arXiv research

The paper disproves the existence of certain subgroups with nontrivial rational abelianization.

Characterizes multitwists in surface mapping class groups.

Multitwists cannot generate all compactly supported mapping class groups on infinite-type surfaces.

This paper analyzes commensurability of the class of surface automorphism generated by two Dehn multitwists. We show pairwise noncommensurability between several classes arising from canonical curve configurations. In addition, we consider the Kenyon-Smillie invariant J of flat surfaces in this setting. We also introdu…

Homomorphisms between pure mapping class groups are classified for certain genus surfaces.

Meridian surfaces in the Euclidean 4-space are two-dimensional surfaces which are one-parameter systems of meridians of a standard rotational hypersurface. On the base of our invariant theory of surfaces we study meridian surfaces with special invariants. In the present paper we give the complete classification of Chen…

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. We call these surfaces meridian surfaces of elliptic or hyperbolic type, respectively. On the base of our invariant theory of surfaces w…

The study classifies timelike meridian surfaces in Minkowski 4-space.

We construct a special class of spacelike surfaces in the Minkowski 4-space which are one-parameter systems of meridians of the rotational hypersurface with lightlike axis and call these surfaces meridian surfaces of parabolic type. They are analogous to the meridian surfaces of elliptic or hyperbolic type. Using the i…

Infinite Klein bottles with 4-fold meridians found.

On a compact oriented surface of genus $g$ with $n\geq 1$ boundary components, $δ_1, δ_2,\ldots, δ_n$, we consider positive factorizations of the boundary multitwist $t_{δ_1} t_{δ_2} \cdots t_{δ_n}$, where $t_{δ_i}$ is the positive Dehn twist about the boundary $δ_i$. We prove that for $g\geq 3$, the boundary multitwis…

Meridian lemma extended to fully alternating links in thickened surfaces.

In the present article we study a special class of surfaces in the four-dimensional Euclidean space, which are one-parameter systems of meridians of the standard rotational hypersurface. They are called meridian surfaces. We show that a meridian surface has a harmonic Gauss map if and only if it is part of a plane. Fur…

We construct a special class of Lorentz surfaces in the pseudo-Euclidean 4-space with neutral metric which are one-parameter systems of meridians of rotational hypersurfaces with lightlike axis and call them meridian surfaces. We give the complete classification of the meridian surfaces with constant Gauss curvature an…

In the present paper we consider a special class of Lorentz surfaces in the four-dimensional pseudo-Euclidean space with neutral metric which are one-parameter systems of meridians of rotational hypersurfaces with timelike or spacelike axis and call them meridian surfaces. We give the complete classification of minimal…

In the present article we study a special class of surfaces in the four-dimensional Euclidean space, which are one-parameter systems of meridians of the standard rotational hypersurface. They are called meridian surfaces. We classified semi-parallel meridian surface in 4-dimensional Euclidean space $E^4$.

We construct a special class of Lorentz surfaces in the pseudo-Euclidean 4-space with neutral metric which are one-parameter systems of meridians of rotational hypersurfaces with timelike or spacelike axis and call them meridian surfaces. We give the complete classification of the meridian surfaces with parallel mean c…

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…

Study of hyperelliptic mapping class groups with applications and profinite completions.

For a handlebody of genus $g\geq6$ it is shown that every automorphism of the complex of separating meridians can be extended to an automorphism on the complex of all meridians and, in consequence, it is geometric.

In this paper, we study meridian surfaces of Weingarten type in Euclidean 4-space E^4. We give the neccessary and sufficient conditions for a meridian surface in E^4 to become Weingarten type.

A marginally trapped surface in the four-dimensional Minkowski space is a spacelike surface whose mean curvature vector is lightlike at each point. We introduce meridian surfaces of parabolic type as one-parameter systems of meridians of a rotational hypersurface with lightlike axis in Minkowski 4-space and find their …

Permutation of Weierstrass points on Veech surfaces in $\mathcal{H}(2)$ classified by discriminant.

Examines how meridians and parallels help in map drawing.

Suppose a genus two handlebody is removed from a 3-manifold M and then a single meridian of the handlebody is restored. The result is a knot or link complement in M and it is natural to ask whether geometric properties of the link complement say something about the meridian that was restored. Here we consider what the …

We derive bounds on the length of the meridian and the cusp volume of hyperbolic knots in terms of the topology of essential surfaces spanned by the knot. We provide an algorithmically checkable criterion that guarantees that the meridian length of a hyperbolic knot is below a given bound. As applications we find knot …

Study timelike meridian surfaces in Minkowski 4-space with specific properties.

We show that in the mapping class group of a surface any relation between Dehn twists of the form T_x^j T_y^k = M (M a multitwist) is the lantern relation, and any relation of the form (T_x T_y)^k = M (where T_x commutes with M) is the 2-chain relation.

Prove integrality of genus-$g$ indices with adjoint Reidemeister torsions for twist knots and meridians.

New foliations show knot meridians are detectable.

In the literature of the study of knot group epimorphisms, the existence of an epimorphism between two given knot groups is mostly (if not always) shown by giving an epimorphism which preserves meridians. A natural question arises: is there an epimorphism preserving meridians whenever a knot group is a homomorphic imag…

Khovanov homology detects specific links.

We apply the invariant theory of surfaces in the four-dimensional Euclidean space to the class of general rotational surfaces with meridians lying in two-dimensional planes. We find all minimal super-conformal surfaces of this class.

A marginally trapped surface in the four-dimensional Minkowski space is a spacelike surface whose mean curvature vector is lightlike at each point. We associate a geometrically determined moving frame field to such a surface and using the derivative formulas for this frame field we obtain seven invariant functions. Our…

It is proved that every disconnected surface-link with meridian-based free fundamental group is a trivial (i.e., an unknotted-unlinked) surface-link. This result is a surface-link version of the author's recent announcement result on smooth unknotting of a surface-knot.

Let S be a closed oriented surface of genus g > 1, and let T denote its Torelli group. First, given a set E of homotopically nontrivial, pairwise disjoint, pairwise nonisotopic simple closed curves on S, we determine precisely when a multitwist on E is an element of T by defining an equivalence relation on E and then a…

The goal of this paper is to offer a comprehensive exposition of the current knowledge about Heegaard splittings of exteriors of knots in the 3-sphere. The exposition is done with a historical perspective as to how ideas developed and by whom. Several new notions are introduced and some facts about them are proved. In …

Link groups can only have certain SU(2) representations.

Formula calculates knot Floer complexes for specific cable knots.

New example shows figure eight knot not smoothly concordant but homology cobordant.

Given any knot k, there exists a hyperbolic knot tilde k with arbitrarily large volume such that the knot group pi k is a quotient of pi tilde k by a map that sends meridian to meridian and longitude to longitude. The knot tilde k can be chosen to be ribbon concordant to k and also to have the same Alexander invariant …

The study characterizes lamination limits and homeomorphisms in 3D handlebodies.

The paper studies loxodromes on twisted surfaces in a specific 3D space.

It is a well-known procedure for constructing a torus knot or link that first we prepare an unknotted torus and meridian disks in the complementary solid tori of it, and second smooth the intersections of the boundary of meridian disks uniformly. Then we obtain a torus knot or link on the unknotted torus and its Seifer…

We compute the Kauffman bracket skein module of the complement of a twist knot, finding that it is free and infinite dimensional. The basis consists of cables of a two-component link, one component of which is a meridian of the knot. The cabling of the meridian can be arbitrarily large while the cabling of the other co…

New reflection groups derived from torus knots with finite meridians.

The study defines new surfaces with specific cut locus properties and provides conditions for their existence.

New 2-spheres of revolution with simple cut locus structures.