- Knot invariant
In the mathematical field of
knot theory , a knot invariant is a quantity (in a broad sense) defined for each knot which is the same for equivalent knots. The equivalence is often given byambient isotopy but can be given byhomeomorphism . Some invariants are indeed numbers, but invariants can range from the simple, such as a yes/no answer, to those as complex as ahomology theory . 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.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 theReidemeister moves .Tricoloring is a particularly simple example. Other examples areknot polynomial s, such as theJones polynomial , which are currently among the most useful invariants for distinguishing knots from one another, though at the time of writing it is not known whether there exists a knot polynomial which distinguishes all knots from each other, or even which distinguishes just theunknot from all other knots.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 thebridge 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 mutants).The complement of a knot itself (as a
topological space ) is known to be a "complete invariant" of the knot by theGordon-Luecke theorem in the sense that it distinguishes the given knot from all other knots up toambient isotopy and mirror image. Some invariants associated with the knot complement include theknot group which is just thefundamental group of the complement. Theknot quandle is also a complete invariant in this sense but it is difficult to determine if two quandles are isomorphic.Most knots are hyperbolic, which means the hyperbolic volume is an invariant for these knots. Volume, and other hyperbolic invariants, have proven very effective, utilized in some of the extensive efforts at knot tabulation.
In recent years, there has been much interest in homological invariants of knots which categorify well-known invariants. Heegaard Floer homology is a
homology theory whoseEuler characteristic is theAlexander 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 calledKhovanov homology whose Euler characteristic is theJones polynomial . This has recently been shown to be useful in obtaining bounds onslice genus whose earlier proofs requiredgauge theory . Khovanov and Rozansky have since defined several other related cohomology theories whose Euler characteristics recover other classical 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
Fary-Milnor theorem states that if thetotal curvature of a knot "K" in satisfies:
where is the curvature at "p", then "K" is an unknot. Therefore, for knotted curves,
:
An example of a "physical" invariant is
ropelength , which is the amount of 1-inch diameter rope needed to realize a particular knot type.Other invariants
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