Tree-like curve: Difference between revisions

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	{{short description\|Type of planar curve with tree-like structure}}		{{short description\|Type of planar curve with tree-like structure}}
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	In ], particularly in ], a '''tree-like curve''' is a ] ] <math>c: S^1 \to \mathbb{R}^2</math> with the property that removing any ] splits the curve into exactly two ] ].<ref name="Shapiro-1997">Shapiro, B. (~~1997~~). ~~"Tree-like curves and their number of inflection points"~~. ~~arXiv:dg-ga/9708009</ref>~~ This property gives these curves a ]-like structure, hence their name. They were first systematically studied by ] ] ] and ] in the 1990s.<ref ~~name="Shapiro-1997"/~~>		In ], particularly in ], a '''tree-like curve''' is a ] ] <math>c: S^1 \to \mathbb{R}^2</math> with the property that removing any ] splits the curve into exactly two ] ]. This property gives these curves a ]-like structure, hence their name. They were first systematically studied by ] ] ] and ] in the 1990s.<ref>{{citation
			\| last = Aicardi \| first = F.
			\| editor-last = Arnol'd \| editor-first = V. I.
			\| contribution = Tree-like curves
			\| isbn = 0-8218-0237-2
			\| location = Providence, Rhode Island
			\| mr = 1310594
			\| pages = 1–31
			\| publisher = American Mathematical Society
			\| series = Advances in Soviet Mathematics
			\| title = Singularities and bifurcations
			\| volume = 21
			\| year = 1994}}</ref><ref name="Shapiro-1997">{{citation
			\| last = Shapiro \| first = Boris
			\| editor-last = Tabachnikov \| editor-first = S.
			\| arxiv = dg-ga/9708009
			\| contribution = Tree-like curves and their number of inflection points
			\| doi = 10.1090/trans2/190/08
			\| isbn = 0-8218-1354-4
			\| location = Providence, Rhode Island
			\| mr = 1738394
			\| pages = 113–129
			\| publisher = American Mathematical Society
			\| series = American Mathematical Society Translations, Series 2
			\| title = Differential and symplectic topology of knots and curves
			\| volume = 190
			\| year = 1999}}</ref>

			For generic curves interpreted as the shadows of ] (that is, ]s from which the over-under relations at each crossing have been erased), the tree-like curves can only be shadows of the ]. As knot diagrams, these represent ]s of ]. Each figure-eight is unknotted and their connected sum remains unknotted. Random curves with few crossings are likely to be tree-like, and therefore random knot diagrams with few crossings are likely to be unknotted.<ref>{{citation
			\| last1 = Cantarella \| first1 = Jason
			\| last2 = Chapman \| first2 = Harrison
			\| last3 = Mastin \| first3 = Matt
			\| doi = 10.1088/1751-8113/49/40/405001
			\| issue = 40
			\| journal = Journal of Physics
			\| mr = 3556174
			\| article-number = 405001
			\| title = Knot probabilities in random diagrams
			\| volume = 49
			\| year = 2016}}</ref>

	==References==		==References==

Latest revision as of 07:25, 14 January 2025

Type of planar curve with tree-like structure

In mathematics, particularly in differential geometry, a tree-like curve is a generic immersion $c:S^{1}\to \mathbb {R} ^{2}$ with the property that removing any double point splits the curve into exactly two disjoint connected components. This property gives these curves a tree-like structure, hence their name. They were first systematically studied by Russian mathematicians Boris Shapiro and Vladimir Arnold in the 1990s.

For generic curves interpreted as the shadows of knots (that is, knot diagrams from which the over-under relations at each crossing have been erased), the tree-like curves can only be shadows of the unknot. As knot diagrams, these represent connected sums of figure-eight curves. Each figure-eight is unknotted and their connected sum remains unknotted. Random curves with few crossings are likely to be tree-like, and therefore random knot diagrams with few crossings are likely to be unknotted.

References

Aicardi, F. (1994), "Tree-like curves", in Arnol'd, V. I. (ed.), Singularities and bifurcations, Advances in Soviet Mathematics, vol. 21, Providence, Rhode Island: American Mathematical Society, pp. 1–31, ISBN 0-8218-0237-2, MR 1310594
Shapiro, Boris (1999), "Tree-like curves and their number of inflection points", in Tabachnikov, S. (ed.), Differential and symplectic topology of knots and curves, American Mathematical Society Translations, Series 2, vol. 190, Providence, Rhode Island: American Mathematical Society, pp. 113–129, arXiv:dg-ga/9708009, doi:10.1090/trans2/190/08, ISBN 0-8218-1354-4, MR 1738394
Cantarella, Jason; Chapman, Harrison; Mastin, Matt (2016), "Knot probabilities in random diagrams", Journal of Physics, 49 (40) 405001, doi:10.1088/1751-8113/49/40/405001, MR 3556174

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