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 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.

scholarly journals R-CHARGE KILLS MONOPOLES

1998 ◽  
Vol 13 (36) ◽  
pp. 2955-2964 ◽  
Author(s):  
BORUT BAJC ◽  
ANTONIO RIOTTO ◽  
GORAN SENJANOVIĆ
Keyword(s):  
High Temperature ◽  
Charge Density ◽  
Gauge Group ◽  
Early Universe ◽  
Gauge Symmetries ◽  
Cosmological Problem ◽  
Large Charge

Large charge density, unlike high temperature, may lead to nonrestoration of global and gauge symmetries. Supersymmetric GUTs with the appealing scenario of unification scale being generated dynamically naturally contain global continuous R-symmetries. We point out that the presence of a large R-charge in the early universe can lead to GUT symmetry nonrestoration. This provides a simple way out of the monopole problem and, at the same time, solves the well-known cosmological problem of the theory being caught permanently in the wrong vacuum with the GUT gauge group unbroken.

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 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 Right–right–left extension of the Standard Model

2016 ◽  
Vol 31 (19) ◽  
pp. 1650117 ◽  
Author(s):  
Gauhar Abbas
Keyword(s):  
Standard Model ◽  
Gauge Group ◽  
Vacuum Expectation ◽  
Expectation Values ◽  
Unique Pattern ◽  
Gauge Sector ◽  
Gauge Symmetries ◽  
The Standard Model ◽  
Higgs Fields ◽  
Scalar Sector

A right–right–left extension of the Standard Model is proposed. In this model, SM gauge group [Formula: see text] is extended to [Formula: see text]. The gauge symmetries [Formula: see text], [Formula: see text] are the mirror counterparts of the [Formula: see text] and [Formula: see text], respectively. Parity is spontaneously broken when the scalar Higgs fields acquire vacuum expectation values (VEVs) in a certain pattern. Parity is restored at the scale of [Formula: see text]. The gauge sector has a unique pattern. The scalar sector of the model is optimum, elegant and unique.

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.

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 The Vafa–Witten theory for gauge group $SU(N)$

1999 ◽  
Vol 3 (5) ◽  
pp. 1201-1225 ◽  
Author(s):  
J. M. F. Labastida ◽  
Carlos Lozano
Keyword(s):  
Gauge Group ◽  
Witten Theory

scholarly journals On the gauge symmetries of the spinning particle

2014 ◽  
Vol 92 (5) ◽  
pp. 411-414
Author(s):  
N. Kiriushcheva ◽  
S.V. Kuzmin ◽  
D.G.C. McKeon
Keyword(s):  
Simple Model ◽  
Gauge Group ◽  
Gauge Transformation ◽  
Group Structure ◽  
Simple Algebra ◽  
Direct Examination ◽  
Spinning Particle ◽  
Gauge Transformations ◽  
Gauge Symmetries ◽  
New Variables

We reconsider the gauge symmetries of the spinning particle by a direct examination of the Lagrangian using a systematic procedure based on the Noether identities. It proves possible to find a set of local bosonic and fermionic gauge transformations that have a simple gauge group structure, which is a true Lie algebra, both for the massless and massive case. This new fermionic gauge transformation of the “position” and “spin” variables in the action decouples from that of the “einbein” and “gravitino”. It is also possible to redefine the fields so that this simple algebra of commutators of the gauge transformations can be derived directly starting from the Lagrangian written in these new variables. We discuss a possible extension of our analysis of this simple model to more complicated cases.

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