Editing Topology (section)

==Applications==

===Biology===

Topology has been used to study various biological systems including molecules and nanostructure (e.g., membraneous objects). In particular, [[circuit topology]] and [[knot theory]] have been extensively applied to classify and compare the topology of folded proteins and nucleic acids. [[Circuit topology]] classifies folded molecular chains based on the pairwise arrangement of their intra-chain contacts and chain crossings. [[Knot theory]], a branch of topology, is used in biology to study the effects of certain enzymes on DNA. These enzymes cut, twist, and reconnect the DNA, causing knotting with observable effects such as slower [[electrophoresis]].<ref>{{Cite book|first=Colin |last= Adams|author-link=Colin Adams (mathematician)|title=The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots |publisher=American Mathematical Society|year=2004|isbn=978-0-8218-3678-1}}</ref>

===Computer science===

[[Topological data analysis]] uses techniques from algebraic topology to determine the large-scale structure of a set (for instance, determining if a cloud of points is spherical or [[torus|toroidal]]). The main method used by topological data analysis is to:
# Replace a set of data points with a family of [[simplicial complex]]es, indexed by a proximity parameter.
# Analyse these topological complexes via algebraic topology – specifically, via the theory of [[persistent homology]].<ref name=carlsson2009>{{cite journal|url=http://www.ams.org/bull/2009-46-02/S0273-0979-09-01249-X/S0273-0979-09-01249-X.pdf|title=Topology and data|author=Gunnar Carlsson|journal=Bulletin of the American Mathematical Society|series=New Series|volume=46|issue=2|date=April 2009|pages=255–308|doi=10.1090/S0273-0979-09-01249-X|doi-broken-date=30 April 2025 |doi-access=free|archive-date=20 March 2009|access-date=30 December 2013|archive-url=https://web.archive.org/web/20090320055948/http://www.ams.org/bull/2009-46-02/S0273-0979-09-01249-X/S0273-0979-09-01249-X.pdf|url-status=live}}</ref>
# Encode the persistent homology of a data set in the form of a parameterized version of a [[Betti number]], which is called a barcode.<ref name=carlsson2009/>

Several branches of [[programming language semantics]], such as [[domain theory]], are formalized using topology. In this context, [[Steve Vickers (computer scientist)|Steve Vickers]], building on work by [[Samson Abramsky]] and Michael B. Smyth, characterizes topological spaces as [[Boolean algebra (structure)|Boolean]] or [[Heyting algebra]]s over open sets, which are characterized as [[semidecidable]] (equivalently, finitely observable) properties.<ref>{{cite book|title=Topology via Logic|last=Vickers|first=Steve|series=Cambridge Tracts in Theoretical Computer Science|publisher=Cambridge University Press|year=1996|isbn=978-0521576512}}</ref>

=== Physics ===
Topology is relevant to physics in areas such as [[condensed matter physics]],<ref>{{cite web|title=The Nobel Prize in Physics 2016|url=https://www.nobelprize.org/nobel_prizes/physics/laureates/2016/press.html|publisher=Nobel Foundation|date=4 October 2016|access-date=12 October 2016|archive-date=6 October 2016|archive-url=https://web.archive.org/web/20161006230055/https://www.nobelprize.org/nobel_prizes/physics/laureates/2016/press.html|url-status=live}}</ref> [[quantum field theory]], [[quantum computing]] and [[physical cosmology]].

The topological dependence of mechanical properties in solids is of interest in the disciplines of [[mechanical engineering]] and [[materials science]]. Electrical and mechanical properties depend on the arrangement and network structures of [[molecules]] and elementary units in materials.<ref>{{cite journal|last1=Stephenson|first1=C.|last2=et.|first2=al.|title=Topological properties of a self-assembled electrical network via ab initio calculation|journal=Sci. Rep.|volume=7|pages=41621|date=2017|doi=10.1038/srep41621|pmid=28155863|pmc=5290745|bibcode=2017NatSR...741621S}}</ref> The [[compressive strength]] of [[Crumpling|crumpled]] topologies is studied in attempts to understand the high strength to weight of such structures that are mostly empty space.<ref>{{cite journal |last1=Cambou| first1=Anne Dominique |last2=Narayanan | first2=Menon| title=Three-dimensional structure of a sheet crumpled into a ball.| journal= Proceedings of the National Academy of Sciences of the United States of America |year=2011 | volume=108 |issue=36 |pages=14741–14745 | doi=10.1073/pnas.1019192108| pmid=21873249 |arxiv=1203.5826|bibcode=2011PNAS..10814741C|pmc=3169141| doi-access=free }}</ref> Topology is of further significance in [[Contact mechanics]] where the dependence of stiffness and friction on the [[Fractal dimension|dimensionality]] of surface structures is the subject of interest with applications in multi-body physics.

A [[topological quantum field theory]] (or topological field theory or TQFT) is a quantum field theory that computes [[topological invariant]]s. Although TQFTs were invented by physicists, they are also of mathematical interest, being related to, among other things, [[knot theory]], the theory of [[four-manifold]]s in algebraic topology, and the theory of [[moduli spaces]] in algebraic geometry. [[Simon Donaldson|Donaldson]], [[Vaughan Jones|Jones]], [[Edward Witten|Witten]], and [[Maxim Kontsevich|Kontsevich]] have all won [[Fields Medal]]s for work related to topological field theory.

The topological classification of [[Calabi–Yau manifold]]s has important implications in [[string theory]], as different manifolds can sustain different kinds of strings.<ref>Yau, S. & Nadis, S.; ''The Shape of Inner Space'', Basic Books, 2010.</ref>

In [[topological quantum computer]]s, the qubits are stored in [[topological properties]], that are by definition invariant with respect to [[homotopies]].<ref>{{cite journal |last=Kitaev |first=Alexei |date=9 July 1997 |title=Fault-tolerant quantum computation by anyons |journal= Annals of Physics|volume=303 |issue=1 |pages=2–30 |doi=10.1016/S0003-4916(02)00018-0 |arxiv=quant-ph/9707021v1|bibcode=2003AnPhy.303....2K |s2cid=11199664 }}</ref>

In cosmology, topology can be used to describe the overall [[shape of the universe]].<ref>''The Shape of Space: How to Visualize Surfaces and Three-dimensional Manifolds'' 2nd ed (Marcel Dekker, 1985, {{isbn|0-8247-7437-X}})</ref> This area of research is commonly known as [[spacetime topology]].

In condensed matter, a relevant application to topological physics comes from the possibility of obtaining a one-way current, which is a current protected from backscattering. It was first discovered in electronics with the famous [[quantum Hall effect]], and then generalized in other areas of physics, for instance in photonics<ref>{{Cite journal |last1=Haldane |first1=F. D. M. |last2=Raghu |first2=S. |date=2008-01-10 |title=Possible Realization of Directional Optical Waveguides in Photonic Crystals with Broken Time-Reversal Symmetry |url=https://link.aps.org/doi/10.1103/PhysRevLett.100.013904 |journal=Physical Review Letters |language=en |volume=100 |issue=1 |pages=013904 |doi=10.1103/PhysRevLett.100.013904 |pmid=18232766 |arxiv=cond-mat/0503588 |bibcode=2008PhRvL.100a3904H |s2cid=44745453 |issn=0031-9007}}</ref> by [[Duncan Haldane|F.D.M Haldane]].

===Robotics===
The possible positions of a [[robot]] can be described by a [[manifold]] called [[Configuration space (physics)|configuration space]].<ref>John J. Craig, ''Introduction to Robotics: Mechanics and Control'', 3rd Ed. Prentice-Hall, 2004</ref> In the area of [[motion planning]], one finds paths between two points in configuration space. These paths represent a motion of the robot's [[joint]]s and other parts into the desired pose.<ref>{{cite book|last=Farber|first=Michael|title=Invitation to Topological Robotics|publisher=European Mathematical Society|date=2008|isbn=978-3037190548}}</ref>

===Games and puzzles===
[[Disentanglement puzzle]]s are based on topological aspects of the puzzle's shapes and components.<ref>{{cite journal |jstor=27642974 |doi=10.2307/27642974 |title= Disentangling Topological Puzzles by Using Knot Theory |first=Mathew |last=Horak |journal=Mathematics Magazine |year=2006 |volume=79 |issue=5 |pages=368–375}}</ref><ref>[http://sma.epfl.ch/~ojangure/Notes.pdf http://sma.epfl.ch/Notes.pdf] {{Webarchive|url=https://web.archive.org/web/20221101044934/https://sma.epfl.ch/~ojangure/Notes.pdf |date=1 November 2022 }} A Topological Puzzle, Inta Bertuccioni, December 2003.</ref><ref>[https://www.futilitycloset.com/2012/06/23/the-figure-8-puzzle/ https://www.futilitycloset.com/the-figure-8-puzzle] {{Webarchive|url=https://web.archive.org/web/20170525030117/http://www.futilitycloset.com/2012/06/23/the-figure-8-puzzle/ |date=25 May 2017 }} The Figure Eight Puzzle, Science and Math, June 2012.</ref>

===Fiber art===
In order to create a continuous join of pieces in a modular construction, it is necessary to create an unbroken path in an order that surrounds each piece and traverses each edge only once. This process is an application of the [[Eulerian path]].<ref>{{cite book|last=Eckman|first=Edie|title=Connect the shapes crochet motifs: creative techniques for joining motifs of all shapes|date=2012|publisher=Storey Publishing|isbn=978-1603429733}}</ref>