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.

Accidental surfaces in knot complements

2000 ◽  
Vol 09 (06) ◽  
pp. 725-733 ◽  
Author(s):  
Kazuhiro Ichihara ◽  
Makoto Ozawa
Keyword(s):  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Incompressible Surface ◽  
Incompressible Surfaces ◽  
Parallel Surface ◽  
Seifert Surfaces ◽  
Knot Complements ◽  
Necessary And Sufficient

It is well known that for many knot classes in the 3-sphere, every closed incompressible surface in their complements contains an essential loop which is isotopic into the boundary of the knot exterior. In this paper, we investigate closed incompressible surfaces in knot complements with this property. We show that if a closed, incompressible, non-boundary-parallel surface in a knot complement has such loops, then they determine the unique slope on the boundary of the knot exterior. Moreover, if the slope is non-meridional, then such loops are mutually isotopic in the surface. As an application, a necessary and sufficient condition for knots to bound totally knotted Seifert surfaces is given.

scholarly journals Normal surfaces in non-compact 3-manifolds

2005 ◽  
Vol 78 (3) ◽  
pp. 305-321 ◽  
Author(s):  
Ensil Kang
Keyword(s):  
Finite Number ◽  
Infinite Number ◽  
Boundary Component ◽  
Unique Feature ◽  
Admissible Solution ◽  
Compact Surface ◽  
Normal Surface ◽  
Ideal Triangulation ◽  
Normal Surfaces ◽  
Ideal Triangulations

AbstractWe extend the normal surface Q-theory to non-compact 3-manifolds with respect to ideal triangulations. An ideal triangulation of a 3-manifold often has a small number of tetrahedra resulting in a system of Q-matching equations with a small number of variables. A unique feature of our approach is that a compact surface F with boundary properly embedded in a non-compact 3-manifold M with an ideal triangulation with torus cusps can be represented by a normal surface in M as follows. A half-open annulus made up of an infinite number of triangular disks is attached to each boundary component of F. The resulting surface , when normalized, will contain only a finite number of Q-disks and thus correspond to an admissible solution to the system of Q-matching equations. The correspondence is bijective.

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 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 KNOTS WITH INFINITELY MANY INCOMPRESSIBLE SEIFERT SURFACES

2008 ◽  
Vol 17 (05) ◽  
pp. 537-551 ◽  
Author(s):  
ROBIN T. WILSON
Keyword(s):  
Infinite Number ◽  
Incompressible Surface ◽  
Seifert Surfaces

We show that a knot in S3 with an infinite number of incompressible Seifert surfaces contains a closed non-peripheral incompressible surface in its complement.

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.

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.

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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