039/c6cs00448b. PMID 27868114. https://pubs.rsc.org/en/content/getauthorversionpdf/C6CS00448B ↩

  11. 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. https://carmamaths.org/resources/jon/Preprints/Talks/M2600/Readings/KnotTheory.pdf ↩

  12. 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. https://www.dk-compmath.jku.at/people/mhodorog/data-mh/january.pdf ↩

  13. Waldhausen, Friedhelm (1968). "On Irreducible 3-Manifolds Which are Sufficiently Large". Annals of Mathematics. 87 (1): 56–88. doi:10.2307/1970594. ISSN 0003-486X. JSTOR 1970594. https://www.jstor.org/stable/1970594 ↩