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knot theory In the mathematical field of topology, knot theory is the study of 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 joined so it cannot ...
, a petal projection of a knot is a
knot diagram In the mathematical field of topology, knot theory is the study of 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 joined so it cannot ...
with a single crossing, at which an odd number of non-nested arcs ("petals") all meet. Because the above-below relation between the branches of a knot at this crossing point is not apparent from the appearance of the diagram, it must be specified separately, as a
permutation In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or p ...
describing the top-to-bottom ordering of the branches. Every knot or link has a petal projection; the minimum number of petals in such a projection defines a
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 by ambient isotopy but can be given by homeomorphism. Some ...
, the petal number of the knot. Petal projections can be used to define the Petaluma model, a family of
probability distribution In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomeno ...
s on knots with a given number of petals, defined by choosing a
random permutation A random permutation is a random ordering of a set of objects, that is, a permutation-valued random variable. The use of random permutations is often fundamental to fields that use randomized algorithms such as coding theory, cryptography, and sim ...
for the branches of a petal diagram.


Petal projection

A petal projection is a description of a knot as a special kind of knot diagram, a two-dimensional self-crossing curve formed by projecting the knot from three dimensions down to a plane. In a petal projection, this diagram has only one crossing point, forming a
topological rose In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ho ...
. Every two branches of the curve that pass through this point cross each other there; branches that meet tangentially without crossing are not allowed. The "petals" formed by arcs of the curve that leave and then return to this crossing point are all non-nested, bounding closed disks that are disjoint except for their common intersection at the crossing point. Beyond this topological description, the precise shape of the curve is unimportant. For instance, curves of this type could be realized algebraically as certain rose curves. However, it is common instead to draw a petal projection using straight line segments across the crossing point, connected at their endpoints by smooth curves to form the petals. In order to specify the above-below relation of the branches of the curve at the crossing point, each branch is labeled with an integer, from 1 to the number of branches, giving its position in the top-down ordering of the branches as would be seen from a three-dimensional viewpoint above the projected diagram. The
cyclic permutation In mathematics, and in particular in group theory, a cyclic permutation (or cycle) is a permutation of the elements of some set ''X'' which maps the elements of some subset ''S'' of ''X'' to each other in a cyclic fashion, while fixing (that is, ...
of these integers, in the radial ordering of the branches around the crossing point, can be used as a purely combinatorial description of the petal projection. In order to form a single knot, rather than a link, a petal projection must have an odd number of branches at its crossing point. Every knot can be represented as a petal projection, for diagrams with a sufficiently large number of petals. The minimum possible number of petals in a petal projection of a given knot defines a
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 by ambient isotopy but can be given by homeomorphism. Some ...
called its petal number.


Petaluma model

The Petaluma model is a random distribution on knots, parameterized by an odd number 2n+1 of petals in a petal diagram, and defined by constructing a petal diagram with this number of petals using a uniformly random permutation on its branches.


Generalization to links

Petal projections, and the petaluma model, can be generalized from knots to links. However, for this generalization, it is no longer possible to guarantee that all petals are non-nested. Instead, the generalized petal projections for links have a different type of standard diagram allowing some nesting of the petals.


References

{{reflist, refs= {{citation , last1 = Adams , first1 = Colin , author1-link = Colin Adams (mathematician) , last2 = Crawford , first2 = Thomas , last3 = DeMeo , first3 = Benjamin , last4 = Landry , first4 = Michael , last5 = Lin , first5 = Alex Tong , last6 = Montee , first6 = MurphyKate , last7 = Park , first7 = Seojung , last8 = Venkatesh , first8 = Saraswathi , last9 = Yhee , first9 = Farrah , arxiv = 1208.5742 , doi = 10.1142/S021821651550011X , issue = 3 , journal = Journal of Knot Theory and its Ramifications , mr = 3342136 , page = 1550011, 30 , title = Knot projections with a single multi-crossing , volume = 24 , year = 2015 {{citation , last1 = Adams , first1 = Colin , author1-link = Colin Adams (mathematician) , last2 = Capovilla-Searle , first2 = Orsola , last3 = Freeman , first3 = Jesse , last4 = Irvine , first4 = Daniel , last5 = Petti , first5 = Samantha , last6 = Vitek , first6 = Daniel , last7 = Weber , first7 = Ashley , last8 = Zhang , first8 = Sicong , arxiv = 1311.0526 , doi = 10.1142/S0218216515500121 , issue = 2 , journal = Journal of Knot Theory and its Ramifications , mr = 3334663 , page = 1550012, 16 , title = Bounds on übercrossing and petal numbers for knots , volume = 24 , year = 2015 {{citation , last1 = Even-Zohar , first1 = Chaim , last2 = Hass , first2 = Joel , author2-link = Joel Hass , last3 = Linial , first3 = Nati , author3-link = Nati Linial , last4 = Nowik , first4 = Tahl , arxiv = 1411.3308 , doi = 10.1007/s00454-016-9798-y , issue = 2 , journal =
Discrete & Computational Geometry '' Discrete & Computational Geometry'' is a peer-reviewed mathematics journal published quarterly by Springer. Founded in 1986 by Jacob E. Goodman and Richard M. Pollack, the journal publishes articles on discrete geometry and computational ge ...
, mr = 3530968 , pages = 274–314 , title = Invariants of random knots and links , volume = 56 , year = 2016
Knot theory