相关论文: Weights of essential surfaces in 2-bridge knot com…

We study surface knots in 4-space by using generic planar projections. These projections have fold points and cusps as their singularities and the image of the singular point set divides the plane into several regions. The width (or the…

In earlier work, the authors introduced a conjecture which, for an orientation-preserving diffeomorphism $\varphi \colon S \to S$ of a surface, connects a certain quantum invariant of $\varphi$ with the hyperbolic volume of its mapping…

A self-transverse immersion of a smooth manifold M^{k+2} in R^{2k+2} has a double point self-intersection set which is the image of an immersion of a smooth surface, the double point self-intersection surface. We prove that this surface may…

This is the last of a series of papers which give a necessary and sufficient condition for two essential simple loops on a 2-bridge sphere in a 2-bridge link complement to be homotopic in the link complement. The first paper of the series…

In this paper we verify that with the exception of the $(2, 2n+1)$ torus knots, positive 2-bridge knots up to 31 crossings do not admit chirally cosmetic surgeries. A knot $K$ admits chirally cosmetic surgeries if there exist surgeries…

We introduce a quantum trace map for an ideally triangulated hyperbolic knot complement $S^3\backslash \mathcal{K}$. The map assigns a quantum operator to each element of Kauffmann Skein module of the 3-manifold. The quantum operator lives…

We discuss 3-manifolds which are cyclic coverings of the 3-sphere, branched over 2-bridge knots and links. Different descriptions of these manifolds are presented: polyhedral, Heegaard diagram, Dehn surgery and coloured graph constructions.…

We give a new characterization of symplectic surfaces in CP^2 via bridge trisections. Specifically, a minimal genus surface in CP^2 is smoothly isotopic to a symplectic surface if and only if it is smoothly isotopic to a surface in…

We will discuss a method for visual presentation of knotted surfaces in the four space, by examining a number and a position of its Morse's critical points. Using this method, we will investigate surface-knot with one critical point of…

Experimental work suggests that the Seifert genus of a knot grows linearly with respect to the crossing number of the knot. In this article, we use a billiard table model for $2$-bridge or rational knots to show that the average genus of a…

We present a topological interpretation of knot and braid contact homology in degree zero, in terms of cords and skein relations. This interpretation allows us to extend the knot invariant to embedded graphs and higher-dimensional knots. We…

In this article we introduce new affinely invariant points---`special parabolic points'---on the parabolic set of a generic surface $M$ in real 4-space, associated with symmetries in the 2-parameter family of reflexions of $M$ in points of…

We define an invariant, which we call surface-complexity, of closed 3-manifolds by means of Dehn surfaces. The surface-complexity of a manifold is a natural number measuring how much the manifold is complicated. We prove that it fulfils…

In a recent work, we analyzed a weighted-residual error estimator for isogeometric boundary element methods in 2D and proposed an adaptive algorithm which steers the local mesh-refinement of the underlying partition as well as the…

We construct knot invariants on the basis of ascribing Euclidean geometric values to a triangulation of sphere S^3 where the knot lies. The main new feature of this construction compared to the author's earlier papers on manifold invariants…

We extend Culler and Shalen's construction of detecting essential surfaces in 3-manifolds to 3-orbifolds. We do so in the setting of the $\mathrm{SL}_2(\mathbb{C})$ character variety, and following Boyer and Zhang in the…

Let K be a knot in the 3-sphere with 2-fold branched covering space M. If for some prime p congruent to 3 mod 4 the p-torsion in the first homology of M is cyclic with odd exponent, then K is of infinite order in the knot concordance group.…

We consider the ring of invariants of n points on the projective line. The space (P^1)^n // PGL_2 is perhaps the first nontrivial example of a Geometry Invariant Theory quotient. The construction depends on the weighting of the n points.…

Torsion polynomials connect the genus of a hyperbolic knot (a topological invariant) with the discrete faithful representation (a geometric invariant). Using a new combinatorial structure of an ideal triangulation of a 3-manifold that…

We give examples showing that Kidwell's inequality for the maximal degree of the Brandt-Lickorish-Millett-Ho polynomial is in general not sharp.