Knot theory: Difference between revisions

Latest revision as of 15:30, 9 December 2024

Knot Theory

Knot theory is a branch of topology, a field of mathematics that studies the properties of space that are preserved under continuous transformations. Knot theory specifically deals with the study of mathematical knots, which are embeddings of a circle in 3-dimensional Euclidean space, \(\mathbb{R}^3\). Unlike the knots in everyday life, mathematical knots have no ends and cannot be untied.

Definition[edit]

A knot is defined as a closed, non-self-intersecting curve that is embedded in three-dimensional space. Formally, a knot is a homeomorphism from the circle \(S^1\) to \(\mathbb{R}^3\). Two knots are considered equivalent if one can be transformed into the other via a continuous deformation, known as an ambient isotopy.

Knot Invariants[edit]

Knot invariants are quantities or algebraic objects that remain unchanged under ambient isotopies of the knot. They are crucial for distinguishing between different knots. Some important knot invariants include:

Knot Polynomials: These include the Alexander polynomial, Jones polynomial, and HOMFLY polynomial. Each of these polynomials assigns a polynomial to a knot in such a way that equivalent knots have the same polynomial.

Knot Group: The fundamental group of the knot complement, which is the set of all loops in the space surrounding the knot, modulo homotopy.

Knot Genus: The minimum genus of any Seifert surface for the knot.

Types of Knots[edit]

Knot theory classifies knots into various types based on their properties:

Trivial Knot: Also known as the unknot, it is a simple loop with no crossings or twists.

Prime Knots: Knots that cannot be decomposed into simpler knots via connected sum operations.

Composite Knots: Knots that can be expressed as the connected sum of two or more nontrivial knots.

Applications[edit]

Knot theory has applications in various fields, including:

Biology: Understanding the structure of DNA and how it knots and unknots during cellular processes.

Chemistry: Studying the topology of molecular structures and the synthesis of molecular knots.

Physics: Analyzing the properties of quantum field theory and string theory.

History[edit]

The study of knots dates back to the 19th century, with significant contributions from mathematicians such as Carl Friedrich Gauss, who developed the Gauss linking integral, and Peter Guthrie Tait, who created the first systematic tables of knots. The modern development of knot theory was greatly influenced by the work of Vaughan Jones, who discovered the Jones polynomial in the 1980s.

Also see[edit]

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-[[File:Tabela_de_nós_matemáticos_01,_crop.jpg|thumb|Tabela_de_nós_matemáticos_01,_crop]] [[file:TrefoilKnot_01.svg|right|thumb|TrefoilKnot_01]] [[file:KellsFol034rXRhoDet3.jpeg|right|thumb|KellsFol034rXRhoDet3]] [[file:Peter_Guthrie_Tait._Mezzotint_by_J._Faed_after_Sir_G._Reid._Wellcome_V0006622.jpg|thumb|Peter_Guthrie_Tait._Mezzotint_by_J._Faed_after_Sir_G._Reid._Wellcome_V0006622]] [[file:Tenfold_Knottiness,_plate_IX.png|thumb|Tenfold_Knottiness,_plate_IX]] [[file:Reidemeister_move_1.png|thumb|Reidemeister_move_1]] [[file:Frame_left]]_|thumb|Frame_left]]_]] '''Knot theory''' is a branch of [[topology]] that studies mathematical [[knots]]. While inspired by knots which appear in daily life in shoelaces and rope, a mathematical knot differs in that the ends are joined together so that it cannot be undone. In mathematical terms, a knot is an embedding of a [[circle]] in 3-dimensional [[Euclidean space]], \(\mathbb{R}^3\). Two knots are considered equivalent if one can be transformed into the other via a continuous deformation, known as an [[isotopy]].
+[[File:Tabela de nós matemáticos 01, crop.jpg|thumb]] [[File:TrefoilKnot 01.svg|thumb]] [[File:KellsFol034rXRhoDet3.jpeg|thumb]] [[File:Peter Guthrie Tait. Mezzotint by J. Faed after Sir G. Reid. Wellcome V0006622.jpg|thumb]] Knot Theory
-== History ==
+Knot theory is a branch of [[topology]], a field of [[mathematics]] that studies the properties of space that are preserved under continuous transformations. Knot theory specifically deals with the study of mathematical knots, which are embeddings of a circle in 3-dimensional Euclidean space, \(\mathbb{R}^3\). Unlike the knots in everyday life, mathematical knots have no ends and cannot be untied.
-The origins of knot theory can be traced back to the 19th century when [[Lord Kelvin]] proposed that atoms were knots in the [[aether]]. This idea led to the first systematic tabulation of knots by [[Peter Guthrie Tait]], [[Charles Niven]], and [[Thomas Kirkman]]. Although Kelvin's theory was eventually discarded, the mathematical study of knots continued to develop.
+== Definition ==
+A '''knot''' is defined as a closed, non-self-intersecting curve that is embedded in three-dimensional space. Formally, a knot is a homeomorphism from the circle \(S^1\) to \(\mathbb{R}^3\). Two knots are considered equivalent if one can be transformed into the other via a continuous deformation, known as an ambient isotopy.
+== Knot Invariants ==
+Knot invariants are quantities or algebraic objects that remain unchanged under ambient isotopies of the knot. They are crucial for distinguishing between different knots. Some important knot invariants include:
+* '''Knot Polynomials''': These include the [[Alexander polynomial]], [[Jones polynomial]], and [[HOMFLY polynomial]]. Each of these polynomials assigns a polynomial to a knot in such a way that equivalent knots have the same polynomial.
+* '''Knot Group''': The fundamental group of the knot complement, which is the set of all loops in the space surrounding the knot, modulo homotopy.
+* '''Knot Genus''': The minimum genus of any [[Seifert surface]] for the knot.
+== Types of Knots ==
+Knot theory classifies knots into various types based on their properties:
-== Basic Concepts ==
+* '''Trivial Knot''': Also known as the unknot, it is a simple loop with no crossings or twists.
-=== Knot Invariants ===
-A key aspect of knot theory is the study of [[knot invariants]], which are properties of knots that remain unchanged under isotopy. Examples of knot invariants include the [[knot group]], the [[Alexander polynomial]], and the [[Jones polynomial]].
-=== Types of Knots ===
+* '''Prime Knots''': Knots that cannot be decomposed into simpler knots via connected sum operations.
-Knots can be classified into various types based on their properties. Some common types include:
-* The [[unknot]], which is a simple loop.
-* The [[trefoil knot]], which is the simplest nontrivial knot.
-* The [[figure-eight knot]], which is the simplest knot with a crossing number of four.
-=== Knot Diagrams ===
+* '''Composite Knots''': Knots that can be expressed as the connected sum of two or more nontrivial knots.
-A [[knot diagram]] is a projection of a knot onto a plane, with information about the over and under crossings. Knot diagrams are useful for visualizing and manipulating knots.
 == Applications ==
-Knot theory has applications in various fields including [[biology]], where it is used to study the structure of [[DNA]] and other molecules, and in [[chemistry]], where it helps in understanding the properties of certain chemical compounds. It also has applications in [[physics]], particularly in the study of [[quantum field theory]] and [[statistical mechanics]].
+Knot theory has applications in various fields, including:
+* '''Biology''': Understanding the structure of [[DNA]] and how it knots and unknots during cellular processes.
+* '''Chemistry''': Studying the topology of molecular structures and the synthesis of [[molecular knots]].
+* '''Physics''': Analyzing the properties of [[quantum field theory]] and [[string theory]].
+== History ==
+The study of knots dates back to the 19th century, with significant contributions from mathematicians such as [[Carl Friedrich Gauss]], who developed the Gauss linking integral, and [[Peter Guthrie Tait]], who created the first systematic tables of knots. The modern development of knot theory was greatly influenced by the work of [[Vaughan Jones]], who discovered the Jones polynomial in the 1980s.
-== Related Pages ==
+== Also see ==
+* [[Braid theory]]
+* [[Link (knot theory)]]
 * [[Topology]]
-* [[Braid theory]]
 * [[Knot polynomial]]
-* [[Link (knot theory)]]
-* [[Knot group]]
-* [[Reidemeister move]]
-== References ==
+{{Knot theory}}
-{{Reflist}}
-== External Links ==
-{{Commons category|Knot theory}}
-{{Wikibooks|Knot theory}}
-{{Wiktionary|knot theory}}
+[[Category:Topology]]
 [[Category:Knot theory]]
-[[Category:Topology]]
-[[Category:Mathematical concepts]]
-[[Category:Mathematical structures]]
-{{math-stub}}