Knot Theory of Complex Plane Curves

By Lee Rudolph. 2004

65 citations recorded by the source corpus.

Publisher Summary The knot theory of complex plane curves draws attraction not only for its own internal results but also for its intriguing relationships and interesting contributions elsewhere in mathematics. Within low-dimensional topology, related subjects include braids, concordance, polynomial invariants, contact geometry, fibered links and open books, and Lefschetz pencils. Within low-dimensional algebraic and analytic geometry, related subjects include embeddings and injections of the complex line in the complex plane, line arrangements, Stein surfaces, and Hilbert's 16th problem. There is experimental evidence that nature favors quasipositive knots. This chapter summarizes studies that relate the knot theory of complex plane curves to other research areas, highlighting the results from few studies. There is an enormous, interesting, and constantly growing literature on braid monodromy and its many offsprings, including applications to line arrangements and the Alexander invariants of complex plane curves.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/d7a9eb4d0fdb767db6cb4eb1e20c721a30008bca

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

NOTIONS OF POSITIVITY AND THE OZSVÁTH–SZABÓ CONCORDANCE INVARIANT (2005, background): “1”
Fibered Transverse Knots and the Bennequin Bound (2008, background): “By writing B as the product of a(B) conjugates of positive generators in the braid group it is easy to construct a ribbon immersed surface in S3 with boundary LB (see, for example, [29]) and hence a slice surface in B4, of genus g = 12(1 − n(B) + a(B)).”
On Knot Floer Homology and Cabling: 2 (2008, background): “8 is a corollary of this relationship and of a connection between contact geometry and the knot theory of complex curves established in [13, 37].”