Knot Theory of Complex Plane Curves

Abstract

(1) Colin Adams, Hyperbolic knots AbstractThe primary objects of study in the "knot theory of complex plane curves" are C-links: links (or knots) cut out of a 3-sphere in C 2 by complex plane curves. There are two very different classes of C-links, transverse and totally tangential. Transverse C-links are naturally oriented. There are many natural classes of examples: links of singularities; links at infinity; links of divides, free divides, tree divides, and graph divides; and-most generally-quasipositive links. Totally tangential C-links are unoriented but naturally framed; they turn out to be precisely the real-analytic Legendrian links, and can profitably be investigated in terms of certain closely associated transverse C-links.The knot theory of complex plane curves is attractive 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 even some experimental evidence that nature favors quasipositive knots.

The Ozsváth-Szabó and Rasmussen Concordance Invariants are not Equal

“…This is due to Livingston, [14]. See also [32]. (4) Quasipositive knots: s(K) = 2τ (K) = 2g 4 (K), where g 4 (K) denotes the smooth slice genus of K. This follows from work of Plamenevskaya [27] for τ and from Plamenevskaya [28] and Shumakovitch [34] for s. See also [7].…”

Lens space surgeries as A’Campo’s divide knots

“…For theory of divide knots, see also Chmutov [12] and Quach Hongler-Weber [36] and "transverse C-links" defined by Rudolph [37]. In [25], Hedden studied some relationship among 4-genus, fibered-ness, (strongly) (quasi-) positivity of braids, and knot Floer homology.…”

Knotted holomorphic discs in

“…In the following, we will give a short description of Rudolph's theory; more details are contained in [7] and [8].…”