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Knots and Surfaces in Real Algebraic and Contact Geometry
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics. (Topology)
2011 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis consists of a summary and three articles. The thesis is devoted to the study of knots and surfaces with additional geometric structures compared to the classical smooth structure.

In Paper I, real algebraic rational knots in real projective space are studied up to rigid isotopy and we show that two real rational algebraic knots of degree at most 5 are rigidly isotopic if, and only if, their degree and encomplexed writhe are equal. We also show that any smooth irreducible knot which admits a plane projection with less than or equal to four crossings has a rational parametrization of degree at most 6. Furthermore, an explicit construction of rational knots of a given degree with arbitrary encomplexed writhe (subject to natural restrictions) is presented.

In Paper II, we construct an invariant of parametrized generic real algebraic surfaces in real projective space which generalizes the Brown invariant of immersed surfaces from smooth topology. The invariant is constructed using the self intersection, which is a real algebraic curve with points of three local characters: an intersection of two real sheets, an intersection of two complex conjugate sheets or a Whitney umbrella. The Brown invariant was expressed through a self linking number of the self intersection by Kirby and Melvin. We extend their definition of this self linking number to the case of parametrized generic real algebraic surfaces.

In Paper III, we give a combinatorial description of the Legendrian differential graded algebra associated to a Legendrian knot in the product of a punctured Riemann surface with the real line. As an application we show that for any nonzero homology class h, and for any integer k there exist k Legendrian knots all representing h which are pairwise smoothly isotopic through a formal Legendrian isotopy but which lie in mutually distinct Legendrian isotopy classes.

Place, publisher, year, edition, pages
Uppsala: Department of Mathematics , 2011. , 17 p.
Series
Uppsala Dissertations in Mathematics, ISSN 1401-2049 ; 72
National Category
Mathematics
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:uu:diva-156908ISBN: 978-91-506-2230-0 (print)OAI: oai:DiVA.org:uu-156908DiVA: diva2:433836
Public defence
2011-09-21, Polhemsalen, Ångströmslaboratoriet, Lägerhyddsvägen 1, Uppsala, 13:15 (English)
Opponent
Supervisors
Available from: 2011-08-31 Created: 2011-08-11 Last updated: 2011-08-31Bibliographically approved
List of papers
1. Real Algebraic Knots of Low Degree
Open this publication in new window or tab >>Real Algebraic Knots of Low Degree
2011 (English)In: Journal of knot theory and its ramifications, ISSN 0218-2165, Vol. 20, no 9, 1285-1309 p.Article in journal (Refereed) Published
Abstract [en]

In this paper we study rational real algebraic knots in RP3. We show that two real algebraic knots of degree d<6 are rigidly isotopic if and only if their degrees and encomplexed writhes are equal. We also show that any irreducible smooth knot which admits a plane projection with less than or equal to four crossings has a rational parametrization of degree d<7. Furthermore an explicit construction of rational knots of a given degree with arbitrary encomplexed writhe (subject to natural restrictions) is presented.

Keyword
Real algebraic knot theory, encomplexed writhe, topology of real algebraic varieties
National Category
Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:uu:diva-156717 (URN)10.1142/S0218216511009248 (DOI)000295271700007 ()
Available from: 2011-08-08 Created: 2011-08-08 Last updated: 2017-12-08Bibliographically approved
2. Encomplexed Brown Invariant of Real Algebraic Surfaces in RP3
Open this publication in new window or tab >>Encomplexed Brown Invariant of Real Algebraic Surfaces in RP3
2013 (English)In: Arkiv för matematik, ISSN 0004-2080, E-ISSN 1871-2487, Vol. 51, no 2, 251-267 p.Article in journal (Other academic) Published
Abstract [en]

We construct an invariant of parametrized generic real algebraic surfaces in RP3 which generalizes the Brown invariant of immersed surfaces from smooth topology. The invariant is constructed using the self intersection, which is a real algebraic curve with points of three local characters: the intersection of two real sheets, the intersection of two complex conjugate sheets or a Whitney umbrella. The Brown invariant was expressed through a self linking number of the self intersection by Kirby and Melvin. We extend the definition of this self linking number to the case of parametrized generic real algebraic surfaces.

National Category
Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:uu:diva-156749 (URN)10.1007/s11512-012-0176-6 (DOI)000323247000003 ()
Available from: 2011-08-09 Created: 2011-08-09 Last updated: 2017-12-08Bibliographically approved
3. Legendrian contact homology in the product of a punctured Riemann surface and the real line
Open this publication in new window or tab >>Legendrian contact homology in the product of a punctured Riemann surface and the real line
2016 (English)In: Journal of the London Mathematical Society, ISSN 0024-6107, E-ISSN 1469-7750, Vol. 94, no 3, 970-992 p.Article in journal (Refereed) Published
Abstract [en]

We give a combinatorial description of the Legendrian differential graded algebra associated to a Legendrian knot in PxR, where P is a punctured Riemann surface. As an application we show that for any integer k and any homology class h in H_1(PxR) there are k Legendrian knots all representing h which are pairwise smoothly isotopic through a formal Legendrian isotopy but which lie in mutually distinct Legendrian isotopy classes.

National Category
Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:uu:diva-156750 (URN)10.1112/jlms/jdw066 (DOI)000392842700014 ()
Note

Submitted

Available from: 2011-08-09 Created: 2011-08-09 Last updated: 2017-06-30Bibliographically approved

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