A new formula for winding number

McIntyre, Margaret; Cairns, Grant

doi:10.1007/bf01264913

Cited by 25 publications

(21 citation statements)

References 9 publications

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“…After performing one circuit of β the tangent vector will have turned anticlockwise through an integer multiple of 2π; we call this the rotation index i of β (at cusps the tangent vector rotates by +π, see DoCarmo [9]). Note: we use the term 'rotation index' in keeping with [9], [18], [19], however some authors use 'index' [1], 'rotation number' [31], [11], 'winding number' [20], [26] or 'turning number' [2]. Lemma 1.…”

Section: Rotation Indexmentioning

confidence: 99%

The conjugate locus on convex surfaces

Waters

2018

Geom Dedicata

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The conjugate locus of a point on a surface is the envelope of geodesics emanating radially from that point. In this paper we show that the conjugate locus of generic points on convex surfaces satisfy a simple relationship between the rotation index and the number of cusps. As a consequence we prove the 'vierspitzensatz': the conjugate locus of a generic point on a convex surface must have at least four cusps. Along the way we prove certain results about evolutes in the plane and we extend the discussion to the existence of 'smooth loops' and geodesic curvature.geodesic and conjugate locus and Jacobi field and rotation index and evolute and convex arXiv:1806.00278v1 [math.DG]

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Section: Rotation Indexmentioning

confidence: 99%

The conjugate locus on convex surfaces

Waters

2018

Geom Dedicata

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“…A formula between Whitney degree [27] and the Alexander numbering was given in [22], which was based on the integral calculus with respect to the Euler characteristic [26]. The Alexander numbering for surfaces in 3-space was used in [11] and [23].…”

Section: Historical Remarksmentioning

confidence: 99%

Alexander numbering of knotted surface diagrams

Carter

Kamada

Saito

2000

Proc. Amer. Math. Soc.

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Abstract. A generic projection of a knotted oriented surface in 4-space divides 3-space into regions. The number of times (counted with sign) that a path from infinity to a given region intersects the projected surface is called the Alexander numbering of the region. The Alexander numbering is extended to branch and triple points of the projections. A formula that relates these indices is presented.

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“…Such a numbering is called an Alexander numbering ([1], [6]). Let Po E S 2 be in a region numbered 0, and ro(70) be the rotation number of To as a planar curve in R 2 ~ ,_q2k{Po }.…”

Section: Examplesmentioning

confidence: 99%

“…(We remark that 7 is null-homologous if and only if mt = m2 (cf. [1], [6]). ) Then we have PROPOSITION 6.4.…”

Section: Examplesmentioning

confidence: 99%

“…Recently, McIntyre and Cairns [6] have also proved a formula for the rotation number. Ours has an advantage in that it gives a way of computing the rotation number from (signed) Gauss word of a curve (for definition see [3], [1] Though the above theorems deal only with homologically nontrivial curves, they also give a method of computing the rotation number of null-homologous curves.…”

Section: Introductionmentioning

confidence: 99%

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