scholarly journals A family of freely slice good boundary links

2019 ◽  
Vol 376 (3-4) ◽  
pp. 1009-1030
Author(s):  
Jae Choon Cha ◽  
Min Hoon Kim ◽  
Mark Powell
Keyword(s):  
Seifert Surface

Abstract We show that every good boundary link with a pair of derivative links on a Seifert surface satisfying a homotopically trivial plus assumption is freely slice. This subsumes all previously known methods for freely slicing good boundary links with two or more components, and provides new freely slice links.

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 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 Knot projections and Coxeter groups

1994 ◽  
Vol 56 (1) ◽  
pp. 1-16 ◽  
Author(s):  
A. M. Brunner ◽  
Y. W. Lee
Keyword(s):  
Coxeter Group ◽  
Coxeter Groups ◽  
Seifert Surface ◽  
Connected Sum

AbstractEvery knot admits a special projection with the property that under the projection discs in the canonical Seifert surface project disjointly. Under an isotopy, such a projection can be turned into a connected sum of what we call inseparable projections. The main result is that if there is no band in an inseparable projection with half-twisting number +1 or −1, then the projection is not a projection of the trivial knot. To prove this a non-cyclic Coxeter group is constructed as a quotient of the knot group. The construction is possibly of interest in itself. The techniques developed are applied to give a criterion to decide when an inseparable projection with 3 discs comes from the trivial knot.

scholarly journals Tight contact structures on Seifert surface complements

2020 ◽  
Vol 13 (2) ◽  
pp. 730-776 ◽  
Author(s):  
Tamás Kálmán ◽  
Daniel V. Mathews
Keyword(s):  
Contact Structures ◽  
Seifert Surface ◽  
Tight Contact

NORMAL SURFACES IN THE FIGURE-8 KNOT COMPLEMENT

2003 ◽  
Vol 12 (02) ◽  
pp. 269-279 ◽  
Author(s):  
ENSIL KANG
Keyword(s):  
Seifert Surface ◽  
Normal Surface ◽  
Surface Theory ◽  
Normal Surfaces ◽  
Geometric Sums

An approach to normal surface theory for non-compact 3-manifolds with respect to ideal pseudo-triangulations is described in [12]. The figure-8 knot complement with a given ideal pseudo-triangulation with two ideal tetrahedra is fully worked out. We construct all Q-fundamental surfaces and the remaining normal surfaces are obtained by linear geometric sums of these. We also construct all almost normal surfaces in the figure-8 knot complement. As a result we obtain an example which does not contain any normal or almost normal surface representing a minimal Seifert surface of the knot. This leads that the figure-8 knot is fibered.

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