Minimal sequences of Reidemeister moves on diagrams of torus knots

Hayashi, Chuichiro; Hayashi, Miwa

doi:10.1090/s0002-9939-2010-10800-6

Cited by 6 publications

(9 citation statements)

References 10 publications

(11 reference statements)

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“…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%

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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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Section: New Resultsmentioning

confidence: 89%

“…Looser bounds are also known for the minimum number of Reidemeister moves needed to reduce a diagram of the unknot [32,42], to separate the components of a split link [36], or to move between two equivalent knot diagrams [19,35].…”

Section: Past Resultsmentioning

confidence: 99%

Untangling Planar Curves

Chang

Erickson

2017

Discrete Comput Geom

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“…Every simple closed curve has defect zero, and any homotopy move changes the defect of a curve by −2, 0, or 2; the various cases are illustrated in Figure 4.1. In Section 4, we compute the defect of the standard planar projection of any p × q torus knot where either p mod q = 1 or q mod p = 1, generalizing earlier results of Hayashi et al [41,43] and Even-Zohar et al [27]. In particular, we show that the standard projection of the p × (p + 1) torus knot, which has p 2 − 1 vertices, has defect 2 p+1 3 .…”

Section: Our Resultsmentioning

confidence: 54%

“…It follows that two doodle equivalent curves are connected by a sequence of only O(n) homotopy moves. 1 Looser bounds are also known for the minimum number of Reidemeister moves needed to reduce a diagram of the unknot [40,52], to separate the components of a split link [42], or to move between two equivalent knot diagrams [21,41].…”

Section: Homotopy Movesmentioning

confidence: 99%

Electrical Reduction, Homotopy Moves, and Defect

Chang¹,

Erickson²

2015

Preprint

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We prove the first nontrivial worst-case lower bounds for two closely related problems. First, Ω(n 3/2 ) degree-1 reductions, series-parallel reductions, and ∆Y transformations are required in the worst case to reduce an n-vertex plane graph to a single vertex or edge. The lower bound is achieved by any planar graph with treewidth Θ( n). Second, Ω(n 3/2 ) homotopy moves are required in the worst case to reduce a closed curve in the plane with n self-intersection points to a simple closed curve. For both problems, the best upper bound known is O(n 2 ), and the only lower bound previously known was the trivial Ω(n).The first lower bound follows from the second using medial graph techniques ultimately due to Steinitz, together with more recent arguments of Noble and Welsh [J. Graph Theory 2000]. The lower bound on homotopy moves follows from an observation by Haiyashi et al. [J. Knot Theory Ramif. 2012] that the standard projections of certain torus knots have large defect, a topological invariant of generic closed curves introduced by Aicardi and Arnold.

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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 sequences of Reidemeister moves on diagrams of torus knots

Cited by 6 publications

References 10 publications

Untangling Planar Curves

Untangling Planar Curves

Electrical Reduction, Homotopy Moves, and Defect

Unknotting number and number of Reidemeister moves needed for unlinking

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