scholarly journals When do links admit homeomorphic C-complexes?

2017 ◽  
Vol 26 (01) ◽  
pp. 1750010 ◽  
Author(s):  
Christopher W. Davis ◽  
Grant Roth
Keyword(s):  
Seifert Surface ◽  
Complete Obstruction ◽  
Linking Number ◽  
Seifert Surfaces

Any two knots admit orientation preserving homeomorphic Seifert surfaces, as can be seen by stabilizing. There is a generalization of a Seifert surface to the setting of links called a [Formula: see text]-complex. In this paper, we ask when two links will admit orientation preserving homeomorphic [Formula: see text]-complexes. In the case of 2-component links, we find that the pairwise linking number provides a complete obstruction. In the case of links with 3 or more components and zero pairwise linking number, Milnor’s triple linking number provides a complete obstruction.

scholarly journals SEIFERT SURFACES, COMMUTATORS AND VASSILIEV INVARIANTS

2007 ◽  
Vol 16 (10) ◽  
pp. 1295-1329
Author(s):  
E. KALFAGIANNI ◽  
XIAO-SONG LIN
Keyword(s):  
Fundamental Group ◽  
Lower Central Series ◽  
Central Series ◽  
Seifert Surface ◽  
Vassiliev Invariants ◽  
Seifert Surfaces

We show that the Vassiliev invariants of a knot K, are obstructions to finding a regular Seifert surface, S, whose complement looks "simple" (e.g. like the complement of a disc) to the lower central series of its fundamental group. We also conjecture a characterization of knots whose invariants of all orders vanish in terms of their Seifert surfaces.

SEIFERT SURFACES IN KNOT COMPLEMENTS

2007 ◽  
Vol 16 (08) ◽  
pp. 1053-1066 ◽  
Author(s):  
ENSIL KANG
Keyword(s):  
Compact Surface ◽  
Seifert Surface ◽  
Normal Surface ◽  
Incompressible Surface ◽  
Ideal Triangulation ◽  
Seifert Surfaces ◽  
Knot Complements

In the ordinary normal surface for a compact 3-manifold, any incompressible, ∂-incompressible, compact surface can be moved by an isotopy to a normal surface [9]. But in a non-compact 3-manifold with an ideal triangulation, the existence of a normal surface representing an incompressible surface cannot be guaranteed. The figure-8 knot complement is presented in a counterexample in [12]. In this paper, we show the existence of normal Seifert surface under some restriction for a given ideal triangulation of the knot complement.

scholarly journals Dijkgraaf–Witten type invariants of Seifert surfaces in 3-manifolds

2017 ◽  
Vol 26 (05) ◽  
pp. 1750026
Author(s):  
I. J. Lee ◽  
D. N. Yetter
Keyword(s):  
Initial Data ◽  
Gauge Group ◽  
Seifert Surface ◽  
Object Categories ◽  
Gauge Symmetries ◽  
Cocycle Condition ◽  
Witten Theory ◽  
Seifert Surfaces ◽  
Combinatorial Construction

We introduce defects, with internal gauge symmetries, on a knot and Seifert surface to a knot into the combinatorial construction of finite gauge-group Dijkgraaf–Witten theory. The appropriate initial data for the construction are certain three object categories, with coefficients satisfying a partially degenerate cocycle condition.

scholarly journals A distance for circular Heegaard splittings

2019 ◽  
Vol 28 (09) ◽  
pp. 1950059
Author(s):  
Kevin Lamb ◽  
Patrick Weed
Keyword(s):  
Critical Points ◽  
Morse Function ◽  
Heegaard Splitting ◽  
Heegaard Splittings ◽  
Seifert Surface ◽  
Singular Foliation ◽  
Incompressible Surfaces ◽  
Minimal Genus ◽  
Seifert Surfaces ◽  
Index 2

For a knot [Formula: see text], its exterior [Formula: see text] has a singular foliation by Seifert surfaces of [Formula: see text] derived from a circle-valued Morse function [Formula: see text]. When [Formula: see text] is self-indexing and has no critical points of index 0 or 3, the regular levels that separate the index-1 and index-2 critical points decompose [Formula: see text] into a pair of compression bodies. We call such a decomposition a circular Heegaard splitting of [Formula: see text]. We define the notion of circular distance (similar to Hempel distance) for this class of Heegaard splitting and show that it can be bounded under certain circumstances. Specifically, if the circular distance of a circular Heegaard splitting is too large: (1) [Formula: see text] cannot contain low-genus incompressible surfaces, and (2) a minimal-genus Seifert surface for [Formula: see text] is unique up to isotopy.

scholarly journals Helicity conservation and twisted Seifert surfaces for superfluid vortices

2017 ◽  
Vol 473 (2200) ◽  
pp. 20160853 ◽  
Author(s):  
Hayder Salman
Keyword(s):  
First Principles ◽  
Velocity Potential ◽  
Macroscopic Description ◽  
Linking Number ◽  
Helicity Conservation ◽  
Seifert Surfaces ◽  
Definition Of ◽  
Superfluid Vortices ◽  
Mathematical Derivation ◽  
The Continuum

Starting from the continuum definition of helicity, we derive from first principles its different contributions for superfluid vortices. Our analysis shows that an internal twist contribution emerges naturally from the mathematical derivation. This reveals that the spanwise vector that is used to characterize the twist contribution must point in the direction of a surface of constant velocity potential. An immediate consequence of the Seifert framing is that the continuum definition of helicity for a superfluid is trivially zero at all times. It follows that the Gauss-linking number is a more appropriate definition of helicity for superfluids. Despite this, we explain how a quasi-classical limit can arise in a superfluid in which the continuum definition for helicity can be used. This provides a clear connection between a microscopic and a macroscopic description of a superfluid as provided by the Hall–Vinen–Bekarevich–Khalatnikov equations. This leads to consistency with the definition of helicity used for classical vortices.

scholarly journals Virtual Seifert surfaces

2019 ◽  
Vol 28 (06) ◽  
pp. 1950039
Author(s):  
Micah Chrisman
Keyword(s):  
Seifert Surface ◽  
Virtual Knot ◽  
Alexander Polynomials ◽  
Seifert Surfaces ◽  
Gauss Diagram ◽  
Thickened Surface

A virtual knot that has a homologically trivial representative [Formula: see text] in a thickened surface [Formula: see text] is said to be an almost classical (AC) knot. [Formula: see text] then bounds a Seifert surface [Formula: see text]. Seifert surfaces of AC knots are useful for computing concordance invariants and slice obstructions. However, Seifert surfaces in [Formula: see text] are difficult to construct. Here, we introduce virtual Seifert surfaces of AC knots. These are planar figures representing [Formula: see text]. An algorithm for constructing a virtual Seifert surface from a Gauss diagram is given. This is applied to computing signatures and Alexander polynomials of AC knots. A canonical genus of AC knots is also studied. It is shown to be distinct from the virtual canonical genus of Stoimenow–Tchernov–Vdovina.

Identifying non-pseudo-alternating knots by using the free factor property of minimal genus Seifert surfaces

2021 ◽  
Author(s):  
Keisuke Himeno ◽  
Masakazu Teragaito
Keyword(s):  
Seifert Surface ◽  
Major Difficulty ◽  
Flat Surfaces ◽  
Minimal Genus ◽  
Pretzel Knots ◽  
Seifert Surfaces ◽  
Manifold Theory ◽  
Knots And Links

Pseudo-alternating knots and links are defined constructively via their Seifert surfaces. By performing Murasugi sums of primitive flat surfaces, such a knot or link is obtained as the boundary of the resulting surface. Conversely, it is hard to determine whether a given knot or link is pseudo-alternating or not. A major difficulty is the lack of criteria to recognize whether a given Seifert surface is decomposable as a Murasugi sum. In this paper, we propose a new idea to identify non-pseudo-alternating knots. Combining with the uniqueness of minimal genus Seifert surface obtained through sutured manifold theory, we demonstrate that two infinite classes of pretzel knots are not pseudo-alternating.

A criterion for detecting inequivalent tunnels for a knot

1990 ◽  
Vol 107 (3) ◽  
pp. 483-491 ◽  
Author(s):  
Tsuyoshi Kobayashi
Keyword(s):  
Seifert Surface ◽  
Oriented Surface ◽  
Seifert Surfaces ◽  
Regular Neighbourhood

Let K be an oriented knot in the 3-sphere S3. An exterior of K is the closure of the complement of a regular neighbourhood of K, and is denoted by E(K). A Seifert surface for K is an oriented surface S( ⊂ S3) without closed components such that ∂S = K. We denote S ∩ E(K) by Ŝ, and we regard S as obtained from Ŝ by a radial extension. S is incompressible if Ŝ is incompressible in E(K). A tunnel for K is an embedded arc τ in S3 such that τ ∪ K = ∂τ. We denote τ ∪ E(K) by τ, and we regard τ as obtained from τ by a radial extension. Let τ1, τ2 be tunnels for K. We say that τ1 and τ2 are homeomorphic if there is a self-homeomorphism f of E(K) such that f(τ1) = τ2. The tunnels τ1 and τ2 are isotopic if τ1 is ambient isotopic to τ2 in E(K). Then the main result of this paper is as follows: Theorem. Let K be a knot in S3, and let τ1, τ2 be tunnels for K. Suppose that there are incompressible Seifert surfaces S1 S2 for K such that S1 ∪ S2 = K, and τi ⊂ Si (i = 1, 2). If τ1 and τ2 are isotopic, then there is an ambient isotopyhτ (0 ≤ t ≤ 1) of S3 such that ht(K) = K, and h1(τ1) = τ2.

scholarly journals LEGENDRIAN RIBBONS IN OVERTWISTED CONTACT STRUCTURES

2009 ◽  
Vol 18 (04) ◽  
pp. 523-529
Author(s):  
SEBASTIAN BAADER ◽  
KAI CIELIEBAK ◽  
THOMAS VOGEL
Keyword(s):  
Contact Structure ◽  
Homology Class ◽  
Contact Manifold ◽  
The Self ◽  
Contact Structures ◽  
Seifert Surface ◽  
Linking Number ◽  
Relative Homology

We show that a null-homologous transverse knot K in the complement of an overtwisted disk in a contact 3-manifold is the boundary of a Legendrian ribbon if and only if it possesses a Seifert surface S such that the self-linking number of K with respect to S satisfies sl (K, S) = -χ(S). In particular, every null-homologous topological knot type in an overtwisted contact manifold can be represented by the boundary of a Legendrian ribbon. Finally, we show that a contact structure is tight if and only if every Legendrian ribbon minimizes genus in its relative homology class.

KNOTTED SEIFERT SURFACES AND UNKNOTTED SEIFERT SURFACES

2008 ◽  
Vol 17 (02) ◽  
pp. 141-155
Author(s):  
YUKIHIRO TSUTSUMI
Keyword(s):  
Seifert Surface ◽  
Minimal Genus ◽  
Seifert Surfaces ◽  
Genus One

It is known that free genus one knots do not admit Seifert surfaces with hyperbolic exteriors. In this paper, for any integer g ≥ 2, we exhibit a knot of genus g which bounds a minimal genus Seifert surface with hyperbolic exterior and a minimal genus free Seifert surface.

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