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.

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 UNKNOTTING VIRTUAL KNOTS WITH GAUSS DIAGRAM FORBIDDEN MOVES

2001 ◽  
Vol 10 (06) ◽  
pp. 931-935 ◽  
Author(s):  
SAM NELSON
Keyword(s):  
Virtual Knot ◽  
Virtual Knots ◽  
Reidemeister Moves ◽  
Gauss Diagram

The forbidden moves can be combined with Gauss diagram Reidemeister moves to obtain move sequences with which we may change any Gauss diagram (and hence any virtual knot) into any other, including in particular the unknotted diagram.

scholarly journals PARITY AND EXOTIC COMBINATORIAL FORMULAE FOR FINITE-TYPE INVARIANTS OF VIRTUAL KNOTS

2012 ◽  
Vol 21 (13) ◽  
pp. 1240001 ◽  
Author(s):  
MICAH WHITNEY CHRISMAN ◽  
VASSILY OLEGOVICH MANTUROV
Keyword(s):  
Finite Type ◽  
Combinatorial Formula ◽  
Knot Invariants ◽  
Virtual Knot ◽  
Additional Information ◽  
Virtual Knots ◽  
Finite Type Invariants ◽  
Gauss Diagram

The present paper produces examples of Gauss diagram formulae for virtual knot invariants which have no analogue in the classical knot case. These combinatorial formulae contain additional information about how a subdiagram is embedded in a virtual knot diagram. The additional information comes from the second author's recently discovered notion of parity. For a parity of flat virtual knots, the new combinatorial formulae are Kauffman finite-type invariants. However, many of the combinatorial formulae possess exotic properties. It is shown that there exists an integer-valued virtualization invariant combinatorial formula of order n for every n (i.e. it is stable under the map which changes the direction of one arrow but preserves the sign). Hence, it is not of Goussarov–Polyak–Viro finite-type. Moreover, every homogeneous Polyak–Viro combinatorial formula admits a decomposition into an "even" part and an "odd" part. For the Gaussian parity, neither part of the formula is of GPV finite-type when it is non-constant on the set of classical knots. In addition, eleven new non-trivial combinatorial formulae of order 2 are presented which are not of GPV finite-type.

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 CROWELL'S DERIVED GROUP AND TWISTED POLYNOMIALS

2006 ◽  
Vol 15 (08) ◽  
pp. 1079-1094 ◽  
Author(s):  
DANIEL S. SILVER ◽  
SUSAN G. WILLIAMS
Keyword(s):  
Knot Theory ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Permutation Representation ◽  
Virtual Knot ◽  
Alexander Polynomials ◽  
Necessary And Sufficient ◽  
New Applications ◽  
Extended Group

The derived group of a permutation representation, introduced by Crowell, unites many notions of knot theory. We survey Crowell's construction, and offer new applications. The twisted Alexander group of a knot is defined. Using it, we obtain twisted Alexander polynomials. Also, we extend a well-known theorem of Neuwirth and Stallings giving necessary and sufficient conditions for a knot to be fibered. Virtual Alexander polynomials provide obstructions for a virtual knot that must vanish if the knot has a diagram with an Alexander numbering. The extended group of a virtual knot is defined, and using it a more sensitive obstruction is obtained.

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.

On Gauss diagrams of periodic virtual knots

2015 ◽  
Vol 24 (10) ◽  
pp. 1540008 ◽  
Author(s):  
Yongju Bae ◽  
In Sook Lee
Keyword(s):  
Virtual Knot ◽  
Virtual Knots ◽  
Gauss Diagram

In this paper, we study the Gauss diagrams for periodic virtual knots (Theorem 3.1) and show that the virtual knot corresponding to a periodic Gauss diagram is equivalent to the periodic virtual knot whose factor is the virtual knot corresponding to the factor Gauss diagram (Theorem 3.2). We give formulae for the writhe polynomial and the affine index polynomial of periodic virtual knots by using those of factor knots (Corollary 4.2, Corollary 4.6).

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.

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.

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