arXiv research

Study shows knots surgered elliptic surfaces admit handle decompositions without 1- and 3-handles.

The purpose of this note is to show that classical cobordism arguments, which go back to the pioneering works of Mandelbaum and Moishezon, provide quick and unified proofs of any knot surgered compact simply-connected 4-manifold X_K becoming diffeomorphic to X after a single stabilization by connected summing with S^2 …

We prove that any symplectic 4-manifold which is not a rational or ruled surface, after sufficiently many blow-ups, admits an arbitrary number of nonisomorphic Lefschetz fibrations of the same genus which cannot be obtained from one another via Luttinger surgeries. This generalizes results of Park and Yun who construct…

By considering non-orientable surfaces in the surgered manifolds, we show that the 10/3- and -10/3-Dehn surgeries on the 2-bridge knot $9_{27} = S(49,19)$ are not cosmetic, i.e., they give mutually non-homeomorphic manifolds. The knot is unknown to have no cosmetic surgeries by previously known results; in particular, …

We show that the distance of a link $K$ with respect to a bridge surface of any genus determines a lower bound on the genus of essential surfaces and Heegaard surfaces in the manifolds that result from non-trivial Dehn surgeries on the knot. In particular, knots with high bridge distance do not admit non-trivial non-hy…

This paper is a continuation on the 2012 paper on "Cutting Twisted Solid Tori (TSTs)", in which we considered twisted solid torus links (tst links). We generalize the notion of tst links to "surgerized tst links": recall that when performing $Φ^μ(n(τ), d(τ), M)$ on a tst $\langle τ\rangle$ where $M$ is odd, we obtain t…

Let K \subset Y be a knot in a three manifold which admits a longitude-framed surgery such that the surgered manifold has first Betti number greater than that of Y. We find a formula which computes the twisted Floer homology of the surgered manifold, in terms of twisted knot Floer homology. Using this, we compute the t…

Two Dehn surgeries on a knot are called purely cosmetic if their surgered manifolds are homeomorphic as oriented manifolds. Gordon conjectured that non-trivial knots in $S^3$ do not admit purely cosmetic surgeries. In this article, we confirm this conjecture for cable knots.

Two Dehn surgeries on a knot are called purely cosmetic if their surgered manifolds are homeomorphic as oriented manifolds. Gordon conjectured that non-trivial knots in $S^3$ do not admit purely cosmetic surgeries. In this article, we confirm this conjecture for connected sums of knots by analysing the JSJ-structures.

Dehn surgery on a knot determines a dual knot in the surgered manifold, the core of the filling torus. We consider duals of knots in $S^3$ that have a lens space surgery. Each dual supports a contact structure. We show that if a universally tight contact structure is supported, then the dual is in the same homology cla…

The study shows conditions for elliptic surfaces without 1-handles.

New invariant connects knot homology and BPS series for plumbed knot complements.

Let E(1)_K denote the closed 4-manifold that is homotopy equivalent (hence homeomorphic) to the rational elliptic surface E(1) and is obtained by performing Fintushel-Stern knot surgery on E(1) using a knot K in S^3. We construct an infinite family of homologous non-isotopic symplectic tori representing a primitive hom…

Let $M$ be a once-punctured torus bundle over $S^1$ with monodromy $h$. We show that, under certain hypotheses on $h$, "most" Dehn-fillings of $M$ (in some cases all but finitely many) are virtually $\mathbb{Z}$-representable. We apply our results to show that surgeries on the figure-eight knot with even numerator are …

If there are any 2-component counterexamples to the Generalized Property R Conjecture, a least genus component of all such counterexamples cannot be a fibered knot. Furthermore, the monodromy of a fibered component of any such counterexample has unexpected restrictions. The simplest plausible counterexample to the Gene…

Recently, Hodgson and Kerckhoff found a small bound on Dehn surgered 3-manifolds from hyperbolic knots not admitting hyperbolic structures using deformations of hyperbolic cone-manifolds. They asked whether the area normalized meridian length squared of maximal tubular neighborhoods of the singular locus of the cone-ma…

We prove that C. Loewner's inequality for the torus is satisfied by all hyperelliptic surfaces X, as well. We first construct the Loewner loops on the (mildly singular) companion tori, locally isometric to X away from the Weierstrass points. The loops are then transplanted to X, and surgered to obtain a Loewner loop on…

Study of torus surgeries on knot traces, finding exotic surfaces and traces.

In this article we construct a family of knot surgery $4$-manifolds admitting arbitrarily many nonisomorphic Lefschetz fibration structures with the same genus fiber. We obtain such families by performing knot surgery on an elliptic surface $E(2)$ using connected sums of fibered knots obtained by Stallings twist from a…

In this article we study Lefschetz fibration structures on knot surgery 4-manifolds obtained from an elliptic surface E(2) using Kanenobu knots $K$. As a result, we get an infinite family of simply connected mutually diffeomorphic 4-manifolds coming from a pair of inequivalent Kanenobu knots. We also obtain an infinite…

In this article we prove that, if $X$ is a smooth $4$-manifold containing an embedded double node neighborhood, all knot surgery $4$-manifolds $X_K$ are mutually diffeomorphic to each other after a connected sum with $\mathbb{CP}^2$. Hence, by applying to the simply connected elliptic surface $E(n)$, we also show that …

This short note presents a simple construction of nonisotopic symplectic tori representing the same primitive homology class in the symplectic 4-manifold E(1)_K, obtained by knot surgery on the rational elliptic surface E(1) with the left-handed trefoil knot K. E(1)_K has the simplest homotopy type among simply-connect…

We show that Bordered Heegaard Floer invariant $\widehat{CFD}$ of a knot complement in $S^3$ is invariant under the elliptic involution on its boundary.

We give a systematic method to calculate some homological data from the global monodromy of a topological elliptic surface. We apply this method to the cases 1) the transcendental lattice of an extremal elliptic K3 surface, 2) the torsion part of Mordell-Weil group of a general elliptic surface, and 3) the Mordell-Weil…

Study finds negatively curved spheres in elliptic surfaces and their modifications.

The paper proves a criterion for L-space knots and their representations.

We study the left-orderability of the fundamental groups of cyclic branched covers of links which admit co-oriented taut foliations. In particular we do this for cyclic branched covers of fibred knots in integer homology $3$-spheres and cyclic branched covers of closed braids. The latter allows us to complete the proof…

For a fixed p, there are only finitely many elliptic 3-manifolds given by p/q-surgery on a knot in S^3. We prove this result by using the Heegaard Floer correction terms (d-invariants) to obstruct elliptic manifolds from arising as knot surgery.

A knot K in the 3-sphere is said to have Property nR if, whenever K is a component of an n-component link L and some integral surgery on L produces the connected sum of n copies of S^1 x S^2, there is a sequence of handle slides on L that converts L into a 0-framed unlink. The Generalized Property R Conjecture is that …

Characterizes totally elliptic surface group representations into Lie groups.

Elliptic surfaces have unique Lefschetz pencils and Calabi-Yau diffeomorphisms.

Let $D$ be any elliptic right cylinder. We prove that every type of knot can be realized as the trajectory of a ball in $D.$ This proves a conjecture of Lamm and gives a new proof of a conjecture of Jones and Przytycki. We use Jacobi's proof of Poncelet's theorem by means of elliptic functions.

Minimal discs count as knot invariants in hyperbolic 4-space.

We show that if a hyperbolic 3-manifold $M$ with a single torus boundary admits two Dehn fillings at distance 5, each of which contains an essential torus, then $M$ is a rational homology solid torus, which is not large in the sense of Wu. Moreover, one of the surgered manifold contains an essential torus which meets t…

Elliptic surfaces without 1-handles proven for specific cases.

Study of 17 surface behaviors and singularities for elliptic Weingarten equations.

Simple constructions of semi-discrete and discrete surfaces using Jacobi elliptic functions.

Estimates heights of special surfaces in warped products.

Study on knots and dynamics on three-sphere, linking bounds, and upper action bounds.

In this short article we give a criterion whether a given minimal symplectic 4-manifold with $b_{2}^{+}=1$ having a torsion-free canonical class is rational or ruled. As a corollary, we confirm that most of homotopy elliptic surfaces $E(1}_{K}$, K is a fibered knot in $S^3$, constructed by R. Fintushel and R. Stern are…

The paper provides a new inequality for 4-manifolds and uses it to study knot sliceness and symplectic embeddings.

In the present paper we consider a special class of spacelike surfaces in the Minkowski 4-space which are one-parameter systems of meridians of the rotational hypersurface with timelike or spacelike axis. They are called meridian surfaces of elliptic or hyperbolic type, respectively. We study these surfaces with respec…

Constructs a new type of metric for elliptic surfaces.

This paper studies symplectic structures on elliptic surfaces with positive Euler number.

Researchers classify cmc surfaces using Jacobi elliptic functions.

In this paper we study the general affine differential geometry of surfaces in affine space $A^3$. For a regular elliptical surface we define a moving frame of minimal order and get the complete system of differential invariants. As an application we classify regular elliptical surfaces of constant curvatures up to aff…

Harer-Kas-Kirby conjectured that every handle decomposition of the elliptic surface E(1)_{2,3} requires both 1- and 3-handles. We prove that the elliptic surface E(n)_{p,q} has a handle decomposition without 1-handles for $n\geq 1$ and (p,q)=(2,3),(2,5),(3,4),(4,5).

This paper constructs explicit trisection diagrams for elliptic surfaces.