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Research Summary.

I work on a variety of problems in geometric and low-dimensional topology.  Some common themes that arise in these projects are: curves on surfaces and more generally loops in manifolds, intersections and linking, and isotopy in the presence of geometric constraints, including contact topology. 


Papers.

My preprints are all on the arxiv.

Accepted and Published articles

[2017] P. Cahn, H. Gluck, H. Nuchi.  Germs of fibrations of spheres by great circles always extend to the whole sphere.  Algebr. Geom. Topol.  To appear.

[2017]  P. Cahn, F. Fanoni, and B. Petri.  Mapping class group orbits of curves with self-intersections. Israel J. Math.  16 Pages.  To appear.

[2017]  P. Cahn. A generalization of Turaev's virtual string cobracket and self-intersections of virtual strings.  Comm. Contemp. Math. 19, no. 4, 1650053 (37 pages).

[2015]  P. Cahn and A. Levi.  Vassiliev invariants of virtual Legendrian knots.  Pacific J. Math. 273, no. 1, pp. 21-46.

[2014]  P. Cahn, V. Chernov, and R. Sadykov.  The number of framings of a knot in a 3-manifold.  J. Knot Theory Ramifications.  23, 1340072 (9 pages).

[2013]  P. Cahn.  A Generalization of the Turaev cobracket and the minimal self-intersection number of a curve on a surface.  New York J. Math.  19, pp. 253-283.

[2013]  P. Cahn and V. Chernov.  Intersections of loops and the Andersen-Mattes-Reshetikhin algebra.  J. London Math. Soc. (2) 87, no. 3, pp. 785-801.

[2012]  P. Cahn, R. Haas, A.G. Helminck, J. Li, and J. Schwartz.  Permutation notations for the exceptional Weyl group F4.  Involve. 5, no. 1, pp.  81-89.

[2008]  M. Albert, J. Bratz, P. Cahn, T. Fargus, N. Haber, E. McMahon, J. Smith, S. Tekansik.  Congr. Numer. 190, pp. 161-171.

Preprints and submitted articles

[2017] P. Cahn and A. Kjuchukova.  Singular branched covers in dimension four.  Preprint available: https://arxiv.org/abs/1710.11562.

[2016] P. Cahn and A. Kjuchukova.  Linking numbers in 3-manifolds.  Preprint available: https://arxiv.org/abs/1611.10330.

[2014]  P. Cahn and V. Chernov.  The classification of V-transverse knots and loose Legendrians.  Preprint available: http://arxiv.org/abs/1405.5725.


Articles in preparation

P. Cahn.  Intersections of arcs and curves on surfaces.

P. Cahn and B. Tosun.  On the classification of nonzero-homologous loose Legendrians.

P. Cahn and A. Kjuchukova.  Linking numbers and branched covers of 4-manifolds.