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.

Knot Theory of Complex Plane Curves

In this paper we discuss the change in contact structures as their supporting open book decompositions have their binding components cabled. To facilitate this and applications we define the notion of a rational open book decomposition that generalizes the standard notion of open book decomposition and allows one to more easily study surgeries on transverse knots. As a corollary to our investigation we are able to show there are Stein fillable contact structures supported by open books whose monodromies cannot be written as a product of positive Dehn twists. We also exhibit several monoids in the mapping class group of a surface that have contact geometric significance.

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Cabling, contact structures and mapping class monoids

We confirm the AJ conjecture [Ga04] that relates the A-polynomial and the colored Jones polynomial for those hyperbolic knots satisfying certain conditions. In particular, we show that the conjecture holds true for some classes of two-bridge knots and pretzel knots. This extends the result of the first author in [Le06] where he established the AJ conjecture for a large class of two-bridge knots, including all twist knots. Along the way, we explicitly calculate the universal character ring of the knot group of the (-2,3,2n+1)-pretzel knot and show that it is reduced for all integers n.

On the AJ conjecture for knots

Visualization of A'Campo's fibered links and unknotting operation

For a 3-manifold with torus boundary admitting an appropriate involution, we show that Khovanov homology provides obstructions to certain exceptional Dehn fillings. For example, given a strongly invertible knot in S3, we give obstructions to lens space surgeries, as well as obstructions to surgeries with finite fundamental group. These obstructions are based on homological width in Khovanov homology, and in the case of finite fundamental group depend on a calculation of the homological width for a family of Montesinos links.

Surgery obstructions from Khovanov homology

Nous etudions ici la relation entre les surfaces de ruban associees aux graphs legendriens et les diagrammes quasi-positifs. Comme application, nous donnons une preuve elementaire qu'une surface fibree est quasi-positive, si et seulement si elle porte la structure de contact standard dans S 3 . Nous repondons aussi a une question de L. Rudolph concernant les mouvements des surfaces quasi-positives.

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Legendrian graphs and quasipositive diagrams

In the early 1990s, Turaev introduced the notion of shadows as a combinatorial presentation of both 4 and 3-manifolds. Later, Costantino–Thurston revealed a strong relation between the Stein factorizations of stable maps of 3-manifolds into the real plane and the shadows of the manifolds. In fact, a shadow can be seen locally as the Stein factorization of a stable map. In this paper, we define the notion of stable map complexity for a compact orientable 3-manifold bounded by (possibly empty) tori counting, with some weights, the minimal number of singular fibers of codimension 2 of stable maps into the real plane, and prove that this number equals the minimal number of vertices of its branched shadows. In consequence, we give a complete characterization of hyperbolic links in the 3-sphere whose exteriors have stable map complexity 1 in terms of Dehn surgeries, and also give an observation concerning the coincidence of the stable map complexity and shadow complexity using estimations of hyperbolic volumes.

Stable maps and branched shadows of 3-manifolds

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Links of oriented divides and fibrations in link exteriors

Links and gordian numbers associated with generic immersions of intervals

We classify Dehn surgeries on (p, q, r) pretzel knots that result in a manifold of finite funda- mental group. The only hyperbolic pretzel knots that admit non-trivial finite surgeries are ( 2,3,7) and ( 2,3,9). Agol and Lackenby's 6-theorem reduces the argument to knots with small indices p, q, r. We treat these using the Culler-Shalen norm of the SL(2, C)-character variety. In particular, we introduce new techniques for demonstrating that boundary slopes are detected by the character variety.

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Masaharu Ishikawa

Papers

Legendrian graphs and quasipositive diagrams

Stable maps and branched shadows of 3-manifolds

Links of oriented divides and fibrations in link exteriors

Links and gordian numbers associated with generic immersions of intervals

Finite surgeries on three-tangle pretzel knots