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Research Summary.

I work on a variety of problems in geometric and low-dimensional topology, including branched coverings of 3- and 4-manifolds and related knot invariants, and contact topology.


Papers.

[2021] P. Cahn and A. Kjuchukova.  Linking numbers in 3-manifolds.  Discrete Comput. Geom., to appear.

[2021] P. Cahn and S. Wang. The Center for Women in Mathematics at Smith College.  To appear in D. Dumbaugh and D Haunsperger (Eds.), Count Me In: Community and Belonging in Mathematics, MAA Classroom Resource Materials.

[2021] P. Cahn and A. Kjuchukova.  Computing ribbon obstructions for colored knots.  Fund. Math. 253, no. 2, pp. 155-173.

[2021] P. Cahn. Classical invariants of Legendrian and transverse knots. In C. Adams, E. Flapan, A. Henrich, L. Kauffman, L. Ludwig, S. Nelson (Eds.), Encyclopedia of Knot Theory, CRC Press.

[2020] P. Cahn and A. Kjuchukova.  The dihedral genus of a knot.  Algebr. Geom. Topol. 20, no. 4, pp. 1939-1963.

[2020] P. Cahn and V. Chernov.  Loose Legendrian and pseudo-Legendrian knots in 3-manifolds. J. Sympl. Geom.18, no. 3, pp. 651-689.

[2018] P. Cahn, H. Gluck, H. Nuchi.  Germs of fibrations of spheres by great circles always extend to the whole sphere.  Algebr. Geom. Topol.  18, no. 3, pp. 1323-1360.

[2018]  P. Cahn, F. Fanoni, and B. Petri.  Mapping class group orbits of curves with self-intersections. Israel J. Math.  223, pp. 53-74.

[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/Submitted/Under Revision

R. Blair, P. Cahn, A. Kjuchukova, J. Meier.  A note on three-fold branched covers of S^4. https://arxiv.org/abs/1909.11788.

P. Cahn and A. Kjuchukova.  Singular branched covers of four-manifolds. https://arxiv.org/abs/1710.11562.


Articles in preparation

P. Cahn and B. Tosun.  On the classification of knots in overtwisted contact structures.

R. Blair, P. Cahn, and A. Kjuchukova.  Blowing up dihedral covers.

P. Cahn, S. Chimgee, E. Catania, O. Del Guercio, J. Kendrick.  An algorithm for computing the p-dihedral linking invariant.