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 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.

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.

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.

Sliceness of alternating pretzel knots and links

2020 ◽  
Vol 282 ◽  
pp. 107317
Author(s):  
Kengo Kishimoto ◽  
Tetsuo Shibuya ◽  
Tatsuya Tsukamoto
Keyword(s):  
Pretzel Knots ◽  
Knots And Links

SURGERY AND FOLIATIONS OF KNOT COMPLEMENTS

1993 ◽  
Vol 02 (04) ◽  
pp. 369-397 ◽  
Author(s):  
JOHN CANTWELL ◽  
LAWRENCE CONLON
Keyword(s):  
Unit Ball ◽  
Lattice Points ◽  
Compact Leaf ◽  
Minimal Genus ◽  
Relative Homology ◽  
Topological Description ◽  
Seifert Surfaces ◽  
Knot Complements ◽  
Interesting Class

An interesting class of knots have complement with a remarkably simple topological description. This class includes all the arborescent knots with only even weights hence, in particular, the two bridge knots and many knots of ten or fewer crossings. For these knots, there are choices of minimal genus Seifert surfaces S such that all taut, depth one foliations of the knot complement, having S as sole compact leaf, can be classified up to isotopy. These foliations correspond exactly to the lattice points over the open faces of the unit ball in a Thurston-like norm on the relative homology of the complement of S.

FIBERED TORTI-RATIONAL KNOTS

2010 ◽  
Vol 19 (10) ◽  
pp. 1291-1353 ◽  
Author(s):  
MIKAMI HIRASAWA ◽  
KUNIO MURASUGI
Keyword(s):  
Arbitrary Number ◽  
Continued Fraction ◽  
Alexander Polynomials ◽  
Minimal Genus ◽  
Seifert Surfaces ◽  
Tunnel Number ◽  
Rational Knots ◽  
Geometric Techniques ◽  
Continued Fraction Expansions ◽  
Satellite Knots

A torti-rational knot, denoted by K(2α, β|r), is a knot obtained from the 2-bridge link B(2α, β) by applying Dehn twists an arbitrary number of times, r, along one component of B(2α, β). We determine the genus of K(2α, β|r) and solve a question of when K(2α, β|r) is fibered. In most cases, the Alexander polynomials determine the genus and fiberedness of these knots. We develop both algebraic and geometric techniques to describe the genus and fiberedness by means of continued fraction expansions of β/2α. Then, we explicitly construct minimal genus Seifert surfaces. As an application, we solve the same question for the satellite knots of tunnel number one.

scholarly journals Colored HOMFLY polynomials for the pretzel knots and links

2015 ◽  
Vol 2015 (7) ◽  
Author(s):  
A. Mironov ◽  
A. Morozov ◽  
A. Sleptsov
Keyword(s):  
Pretzel Knots ◽  
Homfly Polynomials ◽  
Knots And Links
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