Four-dimensional aspects of tight contact 3-manifolds

We conjecture a four-dimensional characterization of tightness: A contact structure on a 3-manifold Y is tight if and only if a slice-Bennequin inequality holds for smoothly embedded surfaces in Y×[0,1]. An affirmative answer to our conjecture would imply an analogue of the Milnor conjecture for torus knots: If a fibered link L induces a tight contact structure on Y, then its fiber surface maximizes the Euler characteristic among all surfaces in Y×[0,1] with boundary L. We provide evidence for both conjectures by proving them for contact structures with nonvanishing Ozsváth–Szabó contact invariant.

Monodromy Action on Unknotting Tunnels in Fiber Surfaces

In a 2012 paper, the second author showed that a tunnel of a tunnel number one, fibered link in S3 can be isotoped to lie as a properly embedded arc in the fiber surface of the link. In this paper we observe that this is true for fibered links in any 3-manifold, we analyze how the arc behaves under the monodromy action, and we show that the tunnel arc is nearly clean, with the possible exception of twisting around the boundary of the fiber.

Virtual covers of links

We use virtual knot theory to detect the non-invertibility of some classical links in [Formula: see text]. These links appear in the study of virtual covers. Briefly, a virtual cover associates a virtual knot [Formula: see text] to a knot [Formula: see text] in a [Formula: see text]-manifold [Formula: see text], under certain hypotheses on [Formula: see text] and [Formula: see text]. Virtual covers of links in [Formula: see text] come from taking [Formula: see text] to be in the complement [Formula: see text] of a fibered link [Formula: see text]. If [Formula: see text] is invertible and [Formula: see text] is “close to” a fiber of [Formula: see text], then [Formula: see text] satisfies a symmetry condition to which some virtual knot polynomials are sensitive. We also discuss virtual covers of links [Formula: see text], where [Formula: see text] is not fibered, but is virtually fibered (in the sense of W. Thurston).

RECURRENCE FORMULAS FOR THE ALEXANDER POLYNOMIALS OF 2-BRIDGE LINKS AND THEIR COVERING LINKS

We give a recurrence formula for calculating the Alexander polynomials of 2-bridge links by using a special type of Conway diagram. As an application we give sufficient and necessary conditions for the periodic covering link over a 2-bridge link to be a fibered link. We also give a recurrence formula to calculate the reduced Alexander polynomials of periodic covering links over 2-bridge links.

n-QUASI-ISOTOPY I: QUESTIONS OF NILPOTENCE

It is well-known that no knot can be cancelled in a connected sum with another knot, whereas every link can be cancelled up to link homotopy in a (componentwise) connected sum with another link. In this paper we address the question whether the noncancellation property of knots holds for (piecewise-linear) links up to some stronger analogue of link homotopy, which still does not distinguish between sufficiently close C0-approximations of a topological link. We introduce a sequence of such increasingly stronger equivalence relations under the name of k-quasi-isotopy, k∈ℕ; all of them are weaker than isotopy (in the sense of Milnor). We prove that every link can be cancelled up to peripheral structure preserving isomorphism of any quotient of the fundamental group, functorially invariant under k-quasi-isotopy; functoriality means that the isomorphism between the quotients for links related by any allowable crossing change fits in the commutative diagram with the fundamental group of the complement to the intermediate singular link. The proof invokes Baer's theorem on the join of subnormal locally nilpotent subgroups. On the other hand, the integral generalized ( lk ≠ 0) Sato–Levine invariant [Formula: see text] is invariant under 1-quasi-isotopy, but is not determined by any quotient of the fundamental group (endowed with the peripheral structure), functorially invariant under 1-quasi-isotopy — in contrast to Waldhausen's theorem.As a byproduct, we use [Formula: see text] to determine the image of the Kirk–Koschorke invariant [Formula: see text] of fibered link maps.

Stallings Twists Which Can be Realized by Plumbing and Deplumbing Hopf Bands

We define a type of Stallings twists, which represents a "complexity" of the twist, and show that the Stallings twists of a certain type on a fiber surface for a fibered link in S3 can be realized by plumbing one Hopf band and deplumbing another Hopf band. Using this technique, we construct stable Hopf plumbings which are not Hopf plumbings with an arbitrary high first Betti number.

CONWAY POLYNOMIAL OF A FIBERED SOLVABLE LINK

We compute Kauffman's potential function (equivalent to the Conway polynomial) for any solvable fibered link. Solvable links are links that can be built from the unknot by iterated cabling and summing. Links that arise in complex algebraic geometry are generally of this type.

ON FRAMED LINK PRESENTATIONS OF SURFACE BUNDLES

In this paper, we show that every closed orientable surface bundle over the circle is represented by a fibered link in the 3-sphere with framings induced by the fibration of the complement.