Related papers: Concordance of Seifert surfaces
Let $LHT$ be a left handed trefoil knot and $K$ be any knot. We define $M_n(K)$ to be the homology $3$-sphere which is represented by a simple link of $LHT$ and $LHT \sharp K$ with framings $0$ and $n$ respectively. Starting with this link,…
By applying Seifert's algorithm to a special alternating diagram of a link L, one obtains a Seifert surface F of L. We show that the support of the sutured Floer homology of the sutured manifold complementary to F is affine isomorphic to…
We define a Floer-homology invariant for knots in an oriented three-manifold, closely related to the holomorphic disk Floer homologies for three-manifolds defined in an earlier paper. We set up basic properties of these invariants,…
This monograph derives direct and concrete relations between colored Jones polynomials and the topology of incompressible spanning surfaces in knot and link complements. Under mild diagrammatic hypotheses that arise naturally in the study…
We show that the non-measure hyperbolicity of K3 surfaces -- which M. Green and P. Griffiths verified for certain cases in 1980 -- holds for all K3 surfaces. As a byproduct, we prove the non-measure hyperbolicity of any Hilbert schemes of…
Similar to knots in S^3, any knot in a lens space has a grid diagram from which one can combinatorially compute all of its knot Floer homology invariants. We give an explicit description of the generators, differentials, and rational Maslov…
We give an upper bound for the number of compact essential orientable non-isotopic surfaces, with Euler characteristic at least some constant $\chi$, properly embedded in a finite-volume hyperbolic 3-manifold $M$, closed or cusped. This…
Some questions are posted at the end of Chapter 16 of Huybrechts' book 'Lectures on K3 Surfaces', concerning the bounded derived category of a K3 surface $D^b(S)$. Let $E$ be a spherical object in $D^b(S)$. The first question asks if there…
Let $N$ be a compact, orientable hyperbolic 3-manifold whose boundary is a connected totally geodesic surface of genus $2$. If $N$ has Heegaard genus at least $5$, then its volume is greater than $2V_{\rm oct}$, where $V_{\rm…
A surface in hyperbolic space $\h^3$ invariant by a group of parabolic isometries is called a parabolic surface. In this paper we investigate parabolic surfaces of $\h^3$ that satisfy a linear Weingarten relation of the form…
We show that any two same-genus, oriented, boundary parallel surfaces bounded by a non-split, alternating link into the 4-ball are smoothly isotopic fixing boundary. In other words, any same-genus Seifert surfaces for a non-split,…
Colding and Gabai have given an effective version of Li's theorem that non-Haken hyperbolic 3-manifolds have finitely many irreducible Heegaard splittings. As a corollary of their work, we show that Haken hyperbolic 3-manifolds have a…
In 2016 Levine showed that there exists a knot in a homology 3-sphere which is not smoothly concordant to any knot in the 3-sphere where one allows concordances in any smooth homology cobordism. Whether the same is true if one allows…
We derive basic differential geometric formulae for surfaces in hyperbolic space represented as envelopes of horospheres. The dual notion of parallel hypersurfaces is also studied. The representation is applied to prove existence and…
The Kashaev-Murakami-Murakami Volume Conjecture connects the hyperbolic volume of a knot complement to the asymptotics of certain evaluations of the colored Jones polynomials of the knot. We introduce a closely related conjecture for…
It was conjectured by Lopez that every closed irreducible non-Haken 3-manifold contains a small knot. In this paper, we give explicit examples of hyperbolic small knots in most closed orientable spherical 3-manifolds other than prism…
We prove that any hyper-K\"{a}hler sixfold $K$ of generalized Kummer type has a naturally associated manifold $Y_K$ of $\mathrm{K}3^{[3]}$-type. It is obtained as crepant resolution of the quotient of $K$ by a group of symplectic…
We show the finiteness of the N\'eron-Severi lattices of complex projective K3 surfaces whose automorphism groups are non-elementary hyperbolic with explicit descriptions, under the assumption that the Picard number $\ge 6$ which is optimal…
We prove that any complete hyperbolic 3--manifold with finitely generated fundamental group, with a single topological end, and which embeds into $\BS^3$ is the geometric limit of a sequence of hyperbolic knot complements in $\BS^3$. In…
Kakimizu complex of a knot is a flag simplicial complex whose vertices correspond to minimal genus Seifert surfaces and edges to disjoint pairs of such surfaces. We discuss a general setting in which one can define a similar complex. We…