Editing Knot invariant

{{Short description|Function of a knot that takes the same value for equivalent knots}}
{{No footnotes|date=May 2019}}
[[Image:Knot table.svg|thumb|[[Prime knot]]s are organized by the crossing number invariant.]]

In the [[mathematics|mathematical]] field of [[knot theory]], a '''knot invariant''' is a quantity (in a broad sense) defined for each [[knot (mathematics)|knot]] which is the same for [[equivalence relation|equivalent]] knots. The equivalence is often given by [[ambient isotopy]] but can be given by [[homeomorphism]].<ref>Schultens, Jennifer (2014). ''Introduction to 3-manifolds'', p.113. American Mathematical Society. {{ISBN|9781470410209}}</ref> Some invariants are indeed numbers (algebraic<ref name="Ricca"/>), but invariants can range from the simple, such as a yes/no answer, to those as complex as a [[homology theory]] (for example, "a ''knot invariant'' is a rule that assigns to any knot {{math|''K''}} a quantity {{math|φ(''K'')}} such that if {{math|''K''}} and {{math|{{em|K'}}}} are equivalent then {{math|φ(''K'') {{=}} φ({{em|K'}})}}."<ref name="Purcell"/>). Research on invariants is not only motivated by the basic problem of distinguishing one knot from another but also to understand fundamental properties of knots and their relations to other branches of mathematics. Knot invariants are thus used in knot classification,<ref name="Purcell">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."</ref><ref name=M&S>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.</ref> both in "enumeration" and "duplication removal".<ref name="Ricca">Ricca, Renzo L.; ed. (2012). ''An Introduction to the Geometry and Topology of Fluid Flows'', p.67. Springer Netherlands. {{ISBN|9789401004466}}.</ref>

{{Quote|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.<ref>Morishita, Masanori (2011). ''Knots and Primes: An Introduction to Arithmetic Topology'', p.16. Springer London. {{ISBN|9781447121589}}. "Likewise," with knot invariants, "a quantity {{math|inv(L) {{=}} inv(L')}} for any two equivalent links {{math|''L''}} and {{math|{{em|L'}}}}."</ref>}}

{{Quote|Typically a knot invariant is a [[combinatorial]]
 quantity 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.<ref>Ault, Shaun V. (2018). ''Understanding Topology: A Practical Introduction'', p.245. Johns Hopkins University Press. {{ISBN|9781421424071}}.</ref>}}

From the modern perspective, it is natural to define a knot invariant from a [[knot diagram]]. Of course, it must be unchanged (that is to say, invariant) under the [[Reidemeister move]]s ("triangular moves"<ref name=M&S/>). [[Tricolorability]] (and ''n''-colorability) is a particularly simple and common example. Other examples are [[knot polynomial]]s, such as the [[Jones polynomial]], which are currently among the most useful invariants for distinguishing knots from one another, though currently it is not known whether there exists a knot polynomial which distinguishes all knots from each other.<ref>{{Cite journal |last1=Horner |first1=Kate |last2=Miller |first2=Mark |last3=Steedb |first3=Jonathan |last4=Sutcliffe |first4=Paul |date=August 20, 2016 |title=Knot theory in modern chemistry |url=https://pubs.rsc.org/en/content/getauthorversionpdf/C6CS00448B |journal=[[Chemical Society Reviews]] |publisher=Royal Society of Chemistry |volume=45 |issue=23 |pages=6409–6658 |doi=10.1039/c6cs00448b|pmid=27868114 }}</ref><ref>{{Cite web |last=Skerritt |first=Matt |date=June 27, 2003 |title=An Introduction to Knot Theory |url=https://carmamaths.org/resources/jon/Preprints/Talks/M2600/Readings/KnotTheory.pdf |url-status=live |archive-url=https://web.archive.org/web/20221119163202/https://carmamaths.org/resources/jon/Preprints/Talks/M2600/Readings/KnotTheory.pdf |archive-date=November 19, 2022 |access-date=November 19, 2022 |website=carmamaths.org |page=22}}</ref><ref>{{Cite web |last=Hodorog |first=Mădălina |date=February 2, 2010 |title=Basic Knot Theory |url=https://www.dk-compmath.jku.at/people/mhodorog/data-mh/january.pdf |url-status=live |archive-url=https://web.archive.org/web/20221119181021/https://www.dk-compmath.jku.at/people/mhodorog/data-mh/january.pdf |archive-date=November 19, 2022 |access-date=November 19, 2022 |website=www.dk-compmath.jku.at/people/mhodorog/ |page=47}}</ref> However, there are invariants which distinguish the [[unknot]] from all other knots, such as [[Khovanov homology]] and [[Floer homology#Heegaard Floer homology|knot Floer homology]].

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 (knot theory)|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, [[knot genus]] is particularly tricky to compute, but can be effective (for instance, in distinguishing [[mutation (knot theory)|mutants]]).

The [[knot complement|complement of a knot]] itself (as a [[topological space]]) is known to be a "complete invariant" of the knot by the [[Gordon–Luecke theorem]] in the sense that it distinguishes the given knot from all other knots up to [[ambient isotopy]] and [[mirror image (knot theory)|mirror image]]. Some invariants associated with the knot complement include the [[knot group]] which is just the [[fundamental group]] of the complement. The [[knot quandle]] is also a complete invariant in this sense but it is difficult to determine if two quandles are isomorphic. The [[peripheral subgroup]] can also work as a complete invariant.<ref>{{Cite journal |last=Waldhausen |first=Friedhelm |date=1968 |title=On Irreducible 3-Manifolds Which are Sufficiently Large |url=https://www.jstor.org/stable/1970594 |journal=Annals of Mathematics |volume=87 |issue=1 |pages=56–88 |doi=10.2307/1970594 |jstor=1970594 |issn=0003-486X}}</ref>

By [[Mostow rigidity theorem|Mostow–Prasad rigidity]], the hyperbolic structure on the complement of a [[hyperbolic link]] is unique, which means the [[hyperbolic volume (knot)|hyperbolic volume]] is an invariant for these knots and links. Volume, and other hyperbolic invariants, have proven very effective, utilized in some of the extensive efforts at [[knot table|knot tabulation]].

In recent years, there has been much interest in [[homology theory|homological]] invariants of knots which [[categorification|categorify]] well-known invariants. [[Floer homology#Heegaard Floer homology|Heegaard Floer homology]] is a [[homology theory]] whose [[Euler characteristic]] is the [[Alexander polynomial]] of the knot. It has been proven effective in deducing new results about the classical invariants. Along a different line of study, there is a combinatorially defined cohomology theory of knots called [[Khovanov homology]] whose Euler characteristic is the [[Jones polynomial]]. This has recently been shown to be useful in obtaining bounds on [[slice genus]] whose earlier proofs required [[gauge theory]]. [[Mikhail Khovanov]] and Lev Rozansky have since defined several other related cohomology theories whose Euler characteristics recover other classical invariants. [[Catharina Stroppel]] gave a representation theoretic interpretation of Khovanov homology by categorifying quantum group invariants.

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 {{math|''K''}} in <math>\R^3</math> satisfies

:<math>\oint_K \kappa \,ds \leq 4\pi,</math>

where {{math|''κ''(''p'')}} is the [[Parametric curve#Curvature|curvature]] at {{math|''p''}}, then {{math|''K''}} is an unknot. Therefore, for knotted curves,

:<math>\oint_K \kappa\,ds > 4\pi.\,</math>

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==
* {{annotated link|Linking number}}
* {{annotated link|Finite type invariant}} (or Vassiliev or Vassiliev–Goussarov invariant)
* {{annotated link|Stick number}}
* {{annotated link|Arnold invariants}}

==Sources==
{{reflist}}

==Further reading==
*{{Cite book |last=Rolfsen |first=Dale |title=Knots and Links |location=Providence, RI |publisher=AMS |year=2003 |isbn=0-8218-3436-3 }}
*{{Cite book |last=Adams |first=Colin Conrad |title=The Knot Book: an Elementary Introduction to the Mathematical Theory of Knots |location=Providence, RI |publisher=AMS |edition=Repr., with corr |year=2004 |isbn=0-8218-3678-1 }}
*{{Cite book |last1=Burde |first1=Gerhard |last2=Zieschang |first2=Heiner |title=Knots |location=New York |publisher=De Gruyter |edition=2nd rev. and extended |year=2002 |isbn=3-11-017005-1 }}

==External links==
*{{cite web|first1=Jae Choon |last1=Cha | first2=Charles|last2= Livingston|url=https://knotinfo.math.indiana.edu/ |title=KnotInfo: Table of Knot Invariants|website=Indiana.edu|access-date= 17 August 2021}}
*{{Knot Atlas|Invariants}}

{{Knot theory|state=collapsed}}

[[Category:Knot invariants| ]]