Embedding knots and links in an open book I: Basic properties

A flat plumbing basket is a Seifert surface consisting of a disk and bands contained in distinct pages of the disk open book decomposition of the 3-sphere. In this paper, we examine close connections between flat plumbing baskets and the contact structure supported by the open book. As an application we give lower bounds for the flat plumbing basket numbers and determine the flat plumbing basket numbers for various knots and links, including the torus links.

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Embedding knots and links in an open book II. Bounds on arc index

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100074181 ◽

1996 ◽

Vol 119 (2) ◽

pp. 309-319 ◽

Cited By ~ 19

Author(s):

Peter R. Cromwell ◽

Ian J. Nutt

Keyword(s):

Finite Number ◽

Polar Coordinates ◽

Open Book ◽

Knot Invariant ◽

Braid Index ◽

Satellite Links ◽

Minimum Number ◽

Arc Index ◽

Knots And Links ◽

Simple Arc

There is an open-book decomposition of the 3-sphere which has open discs as pages and an unknotted circle as the binding. We can think of the 3-sphere as ℝ3 ∪ {∞} and of the circle as the z–axis ∪ {∞}. The pages are then half-planes Hθ at angle θ when the x–y plane has polar coordinates. In their investigation of the braid index of satellite links, Birman and Menasco [B–M] embed the companion knot in finitely many such half-planes so that the knot meets each half-plane in a single simple arc, and therefore meets the axis in a finite number of points. At the end of their paper they mention that the minimum number of planes required to present a given knot in this manner is a knot invariant and that it seems to have escaped attention. (Jósef Przytycki has since pointed out to us that the phenomenon is evident in a one-hundred-year-old paper by H. Brunn[Br].) We call this invariant the arc index of a link and denote it by α(L).

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Knots and Links

10.1017/cbo9780511809767 ◽

2004 ◽

Cited By ~ 99

Author(s):

Peter R. Cromwell

Keyword(s):

Knots And Links

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Long-term retention is greater following closed-book tests than open-book tests

PsycEXTRA Dataset ◽

10.1037/e527352012-770 ◽

2006 ◽

Author(s):

Pooja K. Agarwal ◽

Jeffrey D. Karpicke ◽

Sean H. Kang ◽

Henry L. Roediger ◽

Kathleen B. McDermott

Keyword(s):

Open Book ◽

Term Retention ◽

Long Term Retention

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On the Essence and Tipology of International Public Goods

Voprosy Ekonomiki ◽

10.32609/0042-8736-2005-3-131-141 ◽

2005 ◽

pp. 131-141

Author(s):

V. Mortikov

Keyword(s):

Economic Policy ◽

Social Construction ◽

Public Goods ◽

Public Good ◽

Global Public Goods ◽

Club Goods ◽

Basic Properties ◽

International Public Goods ◽

Regional Public Goods ◽

Joint Products

The basic properties of international public goods are analyzed in the paper. Special attention is paid to the typology of international public goods: pure and impure, excludable and nonexcludable, club goods, regional public goods, joint products. The author argues that social construction of international public good depends on many factors, for example, government economic policy. Aggregation technologies in the supply of global public goods are examined.

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