Knot (mathematics)

{{Short description|Embedding of the circle in three dimensional Euclidean space}} {{About|the mathematical object||Knot (disambiguation)}}

[[Image:Knot table.svg|thumb|350px|right|A table of all [[prime knot]]s with seven [[Crossing number (knot theory)|crossings]] or fewer (not including mirror images)]] [[Image:Example of Knots.svg|180px|thumb|An [[overhand knot]] becomes a [[trefoil knot]] by joining the ends.]] [[Image:Trefle.jpg|thumb|The triangle is associated with the trefoil knot.]] [[File:Celtic 7 4 Knot.jpg|thumb|[[Pretzel]] bread in the shape of a {{math|74}} [[pretzel link|pretzel knot]]]]

In [[mathematics]], a '''knot''' is an [[embedding]] of the [[Circle#Topological_definition|circle]] ({{math|[[n-sphere|''S''{{isup|1}}]]}}) into three-dimensional [[Euclidean space]], {{math|'''R'''3}} (also known as {{math|'''E'''3}}). Often two knots are considered equivalent if they are [[Ambient isotopy|ambient isotopic]], that is, if there exists a continuous deformation of {{math|'''R'''3}} which takes one knot to the other.

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 {{math|''S''{{isup| ''j''}}}} in {{math|''S''{{isup|''n''}}}}, especially in the case {{math|''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#General topology|embedding]] of the [[circle]] ({{math|''S''{{isup|1}}}}) into [[three-dimensional space|three-dimensional]] [[Euclidean space]] ({{math|'''R'''3}}),{{sfnp|Armstrong|1983|page=213}} or the [[3-sphere]] ({{math|''S''{{isup|3}}}}), since the 3-sphere is [[compact space|compact]].<ref>{{harvnb|Cromwell|2004|p=33}}; {{harvnb|Adams|1994|pp=246–250}}</ref>{{refn|group=Note|Note that the 3-sphere is equivalent to {{math|'''R'''3}} with a single point added at infinity (see [[one-point compactification]]).}} Two knots are defined to be equivalent if there is an [[ambient isotopy]] between them.{{sfnp|Cromwell|2004|page=5}}

===Projection===

A knot in {{math|[[Euclidean space|'''R'''3]]}} (or alternatively in the [[3-sphere]], {{math|''S''{{isup|3}}}}), can be projected onto a plane {{math|[[Euclidean plane|'''R'''2]]}} (respectively a [[sphere]] {{math|''S''{{isup|2}}}}). 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 [[Line (geometry)|collinear]]. In this case, by choosing a projection side, one can completely encode the [[regular isotopy|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 [[regular isotopy|isotopy]] of the plane) are called [[Reidemeister moves]].

<gallery class="center" mode="nolines" widths="180px"> Image:Reidemeister_move_1.png|Reidemeister move 1 Image:Reidemeister_move_2.png|Reidemeister move 2 Image:Reidemeister_move_3.png|Reidemeister move 3 </gallery>

==Types of knots== [[Image:Untying a knot.svg|thumb|300px|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 {{math|[[Euclidean space|'''R'''3]]}}.{{sfnp|Adams|1994|page=2}} In the ordinary sense of the word, the unknot is not "knotted" at all. The simplest nontrivial knots are the [[trefoil knot]] ({{math|31}} in the table), the [[figure-eight knot (mathematics)|figure-eight knot]] ({{math|41}}) and the [[cinquefoil knot]] ({{math|51}}).<ref>{{harvnb|Adams|1994|loc=Table 1.1, p. 280}}; {{harvnb|Livingstone|1993|loc=Appendix A: Knot Table, p. 221}}</ref>

Several knots, linked or tangled together, are called [[link (knot theory)|links]]. Knots are links with a single component.

===Tame vs. wild knots=== [[Image:Wild knot.svg|thumb|200px|A wild knot]] A ''polygonal'' knot is a knot whose [[image (mathematics)|image]] in {{math|'''R'''3}} is the [[union (set theory)|union]] of a [[finite set]] of [[line segments]].<ref name="Armstrong">{{harvnb|Armstrong|1983|p=215}}</ref> A ''tame'' knot is any knot equivalent to a polygonal knot.<ref name="Armstrong"/>{{refn|group=Note|A knot is tame if and only if it can be represented as a finite [[polygonal chain|closed polygonal chain]]}} Knots which are not tame are called ''[[Wild knot|wild]]'',<ref name="wild">{{cite book|author=Charles Livingston|title=Knot Theory|url=https://books.google.com/books?id=KXAS3KRZGRMC&q=%22Wild+Knots+and+Unknottings%22&pg=PA11|year=1993|publisher=Cambridge University Press|isbn=978-0-88385-027-5|page=11}}</ref> and can have [[pathological (mathematics)|pathological]] behavior.<ref name="wild"/> In knot theory and [[3-manifold]] theory, often the adjective "tame" is omitted. Smooth knots, for example, are always tame.

===Framed knot=== {{anchor|Framed knot}} A ''framed knot'' is the extension of a tame knot to an embedding of the [[solid torus]] {{math|''D''{{isup|2}} × ''S''{{isup|1}}}} in {{math|''S''{{isup|3}}}}.

The ''framing'' of the knot is the [[linking number]] of the image of the ribbon {{math|''I'' × ''S''{{isup|1}}}} with the knot. A framed knot can be seen as the embedded ribbon and the framing is the (signed) number of twists.<ref>{{cite journal|first=Louis H. |last=Kauffman|author-link=Kauffman|url=http://www.math.uic.edu/~kauffman/IRH.pdf|title=An invariant of regular isotopy|journal=[[Transactions of the American Mathematical Society]]| volume=318|issue=2|year= 1990|pages=417–471|doi=10.1090/S0002-9947-1990-0958895-7|doi-access=free}}</ref> 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.<ref>{{citation | last1 = Elhamdadi| first1 = Mohamed| author1-link = Mohamed Elhamdadi (mathematician) | last2 = Hajij| first2= Mustafa| author2-link = Mustafa Hajij (mathematician) | last3= Istvan| first3= Kyle| author3-link = Kyle Istvan (mathematician) | title = Framed Knots | year = 2019| arxiv = 1910.10257}}.</ref> 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=== [[Image:Knot with borromean rings in jsj decomp small.png|thumb|150px|right|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 [[Gordon–Luecke theorem|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-manifolds|3-manifold theory]].<ref>{{harvnb|Adams|1994|pp=261–2}}</ref>

===JSJ decomposition=== {{Main|JSJ decomposition}}

The [[JSJ decomposition]] and [[geometrization conjecture|Thurston's hyperbolization theorem]] reduces the study of knots in the 3-sphere to the study of various geometric manifolds via ''splicing'' or ''[[satellite knot|satellite operations]]''. In the pictured knot, the JSJ-decomposition splits the complement into the union of three manifolds: two [[trefoil knot|trefoil complements]] and the complement of the [[Borromean rings]]. The trefoil complement has the geometry of {{math|'''H'''2 × '''R'''}}, while the Borromean rings complement has the geometry of {{math|'''H'''3}}.

===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.<ref>{{cite thesis |first=Aaron K. |last=Trautwein |title=Harmonic knots |date=1995 |type=PhD |publisher=University of Iowa |series=Dissertation Abstracts International |volume=56-06 |issue=B |page=3234 |url=https://www.proquest.com/docview/304216894 |oclc=1194821918|id={{ProQuest|304216894}} }}</ref><ref>{{cite book |first=Aaron K. |last=Trautwein |chapter=18. An introduction to Harmonic Knots |editor1-first=Andrzej |editor1-last=Stasiak |editor2-first=Vsevolod |editor2-last=Katritch |editor3-first=Louis H. |editor3-last=Kauffman |title=Ideal Knots |chapter-url=https://books.google.com/books?id=v0NqDQAAQBAJ&pg=PA353 |year=1998 |publisher=World Scientific |isbn=978-981-02-3530-7 |pages=353–363}}</ref>

== Bracket operation (Multiplication of knots) == In a 2020 article (based on results originally obtained in the 1970s), V. M. Nezhinskij and V. V. Nesterenok introduced a binary operation on the set of oriented knot isotopy classes <math>\mathcal{K}</math>, denoted by <math>[\cdot,\cdot] : \mathcal{K} \times \mathcal{K} \to \mathcal{K}</math>.<ref>{{cite journal |last1=Nezhinskij |first1=V. M. |last2=Nesterenok |first2=V. V. |title=Multiplication of Classical Knots |journal=Journal of Mathematical Sciences |volume=252 |pages=517–526 |year=2020 |doi=10.1007/s10958-020-05175-4 |note=Translation from Russian: ''Zapiski Nauchnykh Seminarov POMI'', Vol. 468, 2018}}</ref>

To define the operation for classes <math>\alpha</math> and <math>\beta</math>, representatives are chosen in the left and right half-spaces <math>\mathbb{R}^3_{-}</math> and <math>\mathbb{R}^3_{+}</math> respectively, such that their intersections with the separating plane are specific orthogonal segments. A pair of surfaces <math>V_1</math> and <math>V_2</math> cobounding these segments is constructed, and the operation <math>[\alpha,\beta]</math> is defined as the isotopy class of the smoothed boundary of the union <math>\partial(V_1 \cup V_2)</math>.

The operation is antisymmetric, satisfying <math>[\beta,\alpha] = -[\alpha,\beta]</math>, and it has the standard trivial knot <math>\omega</math> as a right null element: <math>[\alpha,\omega] = \omega</math>. A key topological property of this bracket operation is that it produces knots with a trivial Alexander–Conway polynomial; specifically, <math>\nabla([\alpha,\beta]) = 1</math>. The article also establishes a relationship between this bracket operation and the HOMFLY-PT polynomial <math>\mathcal{P}</math>, expressing <math>\mathcal{P}([\alpha,\beta])</math> as a linear combination of the polynomials of the connected sums and the doubled components <math>D(\alpha)</math> and <math>D(\beta)</math>.

==Applications to graph theory== [[Image:KnotGraphTable.svg|thumb|350px|right|A table of all [[prime knot]]s with up to seven [[Crossing number (knot theory)|crossings]] represented as [[knot diagram]]s with their [[medial graph]]]] ===Medial graph=== {{main|Medial graph}} [[Image:KnotCheckerboard.svg|thumb|150px|left]] [[Image:KnotSignedGraph.svg|thumb|150px|left|The signed planar graph associated with a knot diagram.]] [[Image:KnotSignedEdge+.png|thumb|150px|right|Left guide]] [[Image:KnotSignedEdge-.png|thumb|150px|right|Right guide]]

Another convenient representation of knot diagrams <ref>{{cite book |first=Colin C. |last=Adams |chapter=§2.4 Knots and Planar Graphs |title=The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots |chapter-url=https://books.google.com/books?id=RqiMCgAAQBAJ&pg=PA51 |year=2004 |publisher=American Mathematical Society |isbn=978-0-8218-3678-1 |pages=51–55}} </ref><ref>[http://www.entrelacs.net/-Celtic-Knotwork-The-ultimate- Entrelacs.net tutorial]</ref> was introduced by [[Peter Tait (physicist)|Peter Tait]] in 1877.<ref>{{cite journal|last1=Tait|first1=Peter G.|author-link1=Peter Tait (physicist)|title=On Knots I|journal=Proceedings of the Royal Society of Edinburgh|date=1876–1877|volume=28|issue=1 |pages=145–190|doi=10.1017/S0080456800090633|url=https://zenodo.org/record/1844250|quote=Revised May 11, 1877.}}</ref><ref>{{cite journal|last1=Tait|first1=Peter G.|author-link1=Peter Tait (physicist)|title=On Links (Abstract)|journal=Proceedings of the Royal Society of Edinburgh|date=1876–1877|volume=9|issue=98|pages=321–332|doi=10.1017/S0370164600032363|url=https://zenodo.org/record/1677249}}</ref>

Any knot diagram defines a [[planar graph|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 (mathematics)|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 [[Chiral knot|the knot in a mirror]].

===Linkless and knotless embedding=== {{main|linkless embedding}} [[File:Petersen family.svg|thumb|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 embedding]]s and [[knotless embedding]]s. A linkless embedding is an embedding of the graph with the property that any two cycles are [[unlink]]ed; a knotless embedding is an embedding of the graph with the property that any single cycle is [[unknot]]ted. 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.<ref>{{citation | last1 = Robertson | first1 = Neil | author1-link = Neil Robertson (mathematician) | last2 = Seymour | first2 = Paul | author2-link = Paul Seymour (mathematician) | last3 = Thomas | first3 = Robin | author3-link = Robin Thomas (mathematician) | contribution = A survey of linkless embeddings | editor1-last = Robertson | editor1-first = Neil | editor1-link = Neil Robertson (mathematician) | editor2-last = Seymour | editor2-first = Paul | editor2-link = Paul Seymour (mathematician) | pages = 125–136 | publisher = American Mathematical Society | series = Contemporary Mathematics | title = Graph Structure Theory: Proc. AMS–IMS–SIAM Joint Summer Research Conference on Graph Minors | url = http://people.math.gatech.edu/~thomas/PAP/linklsurvey.pdf | volume = 147 | year = 1993}}.</ref> A full characterization of the graphs with knotless embeddings is not known, but the [[complete graph]] {{math|''K''7}} is one of the minimal forbidden graphs for knotless embedding: no matter how {{math|''K''7}} is embedded, it will contain a cycle that forms a [[trefoil knot]].<ref>{{citation | last = Ramirez Alfonsin | first = J. L. | doi = 10.1007/PL00009446 | issue = 1 | journal = Discrete and Computational Geometry | pages = 149–158 | title = Spatial graphs and oriented matroids: the trefoil | volume = 22 | year = 1999| doi-access = free }}.</ref>

==Generalization== {{refimprove section|date=December 2011}}

In contemporary mathematics the term ''knot'' is sometimes used to describe a more general phenomenon related to embeddings. Given a manifold {{math|''M''}} with a submanifold {{math|''N''}}, one sometimes says {{math|''N''}} can be knotted in {{math|''M''}} if there exists an embedding of {{math|''N''}} in {{math|''M''}} which is not isotopic to {{math|''N''}}. Traditional knots form the case where {{math|''N'' {{=}} ''S''{{isup|1}}}} and {{math|''M'' {{=}} '''R'''3}} or {{math|''M'' {{=}} ''S''{{isup|3}}}}.<ref>{{cite book|last1=Carter|first1=J. Scott|last2=Saito|first2=Masahico|isbn=0-8218-0593-2|mr=1487374|publisher=American Mathematical Society |series=Mathematical Surveys and Monographs|title=Knotted Surfaces and their Diagrams|volume=55|year=1998}}</ref><ref>{{cite book|last=Kamada|first=Seiichi|doi=10.1007/978-981-10-4091-7|isbn=978-981-10-4090-0|mr=3588325|publisher=Springer |series=Springer Monographs in Mathematics|title=Surface-Knots in 4-Space|year=2017}}</ref>

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.<ref>{{cite book|last1=Hocking|first1=John G.|last2=Young|first2=Gail S.|edition=2nd|isbn=0-486-65676-4 |mr=1016814|page=175|publisher=Dover Publications|title=Topology|url=https://books.google.com/books?id=EbvCAgAAQBAJ&pg=PA175|year=1988}}</ref> [[Alexander's theorem]] states that the 2-sphere does not smoothly (or PL or tame topologically) knot in the 3-sphere.<ref>{{cite book|last=Calegari|first=Danny|isbn=978-0-19-857008-0|mr=2327361|page=161|publisher=Oxford University Press|series=Oxford Mathematical Monographs|title=Foliations and the geometry of 3-manifolds|url=https://books.google.com/books?id=ks8TDAAAQBAJ&pg=PA161|year=2007}}</ref> In the tame topological category, it's known that the {{math|''n''}}-sphere does not knot in the {{math|''n'' + 1}}-sphere for all {{math|''n''}}. This is a theorem of [[Morton Brown]], [[Barry Mazur]], and [[Marston Morse]].<ref>{{cite journal|last=Mazur|first=Barry|doi=10.1090/S0002-9904-1959-10274-3|journal=Bulletin of the American Mathematical Society|mr=117693|pages=59–65|title=On embeddings of spheres|volume=65|year=1959|issue=2|doi-access=free}} {{cite journal|last=Brown|first=Morton|doi=10.1090/S0002-9904-1960-10400-4|journal=Bulletin of the American Mathematical Society|mr=117695|pages=74–76|title=A proof of the generalized Schoenflies theorem|volume=66|year=1960|issue=2|doi-access=free}} {{cite journal|last=Morse|first=Marston|doi=10.1090/S0002-9904-1960-10420-X|journal=Bulletin of the American Mathematical Society|mr=117694|pages=113–115|title=A reduction of the Schoenflies extension problem|volume=66|year=1960|issue=2|doi-access=free}}</ref> The [[Alexander horned sphere]] is an example of a knotted 2-sphere in the 3-sphere which is not tame.<ref>{{cite journal | last1=Alexander | first1=J. W. | author1-link=James Waddell Alexander II | title=An Example of a Simply Connected Surface Bounding a Region which is not Simply Connected | publisher=National Academy of Sciences | year=1924 | journal=[[Proceedings of the National Academy of Sciences|Proceedings of the National Academy of Sciences of the United States of America]] | issn=0027-8424 | volume=10 | issue=1 | pages= 8–10 | doi=10.1073/pnas.10.1.8| pmc=1085500 | pmid=16576780| bibcode=1924PNAS...10....8A |jstor=84202 | doi-access=free }}</ref> In the smooth category, the {{math|''n''}}-sphere is known not to knot in the {{math|''n'' + 1}}-sphere provided {{math|''n'' ≠ 3}}. The case {{math|''n'' {{=}} 3}} is a long-outstanding problem closely related to the question: does the 4-ball admit an [[exotic sphere|exotic smooth structure]]?

[[André Haefliger]] proved that there are no smooth {{math|''j''}}-dimensional knots in {{math|''S''{{isup|''n''}}}} provided {{math|2''n'' − 3''j'' − 3 > 0}}, and gave further examples of knotted spheres for all {{math|''n'' > ''j'' ≥ 1}} such that {{math|2''n'' − 3''j'' − 3 {{=}} 0}}. {{math|''n'' − ''j''}} is called the [[codimension]] of the knot. An interesting aspect of Haefliger's work is that the isotopy classes of embeddings of {{math|''S''{{isup| ''j''}}}} in {{math|''S''{{isup|''n''}}}} 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|''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 [[Whitney embedding theorem#Isotopy versions|generalization to manifolds]].

==See also== *{{annotated link|Knot theory}} *{{annotated link|Knot invariant}} *{{annotated link|List of mathematical knots and links}}

==Notes== {{reflist|group=Note}}

==References== {{reflist}}

==Bibliography== {{refbegin}} *{{cite book |first=Colin C. |last=Adams |title=The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots |publisher=W. H. Freeman |year= 1994 |url=https://books.google.com/books?id=8MXDQgAACAAJ |isbn=978-0-7167-2393-6 }} *{{cite book |last=Armstrong |first=M. A.| year=1983 |orig-year=1979 |publisher=Springer |series=[[Undergraduate Texts in Mathematics]] |title=Basic Topology |isbn=0-387-90839-0 }} *{{cite book | last = Cromwell | first = Peter R. | doi = 10.1017/CBO9780511809767 | isbn = 0-521-83947-5 | mr = 2107964 | publisher = Cambridge University Press | title = Knots and Links | year = 2004 }} *{{cite book |first1=David W. |last1=Farmer |first2=Theodore B. |last2=Stanford |title=Knots and Surfaces: A Guide to Discovering Mathematics |year=1995 |url=https://books.google.com/books?id=q7XYDp22fTsC&pg=PP5 |publisher=American Mathematical Society |isbn=978-0-8218-7265-9 }} *{{cite book |first=Charles |last=Livingstone |title=Knot Theory |publisher=The Mathematical Association of America |year=1993 |isbn=9780883850008 |volume=24 |series=Mathematical Association of America Textbooks |pages= |url=https://books.google.com/books?id=2TDYvgEACAAJ&pg=PP1 }} {{refend}}

==External links== {{commons category|Knots (knot theory)}} * {{Knot Atlas|Main_Page}} * [http://www.map.mpim-bonn.mpg.de/Embeddings_in_Euclidean_space:_an_introduction_to_their_classification The Manifold Atlas Project]

[[Category:Knots (knot theory)| ]]