Knot (mathematics)

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A table of all prime knots with seven crossings or fewer (not including mirror images) Knot table.svg
A table of all prime knots with seven crossings or fewer (not including mirror images)
An overhand knot becomes a trefoil knot by joining the ends. Example of Knots.svg
An overhand knot becomes a trefoil knot by joining the ends.
The triangle is associated with the trefoil knot. Trefle.jpg
The triangle is associated with the trefoil knot.
Pretzel bread in the shape of a 74 pretzel knot Celtic 7 4 Knot.jpg
Pretzel bread in the shape of a 74 pretzel knot

In mathematics, a knot is an embedding of the circle ( S1 ) into three-dimensional Euclidean space, R3 (also known as E3). Often two knots are considered equivalent if they are ambient isotopic, that is, if there exists a continuous deformation of R3 which takes one knot to the other.

Contents

A crucial difference between the standard mathematical and conventional notions of a knot is that mathematical knots are closed — there are no ends to tie or untie on a mathematical knot. Physical properties such as friction and thickness also do not apply, although there are mathematical definitions of a knot that take such properties into account. The term knot is also applied to embeddings of S j in Sn, especially in the case j = n − 2. The branch of mathematics that studies knots is known as knot theory and has many relations to graph theory.

Formal definition

A knot is an embedding of the circle (S1) into three-dimensional Euclidean space (R3), [1] or the 3-sphere (S3), since the 3-sphere is compact. [2] [Note 1] Two knots are defined to be equivalent if there is an ambient isotopy between them. [3]

Projection

A knot in R3 (or alternatively in the 3-sphere, S3), can be projected onto a plane  R2 (respectively a sphere  S2). This projection is almost always regular, meaning that it is injective everywhere, except at a finite number of crossing points, which are the projections of only two points of the knot, and these points are not collinear. In this case, by choosing a projection side, one can completely encode the isotopy class of the knot by its regular projection by recording a simple over/under information at these crossings. In graph theory terms, a regular projection of a knot, or knot diagram is thus a quadrivalent planar graph with over/under-decorated vertices. The local modifications of this graph which allow to go from one diagram to any other diagram of the same knot (up to ambient isotopy of the plane) are called Reidemeister moves.

Types of knots

A knot can be untied if the loop is broken. Untying a knot.svg
A knot can be untied if the loop is broken.

The simplest knot, called the unknot or trivial knot, is a round circle embedded in R3 . [4] In the ordinary sense of the word, the unknot is not "knotted" at all. The simplest nontrivial knots are the trefoil knot (31 in the table), the figure-eight knot (41) and the cinquefoil knot (51). [5]

Several knots, linked or tangled together, are called links. Knots are links with a single component.

Tame vs. wild knots

A wild knot Wild knot.svg
A wild knot

A polygonal knot is a knot whose image in R3 is the union of a finite set of line segments. [6] A tame knot is any knot equivalent to a polygonal knot. [6] [Note 2] Knots which are not tame are called wild , [7] and can have pathological behavior. [7] In knot theory and 3-manifold theory, often the adjective "tame" is omitted. Smooth knots, for example, are always tame.

Framed knot

A framed knot is the extension of a tame knot to an embedding of the solid torus D2 × S1 in S3.

The framing of the knot is the linking number of the image of the ribbon I × S1 with the knot. A framed knot can be seen as the embedded ribbon and the framing is the (signed) number of twists. [8] This definition generalizes to an analogous one for framed links. Framed links are said to be equivalent if their extensions to solid tori are ambient isotopic.

Framed link diagrams are link diagrams with each component marked, to indicate framing, by an integer representing a slope with respect to the meridian and preferred longitude. A standard way to view a link diagram without markings as representing a framed link is to use the blackboard framing. This framing is obtained by converting each component to a ribbon lying flat on the plane. A type I Reidemeister move clearly changes the blackboard framing (it changes the number of twists in a ribbon), but the other two moves do not. Replacing the type I move by a modified type I move gives a result for link diagrams with blackboard framing similar to the Reidemeister theorem: Link diagrams, with blackboard framing, represent equivalent framed links if and only if they are connected by a sequence of (modified) type I, II, and III moves. Given a knot, one can define infinitely many framings on it. Suppose that we are given a knot with a fixed framing. One may obtain a new framing from the existing one by cutting a ribbon and twisting it an integer multiple of 2π around the knot and then glue back again in the place we did the cut. In this way one obtains a new framing from an old one, up to the equivalence relation for framed knots„ leaving the knot fixed. [9] The framing in this sense is associated to the number of twists the vector field performs around the knot. Knowing how many times the vector field is twisted around the knot allows one to determine the vector field up to diffeomorphism, and the equivalence class of the framing is determined completely by this integer called the framing integer.

Knot complement

A knot whose complement has a non-trivial JSJ decomposition Knot with borromean rings in jsj decomp small.png
A knot whose complement has a non-trivial JSJ decomposition

Given a knot in the 3-sphere, the knot complement is all the points of the 3-sphere not contained in the knot. A major theorem of Gordon and Luecke states that at most two knots have homeomorphic complements (the original knot and its mirror reflection). This in effect turns the study of knots into the study of their complements, and in turn into 3-manifold theory. [10]

JSJ decomposition

The JSJ decomposition and Thurston's hyperbolization theorem reduces the study of knots in the 3-sphere to the study of various geometric manifolds via splicing or satellite operations . In the pictured knot, the JSJ-decomposition splits the complement into the union of three manifolds: two trefoil complements and the complement of the Borromean rings. The trefoil complement has the geometry of H2 × R, while the Borromean rings complement has the geometry of H3.

Harmonic knots

Parametric representations of knots are called harmonic knots. Aaron Trautwein compiled parametric representations for all knots up to and including those with a crossing number of 8 in his PhD thesis. [11] [12]

Applications to graph theory

A table of all prime knots with up to seven crossings represented as knot diagrams with their medial graph KnotGraphTable.svg
A table of all prime knots with up to seven crossings represented as knot diagrams with their medial graph

Medial graph

KnotCheckerboard.svg
The signed planar graph associated with a knot diagram. KnotSignedGraph.svg
The signed planar graph associated with a knot diagram.
Left guide KnotSignedEdge+.png
Left guide
Right guide KnotSignedEdge-.png
Right guide

Another convenient representation of knot diagrams [13] [14] was introduced by Peter Tait in 1877. [15] [16]

Any knot diagram defines a plane graph whose vertices are the crossings and whose edges are paths in between successive crossings. Exactly one face of this planar graph is unbounded; each of the others is homeomorphic to a 2-dimensional disk. Color these faces black or white so that the unbounded face is black and any two faces that share a boundary edge have opposite colors. The Jordan curve theorem implies that there is exactly one such coloring.

We construct a new plane graph whose vertices are the white faces and whose edges correspond to crossings. We can label each edge in this graph as a left edge or a right edge, depending on which thread appears to go over the other as we view the corresponding crossing from one of the endpoints of the edge. Left and right edges are typically indicated by labeling left edges + and right edges –, or by drawing left edges with solid lines and right edges with dashed lines.

The original knot diagram is the medial graph of this new plane graph, with the type of each crossing determined by the sign of the corresponding edge. Changing the sign of every edge corresponds to reflecting the knot in a mirror.

Linkless and knotless embedding

The seven graphs in the Petersen family. No matter how these graphs are embedded into three-dimensional space, some two cycles will have nonzero linking number. Petersen family.svg
The seven graphs in the Petersen family. No matter how these graphs are embedded into three-dimensional space, some two cycles will have nonzero linking number.

In two dimensions, only the planar graphs may be embedded into the Euclidean plane without crossings, but in three dimensions, any undirected graph may be embedded into space without crossings. However, a spatial analogue of the planar graphs is provided by the graphs with linkless embeddings and knotless embeddings. A linkless embedding is an embedding of the graph with the property that any two cycles are unlinked; a knotless embedding is an embedding of the graph with the property that any single cycle is unknotted. The graphs that have linkless embeddings have a forbidden graph characterization involving the Petersen family, a set of seven graphs that are intrinsically linked: no matter how they are embedded, some two cycles will be linked with each other. [17] A full characterization of the graphs with knotless embeddings is not known, but the complete graph K7 is one of the minimal forbidden graphs for knotless embedding: no matter how K7 is embedded, it will contain a cycle that forms a trefoil knot. [18]

Generalization

In contemporary mathematics the term knot is sometimes used to describe a more general phenomenon related to embeddings. Given a manifold M with a submanifold N, one sometimes says N can be knotted in M if there exists an embedding of N in M which is not isotopic to N. Traditional knots form the case where N = S1 and M = R3 or M = S3. [19] [20]

The Schoenflies theorem states that the circle does not knot in the 2-sphere: every topological circle in the 2-sphere is isotopic to a geometric circle. [21] Alexander's theorem states that the 2-sphere does not smoothly (or PL or tame topologically) knot in the 3-sphere. [22] In the tame topological category, it's known that the n-sphere does not knot in the n + 1-sphere for all n. This is a theorem of Morton Brown, Barry Mazur, and Marston Morse. [23] The Alexander horned sphere is an example of a knotted 2-sphere in the 3-sphere which is not tame. [24] In the smooth category, the n-sphere is known not to knot in the n + 1-sphere provided n ≠ 3. The case n = 3 is a long-outstanding problem closely related to the question: does the 4-ball admit an exotic smooth structure?

André Haefliger proved that there are no smooth j-dimensional knots in Sn provided 2n − 3j − 3 > 0, and gave further examples of knotted spheres for all n > j ≥ 1 such that 2n − 3j − 3 = 0. n − j is called the codimension of the knot. An interesting aspect of Haefliger's work is that the isotopy classes of embeddings of S j in Sn form a group, with group operation given by the connect sum, provided the co-dimension is greater than two. Haefliger based his work on Stephen Smale's h-cobordism theorem. One of Smale's theorems is that when one deals with knots in co-dimension greater than two, even inequivalent knots have diffeomorphic complements. This gives the subject a different flavour than co-dimension 2 knot theory. If one allows topological or PL-isotopies, Christopher Zeeman proved that spheres do not knot when the co-dimension is greater than 2. See a generalization to manifolds.

See also

Notes

  1. ↑ Note that the 3-sphere is equivalent to R3 with a single point added at infinity (see one-point compactification).
  2. ↑ A knot is tame if and only if it can be represented as a finite closed polygonal chain

Related Research Articles

In graph theory, a planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints. In other words, it can be drawn in such a way that no edges cross each other. Such a drawing is called a plane graph or planar embedding of the graph. A plane graph can be defined as a planar graph with a mapping from every node to a point on a plane, and from every edge to a plane curve on that plane, such that the extreme points of each curve are the points mapped from its end nodes, and all curves are disjoint except on their extreme points.

<span class="mw-page-title-main">Complete graph</span> Graph in which every two vertices are adjacent

In the mathematical field of graph theory, a complete graph is a simple undirected graph in which every pair of distinct vertices is connected by a unique edge. A complete digraph is a directed graph in which every pair of distinct vertices is connected by a pair of unique edges.

<span class="mw-page-title-main">Knot theory</span> Study of mathematical knots

In 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 be undone, the simplest knot being a ring. In mathematical language, a knot is an embedding of a circle in 3-dimensional Euclidean space, . Two mathematical knots are equivalent if one can be transformed into the other via a deformation of upon itself ; these transformations correspond to manipulations of a knotted string that do not involve cutting it or passing it through itself.

<span class="mw-page-title-main">Knot invariant</span> Function of a knot that takes the same value for equivalent knots

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 invariants are indeed numbers (algebraic), 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 K a quantity φ(K) such that if K and K' are equivalent then φ(K) = φ(K')."). 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, both in "enumeration" and "duplication removal".

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.

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.

<span class="mw-page-title-main">Linking number</span> Numerical invariant that describes the linking of two closed curves in three-dimensional space

In mathematics, the linking number is a numerical invariant that describes the linking of two closed curves in three-dimensional space. Intuitively, the linking number represents the number of times that each curve winds around the other. In Euclidean space, the linking number is always an integer, but may be positive or negative depending on the orientation of the two curves.

<span class="mw-page-title-main">Geometric topology</span> Branch of mathematics studying (smooth) functions of manifolds

In mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another.

<span class="mw-page-title-main">Low-dimensional topology</span> Branch of topology

In mathematics, low-dimensional topology is the branch of topology that studies manifolds, or more generally topological spaces, of four or fewer dimensions. Representative topics are the structure theory of 3-manifolds and 4-manifolds, knot theory, and braid groups. This can be regarded as a part of geometric topology. It may also be used to refer to the study of topological spaces of dimension 1, though this is more typically considered part of continuum theory.

In topological graph theory, a mathematical discipline, a linkless embedding of an undirected graph is an embedding of the graph into three-dimensional Euclidean space in such a way that no two cycles of the graph are linked. A flat embedding is an embedding with the property that every cycle is the boundary of a topological disk whose interior is disjoint from the graph. A linklessly embeddable graph is a graph that has a linkless or flat embedding; these graphs form a three-dimensional analogue of the planar graphs. Complementarily, an intrinsically linked graph is a graph that does not have a linkless embedding.

In mathematics, the Kirby calculus in geometric topology, named after Robion Kirby, is a method for modifying framed links in the 3-sphere using a finite set of moves, the Kirby moves. Using four-dimensional Cerf theory, he proved that if M and N are 3-manifolds, resulting from Dehn surgery on framed links L and J respectively, then they are homeomorphic if and only if L and J are related by a sequence of Kirby moves. According to the Lickorish–Wallace theorem any closed orientable 3-manifold is obtained by such surgery on some link in the 3-sphere.

<span class="mw-page-title-main">Reidemeister move</span> One of three types of isotopy-preserving local changes to a knot diagram

In the mathematical area of knot theory, a Reidemeister move is any of three local moves on a link diagram. Kurt Reidemeister (1927) and, independently, James Waddell Alexander and Garland Baird Briggs (1926), demonstrated that two knot diagrams belonging to the same knot, up to planar isotopy, can be related by a sequence of the three Reidemeister moves.

<span class="mw-page-title-main">Link (knot theory)</span> Collection of knots which do not intersect, but may be linked

In mathematical knot theory, a link is a collection of knots which do not intersect, but which may be linked together. A knot can be described as a link with one component. Links and knots are studied in a branch of mathematics called knot theory. Implicit in this definition is that there is a trivial reference link, usually called the unlink, but the word is also sometimes used in context where there is no notion of a trivial link.

<span class="mw-page-title-main">Alternating knot</span>

In knot theory, a knot or link diagram is alternating if the crossings alternate under, over, under, over, as one travels along each component of the link. A link is alternating if it has an alternating diagram.

Colin de Verdière's invariant is a graph parameter for any graph G, introduced by Yves Colin de Verdière in 1990. It was motivated by the study of the maximum multiplicity of the second eigenvalue of certain Schrödinger operators.

<span class="mw-page-title-main">Unknotting problem</span> Determining whether a knot is the unknot

In mathematics, the unknotting problem is the problem of algorithmically recognizing the unknot, given some representation of a knot, e.g., a knot diagram. There are several types of unknotting algorithms. A major unresolved challenge is to determine if the problem admits a polynomial time algorithm; that is, whether the problem lies in the complexity class P.

In knot theory, a virtual knot is a generalization of knots in 3-dimensional Euclidean space, R3, to knots in thickened surfaces modulo an equivalence relation called stabilization/destabilization. Here is required to be closed and oriented. Virtual knots were first introduced by Kauffman (1999).

<span class="mw-page-title-main">Tricolorability</span> Property in knot theory

In the mathematical field of knot theory, the tricolorability of a knot is the ability of a knot to be colored with three colors subject to certain rules. Tricolorability is an isotopy invariant, and hence can be used to distinguish between two different (non-isotopic) knots. In particular, since the unknot is not tricolorable, any tricolorable knot is necessarily nontrivial.

In the mathematical field of low-dimensional topology, a clasper is a surface in a 3-manifold on which surgery can be performed.

<span class="mw-page-title-main">Apex graph</span> Graph which can be made planar by removing a single node

In graph theory, a branch of mathematics, an apex graph is a graph that can be made planar by the removal of a single vertex. The deleted vertex is called an apex of the graph. It is an apex, not the apex because an apex graph may have more than one apex; for example, in the minimal nonplanar graphs K5 or K3,3, every vertex is an apex. The apex graphs include graphs that are themselves planar, in which case again every vertex is an apex. The null graph is also counted as an apex graph even though it has no vertex to remove.

References

  1. ↑ Armstrong (1983), p. 213.
  2. ↑ Cromwell 2004 , p. 33; Adams 1994 , pp. 246–250
  3. ↑ Cromwell (2004), p. 5.
  4. ↑ Adams (1994), p. 2.
  5. ↑ Adams 1994 , Table 1.1, p. 280; Livingstone 1993 , Appendix A: Knot Table, p. 221
  6. 1 2 Armstrong 1983 , p. 215
  7. 1 2 Charles Livingston (1993). Knot Theory. Cambridge University Press. p. 11. ISBN   978-0-88385-027-5.
  8. ↑ Kauffman, Louis H. (1990). "An invariant of regular isotopy" (PDF). Transactions of the American Mathematical Society . 318 (2): 417–471. doi: 10.1090/S0002-9947-1990-0958895-7 .
  9. ↑ Elhamdadi, Mohamed; Hajij, Mustafa; Istvan, Kyle (2019), Framed Knots, arXiv: 1910.10257 .
  10. ↑ Adams 1994 , pp. 261–2
  11. ↑ Trautwein, Aaron K. (1995). Harmonic knots (PhD). Dissertation Abstracts International. Vol. 56–06. University of Iowa. p. 3234. OCLC   1194821918. ProQuest   304216894.
  12. ↑ Trautwein, Aaron K. (1998). "18. An introduction to Harmonic Knots". In Stasiak, Andrzej; Katritch, Vsevolod; Kauffman, Louis H. (eds.). Ideal Knots. World Scientific. pp. 353–363. ISBN   978-981-02-3530-7.
  13. ↑ Adams, Colin C. (2004). "§2.4 Knots and Planar Graphs". The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. American Mathematical Society. pp. 51–55. ISBN   978-0-8218-3678-1.
  14. ↑ Entrelacs.net tutorial
  15. ↑ Tait, Peter G. (1876–1877). "On Knots I". Proceedings of the Royal Society of Edinburgh. 28: 145–190. doi:10.1017/S0080456800090633. Revised May 11, 1877.
  16. ↑ Tait, Peter G. (1876–1877). "On Links (Abstract)". Proceedings of the Royal Society of Edinburgh. 9 (98): 321–332. doi:10.1017/S0370164600032363.
  17. ↑ Robertson, Neil; Seymour, Paul; Thomas, Robin (1993), "A survey of linkless embeddings", in Robertson, Neil; Seymour, Paul (eds.), Graph Structure Theory: Proc. AMS–IMS–SIAM Joint Summer Research Conference on Graph Minors (PDF), Contemporary Mathematics, vol. 147, American Mathematical Society, pp. 125–136.
  18. ↑ Ramirez Alfonsin, J. L. (1999), "Spatial graphs and oriented matroids: the trefoil", Discrete and Computational Geometry, 22 (1): 149–158, doi: 10.1007/PL00009446 .
  19. ↑ Carter, J. Scott; Saito, Masahico (1998). Knotted Surfaces and their Diagrams. Mathematical Surveys and Monographs. Vol. 55. American Mathematical Society. ISBN   0-8218-0593-2. MR   1487374.
  20. ↑ Kamada, Seiichi (2017). Surface-Knots in 4-Space. Springer Monographs in Mathematics. Springer. doi:10.1007/978-981-10-4091-7. ISBN   978-981-10-4090-0. MR   3588325.
  21. ↑ Hocking, John G.; Young, Gail S. (1988). Topology (2nd ed.). Dover Publications. p. 175. ISBN   0-486-65676-4. MR   1016814.
  22. ↑ Calegari, Danny (2007). Foliations and the geometry of 3-manifolds. Oxford Mathematical Monographs. Oxford University Press. p. 161. ISBN   978-0-19-857008-0. MR   2327361.
  23. ↑ Mazur, Barry (1959). "On embeddings of spheres". Bulletin of the American Mathematical Society. 65 (2): 59–65. doi: 10.1090/S0002-9904-1959-10274-3 . MR   0117693.Brown, Morton (1960). "A proof of the generalized Schoenflies theorem". Bulletin of the American Mathematical Society. 66 (2): 74–76. doi: 10.1090/S0002-9904-1960-10400-4 . MR   0117695.Morse, Marston (1960). "A reduction of the Schoenflies extension problem". Bulletin of the American Mathematical Society. 66 (2): 113–115. doi: 10.1090/S0002-9904-1960-10420-X . MR   0117694.
  24. ↑ Alexander, J. W. (1924). "An Example of a Simply Connected Surface Bounding a Region which is not Simply Connected". Proceedings of the National Academy of Sciences of the United States of America . National Academy of Sciences. 10 (1): 8–10. Bibcode:1924PNAS...10....8A. doi: 10.1073/pnas.10.1.8 . ISSN   0027-8424. JSTOR   84202. PMC   1085500 . PMID   16576780.

Bibliography

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