Classifying Immersed Curves

Carter, J. Scott

doi:10.2307/2047890

Cited by 14 publications

(34 citation statements)

References 5 publications

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“…With the notation above w γ = w γ (k) for every barycentric subdivision, (Σ (k) , γ (k) ), of (Σ, γ). The proof of the following lemma can be found in [2] and [3]. We assume without loss of generality that signed Gauss paragraphs are of the from w = {w 1 , ..., w n }, with w i = a e i 1 i 1 ...a e i k(i) i k(i) , and every w i and w j are not disjoints.…”

Section: Signed Gauss Paragraphsmentioning

confidence: 99%

“…He noted that if we label the crossings points of an oriented normal curve γ on the plane, then the word formed with the letters that we met when following γ has the same characteristics as Gauss words . In 1991, J. S. Carter [2] introduced the concept of signed Gauss paragraphs to classify the stable geotopy class of immersed curves on orientable and compact surfaces. The construction of these signed paragraphs is similar to the one of Gauss words, but Carter considered, not only the number of components of the normal curve, but also the way in which the curve meets itself.…”

Section: Introductionmentioning

confidence: 99%

“…is an infinite collection of realization of w, such that the number of components of the boundary of any of these P L-surfaces, Ω (k) w , is equal to the number of Carter's circles of w as defined in [2]. Therefore, by gluing discs to the boundary of any Ω…”

Section: Introductionmentioning

confidence: 99%

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Characterization of signed Gauss paragraphs and skew-symmetric graded matrices

Rodríguez-Nieto

2018

J. Knot Theory Ramifications

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In this paper we use theory of embedded graphs on oriented and compact P Lsurfaces to construct minimal realizations of signed Gauss paragraphs. We prove that the genus of the ambient surface of these minimal realizations can be seen as a function of the maximum number of Carter's circles. For the case of signed Gauss words, we use a generating set of H 1 (S w , Z), given in [3], and the intersection pairing of immersed P Lnormal curves to present a short solution of the signed Gauss word problem. Moreover, we define the join operation on signed Gauss paragraphs to produce signed Gauss words such that both can be realized on the same minimal genus P L-surface.

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Section: Signed Gauss Paragraphsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Characterization of signed Gauss paragraphs and skew-symmetric graded matrices

Rodríguez-Nieto

2018

J. Knot Theory Ramifications

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“…Given a Gauss diagram D, we review the construction of the Carter surface Σ. Our discussion is based on [Car91], and the analogous construction for virtual knot diagrams can be found in [KK00]. Suppose that D has n chords on the core circle O, which is oriented counterclockwise.…”

Section: Gauss Diagramsmentioning

confidence: 99%

Concordance group of virtual knots

Bodén

Nagel²

2017

Proc. Amer. Math. Soc.

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We introduce Tristram-Levine signatures of virtual knots and use them to investigate virtual knot concordance. The signatures are defined first for almost classical knots, which are virtual knots admitting homologically trivial representations. The signatures and ω-signatures are shown to give bounds on the topological slice genus of almost classical knots, and they are applied to address a recent question of Dye, Kaestner, and Kauffman on the virtual slice genus of classical knots. A conjecture on the topological slice genus is formulated and confirmed for all classical knots with up to 11 crossings and for 2150 out of 2175 of the 12 crossing knots.The Seifert pairing is used to define directed Alexander polynomials, which we show satisfy a Fox-Milnor criterion when the almost classical knot is slice. We introduce virtual disk-band surfaces and use them to establish realization theorems for Seifert matrices of almost classical knots. As a consequence, we deduce that any integral polynomial ∆(t) satisfying ∆(1) = 1 occurs as the Alexander polynomial of an almost classical knot.In the last section, we use parity projection and Turaev's coverings of knots to extend the Tristram-Levine signatures to all virtual knots. A key step is a theorem saying that parity projection preserves concordance of virtual knots. This theorem implies that the signatures, ω-signatures, and Fox-Milnor criterion can be lifted to give slice obstructions for all virtual knots. There are 76 almost classical knots with up to six crossings, and we use our invariants to determine the slice status for all of them and the slice genus for all but four.

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“…the set of doodles on surfaces and the set of virtual doodles. For details and related topics on doodles, refer to [1,2,3,6,7,10,11].…”

Section: Introductionmentioning

confidence: 99%

Colorings and doubled colorings of virtual doodles

Bartholomew

Fenn

Kamada

et al. 2019

Topology and its Applications

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A virtual doodle is an equivalence class of virtual diagrams under an equivalence relation generated by flat version of classical Reidemesiter moves and virtual Reidemsiter moves such that Reidemeister moves of type 3 are forbidden. In this paper we discuss colorings of virtual diagrams using an algebra, called a doodle switch, and define an invariant of virtual doodles. Besides usual colorings of diagrams, we also introduce doubled colorings.

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Classifying Immersed Curves

Cited by 14 publications

References 5 publications

Characterization of signed Gauss paragraphs and skew-symmetric graded matrices

Characterization of signed Gauss paragraphs and skew-symmetric graded matrices

Concordance group of virtual knots

Colorings and doubled colorings of virtual doodles

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