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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.

[20--] P. Cahn and V. Chernov.  Loose Legendrian and pseudo-Legendrian knots in 3-manifolds. J. Sympl. Geom., to appear.

[20--] P. Cahn. Classical invariants of Legendrian and transverse knots.  To appear in A Concise Encyclopedia of Knot Theory.

[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.  The dihedral genus of a knot.  https://arxiv.org/abs/1812.10842.

P. Cahn and A. Kjuchukova.  Computing ribbon obstructions for colored knots.  https://arxiv.org/abs/1812.09553.

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

P. Cahn and A. Kjuchukova.  Linking numbers in 3-manifolds.   https://arxiv.org/abs/1611.10330.


Articles in preparation

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

R. Blair, P. Cahn, and A. Kjuchukova.  Odd indefinite four-manifolds as three-fold branched covers of the four sphere.