Knot projections with a single multi-crossing

Adams, Colin; Crawford, Thomas O.; DeMeo, Benjamin; Landry, Michael R.; Lin, A. T.; Montee, MurphyKate; Park, Seojung; Venkatesh, Saraswathi; Yhee, Farrah

doi:10.48550/arxiv.1208.5742

Cited by 1 publication

(2 citation statements)

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“…We adopt a standard sign convention for vertices originally proposed by Gauss [33]. 2 A vertex is positive if the first traversal through the vertex crosses the second traversal from right to left, and negative otherwise. Equivalently, a vertex x is positive if the winding number of a point moving along the curve increases the first time it reaches x after leaving the basepoint.…”

Section: Closed Curvesmentioning

confidence: 99%

“…Even-Zohar et al [27] studied the distribution of the Casson invariant for two models of random knots, the Petaluma model of Adams et al [1,2], which uses singular one-vertex diagrams consisting of 2p + 1 disjoint non-nested loops for some integer p, and the star model, which uses (a polygonal version of) the flat torus knot T (p, 2p + 1) for some integer p. Even-Zohar et al prove that the expected value of the Casson invariant is We hope to expand on these initial observations in a future paper.…”

Section: Implications For Random Knotsmentioning

confidence: 99%

See 1 more Smart Citation

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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Section: Closed Curvesmentioning

confidence: 99%