Research Summary.
I work on a variety of problems in geometric and low-dimensional
topology, including trisections of 4-manifolds, branched coverings of 3- and 4-manifolds and
related knot invariants, and contact topology.
My work is currently supported by an NSF CAREER grant.
Code for computational projects is available here.
Publications.
[2022] R. Blair, P. Cahn, A. Kjuchukova, J. Meier. A note on three-fold branched covers of S^4. To appear in Annales de l’Institut Fourier.
[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
P. Cahn, S. Chimgee, E. Catania, O. Del Guercio, J. Kendrick. An algorithm for computing the p-dihedral linking invariant. https://arxiv.org/abs/2112.14790
P. Cahn and A. Kjuchukova. Singular branched covers of four-manifolds. https://arxiv.org/abs/1710.11562.