Minimal Unknotting Sequences of Reidemeister Moves Containing Unmatched Rii Moves

Hayashi, Chuichiro; Hayashi, Miwa; Sawada, Minori; Yamada, Sayaka

doi:10.1142/s021821651250099x

Cited by 10 publications

(9 citation statements)

References 14 publications

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“…The lemma follows immediately by induction. Either of the previous lemmas imply the following lower bound, which is also implicit in the work of Hayashi et al [37]. Theorem 2.4.…”

Section: Proofmentioning

confidence: 73%

“…Each homotopy move changes the defect of a closed curve by at most 2. The lower bound therefore follows from constructions of Hayashi et al [35,37] and Even-Zohar et al [22] of closed curves with defect Ω(n 3/2 ). We simplify and generalize their results by computing the defect of the standard planar projection of any p ×q torus knot where either p mod q = 1 or q mod p = 1.…”

Section: New Resultsmentioning

confidence: 89%

“…We call these curves flat torus knots. Hayashi et al [37,Proposition 3.1] proved that for any integer q, the flat torus knot T (q + 1, q) has defect −2 q 3 . Even-Zohar et al [22] used a star-polygon representation of the curve T (p, 2p + 1) as the basis for a universal model of random knots; in our notation, they proved that defect(T (p, 2p + 1)) = 4 p+1 3 for any integer p. In this section we simplify and generalize both of these results to all flat torus knots T (p, q) where either q mod p = 1 or p mod q = 1.…”

Section: Flat Torus Knotsmentioning

confidence: 99%

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Untangling Planar Curves

Chang

Erickson

2017

Discrete Comput Geom

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Any generic closed curve in the plane can be transformed into a simple closed curve by a finite sequence of local transformations called homotopy moves. We prove that simplifying a planar closed curve with n self-crossings requires Θ(n 3/2 ) homotopy moves in the worst case. Our algorithm improves the best previous upper bound O(n 2 ), which is already implicit in the classical work of Steinitz; the matching lower bound follows from the construction of closed curves with large defect, a topological invariant of generic closed curves introduced by Aicardi and Arnold. Our lower bound also implies that Ω(n 3/2 ) facial electrical transformations are required to reduce any plane graph with treewidth Ω( n) to a single vertex, matching known upper bounds for rectangular and cylindrical grid graphs. More generally, we prove that transforming one immersion of k circles with at most n self-crossings into another requires Θ(n 3/2 + nk + k 2 ) homotopy moves in the worst case. Finally, we prove that transforming one noncontractible closed curve to another on any orientable surface requires Ω(n 2 ) homotopy moves in the worst case; this lower bound is tight if the curve is homotopic to a simple closed curve.

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“…The lemma follows immediately by induction. Either of the previous lemmas imply the following lower bound, which is also implicit in the work of Hayashi et al [37]. Theorem 2.4.…”

Section: Proofmentioning

confidence: 73%

Section: New Resultsmentioning

confidence: 89%

Section: Flat Torus Knotsmentioning

confidence: 99%

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Untangling Planar Curves

Chang

Erickson

2017

Discrete Comput Geom

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“…12c for n ¼ 4. Following Hayashi et al (2012), they show that the expected Casson invariant of Tðn þ 1; nÞ is HðÀn 3 Þ. It is conceivable that this latter model contains all knots as well.…”

Section: Crisscross Constructionsmentioning

confidence: 95%

Models of random knots

Even-Zohar

2017

J Appl. and Comput. Topology

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The study of knots and links from a probabilistic viewpoint provides insight into the behavior of ''typical'' knots, and opens avenues for new constructions of knots and other topological objects with interesting properties. The knotting of random curves arises also in applications to the natural sciences, such as in the context of the structure of polymers. We present here several known and new randomized models of knots and links. We review the main known results on the knot distribution in each model. We discuss the nature of these models and the properties of the knots they produce. Of particular interest to us are finite type invariants of random knots, and the recently studied Petaluma model. We report on rigorous results and numerical experiments concerning the asymptotic distribution of such knot invariants. Our approach raises questions of universality and classification of the various random knot models.

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“…There are several studies of lower bounds for the number of Reidemeister moves connecting two knot diagrams of the same knot. See [4], [2], [7], [8], [5], [6]. In particular, Hass and Nowik introduced in [7] a certain knot diagram invariant I lk by using the smoothing operation and the linking number.…”

Section: Introductionmentioning

confidence: 99%

Unknotting number and number of Reidemeister moves needed for unlinking

Hayashi

Nowik

2012

Topology and its Applications

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Using unknotting number, we introduce a link diagram invariant of Hass and Nowik type, which changes at most by 2 under a Reidemeister move. As an application, we show that a certain infinite sequence of diagrams of the trivial two-component link need quadratic number of Reidemeister moves for being unknotted with respect to the number of crossings. Assuming a certain conjecture on unknotting numbers of a certain series of composites of torus knots, we show that the above diagrams need quadratic number of Reidemeister moves for being splitted. Mathematics Subject Classification 2010: 57M25.

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Minimal Unknotting Sequences of Reidemeister Moves Containing Unmatched Rii Moves

Cited by 10 publications

References 14 publications

Untangling Planar Curves

Untangling Planar Curves

Models of random knots

Unknotting number and number of Reidemeister moves needed for unlinking

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