Complexity of virtual multistrings

A virtual[Formula: see text]-string [Formula: see text] consists of a closed, oriented surface [Formula: see text] and a collection [Formula: see text] of [Formula: see text] oriented, closed curves immersed in [Formula: see text]. We consider virtual [Formula: see text]-strings up to virtual homotopy, i.e. stabilizations, destabilizations, stable homeomorphism, and homotopy. Recently, Cahn proved that any virtual 1-string can be virtually homotoped to a minimally filling and crossing-minimal representative by monotonically decreasing both genus and the number of self-intersections. We generalize her result to the case of non-parallel virtual [Formula: see text]-strings. Cahn also proved that any two crossing-irreducible representatives of a virtual 1-string are related by isotopy, Type 3 moves, stabilizations, destabilizations, and stable homeomorphism. Kadokami claimed that this held for virtual [Formula: see text]-strings in general, but Gibson found a counterexample for 5-strings. We show that Kadokami’s statement holds for non-parallel [Formula: see text]-strings and exhibit a counterexample for general virtual 3-strings.

Addendum: "(1, 2) and weak (1, 3) homotopies on knot projections"

10.1142/s0218216514910015 ◽

2014 ◽

Vol 23 (08) ◽

pp. 1491001 ◽

Cited By ~ 1

Author(s):

Noboru Ito ◽

Yusuke Takimura

Keyword(s):

Personal Communication ◽

Finite Collection ◽

Closed Curves ◽

Oriented Surface ◽

Reidemeister Moves ◽

After this paper was published, the following information about doodles was pointed out by Roger Fenn. A doodle was introduced by Fenn and Taylor [2], which is a finite collection of closed curves without triple intersections on a closed oriented surface considered up to the second flat Reidemeister moves with the condition (*) that each component has no self-intersections. Khovanov [4] introduced doodle groups, and for his process, he considered doodles under a more generalized setting (i.e. removing the condition (*) and permitting the first flat Reidemeister moves). He showed [4, Theorem 2.2], a result similar to our [3, Theorem 2.2(c)]. He also pointed out that [1, Corollary 2.8.9] gives a result similar to [4, Theorem 2.2]. The authors first noticed the above results by Fenn and Khovanov via personal communication with Fenn, and therefore, the authors would like to thank Roger Fenn for these references.

COMPLEX FENCHEL-NIELSEN COORDINATES FOR QUASI-FUCHSIAN STRUCTURES

International Journal of Mathematics ◽

10.1142/s0129167x94000140 ◽

1994 ◽

Vol 05 (02) ◽

pp. 239-251 ◽

Cited By ~ 18

Author(s):

SER PEOW TAN

Keyword(s):

Teichmüller Space ◽

Teichmuller Space ◽

Closed Curves ◽

Oriented Surface ◽

Hyperbolic Structures ◽

Embedded Image ◽

Totally Real ◽

Real Subspace ◽

Let Fg be a closed oriented surface of genus g ≥ 2 and let [Formula: see text] be the space of marked quasi-fuchsian structures on Fg. Let [Formula: see text] be a set of non-intersecting, non-trivial simple closed curves on Fg that cuts Fg into pairs of pants components. In this note, we construct global complex coordinates for [Formula: see text] relative to [Formula: see text] giving an embedding of [Formula: see text] into [Formula: see text]. The totally real subspace of [Formula: see text] with respect to these coordinates is the Teichmüller Space [Formula: see text] of marked hyperbolic structures on Fg, the coordinates reduce to the usual Fenchel-Nielsen coordinates for [Formula: see text] relative to [Formula: see text]. Various properties of the embedded image are studied.

GEOMETRY OF THE HOMOLOGY CURVE COMPLEX

Journal of Topology and Analysis ◽

10.1142/s179352531250001x ◽

2012 ◽

Vol 04 (03) ◽

pp. 335-359 ◽

Cited By ~ 1

Author(s):

INGRID IRMER

Keyword(s):

Homotopy Class ◽

Intersection Number ◽

Simple Algorithm ◽

Curve Complex ◽

Oriented Surface ◽

Immersed Surfaces ◽

Minimal Genus ◽

Suppose S is a closed, oriented surface of genus at least two. This paper investigates the geometry of the homology multicurve complex, [Formula: see text], of S; a complex closely related to complexes studied by Bestvina–Bux–Margalit and Hatcher. A path in [Formula: see text] corresponds to a homotopy class of immersed surfaces in S × I. This observation is used to devise a simple algorithm for constructing quasi-geodesics connecting any two vertices in [Formula: see text], and for constructing minimal genus surfaces in S × I. It is proven that for g ≥ 3 the best possible bound on the distance between two vertices in [Formula: see text] depends linearly on their intersection number, in contrast to the logarithmic bound obtained in the complex of curves. For g ≥ 4 it is shown that [Formula: see text] is not δ-hyperbolic.

Filling of closed surfaces

Journal of Topology and Analysis ◽

10.1142/s1793525318500309 ◽

2018 ◽

Vol 10 (04) ◽

pp. 897-913 ◽

Cited By ~ 2

Author(s):

Bidyut Sanki

Keyword(s):

Disjoint Union ◽

Mapping Class ◽

Constructive Proof ◽

Closed Curves ◽

Closed Surfaces ◽

Minimal Filling ◽

Minimum Number ◽

Genus Formula ◽

Positive Integers ◽

Let [Formula: see text] denote a closed oriented surface of genus [Formula: see text]. A set of simple closed curves is called a filling of [Formula: see text] if its complement is a disjoint union of discs. The mapping class group [Formula: see text] of genus [Formula: see text] acts on the set of fillings of [Formula: see text]. The union of the curves in a filling forms a graph on the surface which is a so-called decorated fat graph. It is a fact that two fillings of [Formula: see text] are in the same [Formula: see text]-orbit if and only if the corresponding fat graphs are isomorphic. We prove that any filling of [Formula: see text] whose complement is a single disc (i.e. a so-called minimal filling) has either three or four closed curves and in each of these two cases, there is a unique such filling up to the action of [Formula: see text]. We provide a constructive proof to show that the minimum number of discs in the complement of a filling pair of [Formula: see text] is two. Finally, given positive integers [Formula: see text] and [Formula: see text] with [Formula: see text], we construct a filling pair of [Formula: see text] such that the complement is a union of [Formula: see text] topological discs.

On braid monodromies of non-simple braided surfaces

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s030500410007482x ◽

1996 ◽

Vol 120 (2) ◽

pp. 237-245 ◽

Cited By ~ 5

Author(s):

Seiichi Kamada

Keyword(s):

Knot Theory ◽

Piecewise Linear ◽

Tubular Neighbourhood ◽

Oriented Surface ◽

Covering Map ◽

Branched Covering ◽

Ambient Isotopy ◽

Linear Category ◽

Similar Notion ◽

A braided surface of degree m is a compact oriented surface S embedded in a bidisk such that is a branched covering map of degree m and , where is the projection. It was defined L. Rudolph [14, 16] with some applications to knot theory, cf. [13, 14, 15, 16, 17, 18]. A similar notion was defined O. Ya. Viro: A (closed) 2-dimensional braid in R4 is a closed oriented surface F embedded in R4 such that and pr2 │F: F → S2 is a branched covering map, where is the tubular neighbourhood of a standard 2-sphere in R4. It is related to 2-knot theory, cf. [8, 9, 10]. Braided surfaces and 2-dimensional braids are called simple if their associated branched covering maps are simple. Simple braided surfaces and simple 2-dimensional braids are investigated in some articles, [5, 8, 9, 14, 16], etc. This paper treats of non-simple braided surfaces in the piecewise linear category. For braided surfaces a natural weak equivalence relation, called braid ambient isotopy, appears essentially although it is not important for classical dimensionai braids Artin's argument [1].

Generating the mapping class groups by torsions

10.1142/s0218216517500377 ◽

2017 ◽

Vol 26 (07) ◽

pp. 1750037

Author(s):

Xiaoming Du

Keyword(s):

Mapping Class Group ◽

Class Group ◽

Mapping Class Groups ◽

Mapping Class ◽

Class Groups ◽

Oriented Surface ◽

Genus Formula ◽

Let [Formula: see text] be a closed oriented surface of genus [Formula: see text] and let [Formula: see text] be the mapping class group. When the genus is at least 3, [Formula: see text] can be generated by torsion elements. We prove the following results: For [Formula: see text], [Formula: see text] can be generated by four torsion elements. Three generators are involutions and the fourth one is an order three element. [Formula: see text] can be generated by five torsion elements. Four generators are involutions and the fifth one is an order three element.

On stable commutator lengths of Dehn twists along separating curves

10.1142/s0218216517500560 ◽

2017 ◽

Vol 26 (10) ◽

pp. 1750056 ◽

Cited By ~ 1

Author(s):

Naoyuki Monden ◽

Kazuya Yoshihara

Keyword(s):

Mapping Class Group ◽

Class Group ◽

Upper Bounds ◽

Mapping Class ◽

Oriented Surface ◽

Dehn Twists ◽

We give new upper bounds on the stable commutator lengths of Dehn twists along separating curves in the mapping class group of a closed oriented surface. The estimates of these upper bounds are [Formula: see text], where [Formula: see text] is the genus of the surface.

BIRMAN'S CONJECTURE FOR SINGULAR BRAIDS ON CLOSED SURFACES

10.1142/s0218216504003512 ◽

2004 ◽

Vol 13 (07) ◽

pp. 895-915

Author(s):

LUIS PARIS

Keyword(s):

Braid Group ◽

Oriented Surface ◽

Vassiliev Invariants ◽

Closed Surfaces ◽

Definition Of ◽

Let M be a closed oriented surface of genus g≥1, let Bn(M) be the braid group of M on n strings, and let SBn(M) be the corresponding singular braid monoid. Our purpose in this paper is to prove that the desingularization map η : SBn(M)→ℤ[Bn(M)], introduced in the definition of the Vassiliev invariants (for braids on surfaces), is injective.

Representations of the fundamental group of a closed oriented surface in

Topology ◽

10.1016/j.top.2003.10.010 ◽

2004 ◽

Vol 43 (4) ◽

pp. 831-855 ◽

Cited By ~ 4

Author(s):

O GARCIAPRADA

Keyword(s):

Fundamental Group ◽

Oriented Surface ◽

Khovanov–Lipshitz–Sarkar homotopy type for links in thickened higher genus surfaces

10.1142/s0218216521500528 ◽

2021 ◽

Author(s):

Louis H. Kauffman ◽

Igor Mikhailovich Nikonov ◽

Eiji Ogasa

Keyword(s):

Homotopy Type ◽

Khovanov Homology ◽

Stable Homotopy ◽

Product Manifold ◽

Oriented Surface ◽

Steenrod Square ◽

Higher Genus ◽

Genus Formula ◽

Thickened Surface ◽

We discuss links in thickened surfaces. We define the Khovanov–Lipshitz–Sarkar stable homotopy type and the Steenrod square for the homotopical Khovanov homology of links in thickened surfaces with genus [Formula: see text]. A surface means a closed oriented surface unless otherwise stated. Of course, a surface may or may not be the sphere. A thickened surface means a product manifold of a surface and the interval. A link in a thickened surface (respectively, a 3-manifold) means a submanifold of a thickened surface (respectively, a 3-manifold) which is diffeomorphic to a disjoint collection of circles. Our Khovanov–Lipshitz–Sarkar stable homotopy type and our Steenrod square of links in thickened surfaces with genus [Formula: see text] are stronger than the homotopical Khovanov homology of links in thickened surfaces with genus [Formula: see text]. It is the first meaningful Khovanov–Lipshitz–Sarkar stable homotopy type of links in 3-manifolds other than the 3-sphere. We point out that our theory has a different feature in the torus case.