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The spiral index of knots

Published online by Cambridge University Press:  04 June 2010

COLIN ADAMS ,
RACHEL HUDSON ,
RALPH MORRISON ,
WILLIAM GEORGE ,
LAURA STARKSTON ,
SAMUEL TAYLOR  and
OLGA TURANOVA
COLIN ADAMS
Affiliation:
Department of Mathematics, Bronfman Science Center, Williams College, Williamstown, MA 01267, U.S.A. e-mail: colinC.Adams@williams.edu, 10rah@williams.edu, 10rem@williams.edu
RACHEL HUDSON
Affiliation:
Department of Mathematics, Bronfman Science Center, Williams College, Williamstown, MA 01267, U.S.A. e-mail: colinC.Adams@williams.edu, 10rah@williams.edu, 10rem@williams.edu
RALPH MORRISON
Affiliation:
Department of Mathematics, Bronfman Science Center, Williams College, Williamstown, MA 01267, U.S.A. e-mail: colinC.Adams@williams.edu, 10rah@williams.edu, 10rem@williams.edu
WILLIAM GEORGE
Affiliation:
Department of Mathematics, University of Toronto, Toronto, Ontario, Canada M5S 2E4. e-mail: wgeorge@math.toronto.edu
LAURA STARKSTON
Affiliation:
Department of Mathematics, Harvard University, 1 Oxford St., Cambridge, MA 02138, U.S.A. e-mail: lstarkst@fas.harvard.edu
SAMUEL TAYLOR
Affiliation:
Department of Mathematics, University of Texas, 1 University Station C1200, Austin, TX 78712, U.S.A. e-mail: taylor29@tenj.edu
OLGA TURANOVA
Affiliation:
Department of Mathematics, University of Chicago, 5734 S. University Ave., Chicago, IL 60637, U.S.A. e-mail: ot2130@barnard.edu
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Abstract

In this paper, we introduce two new invariants that are closely related to Milnor's curvature-torsion invariant. The first, a particularly natural invariant called the spiral index of a knot, captures the number of local maxima in a knot projection that is free of inflection points. This invariant is sandwiched between the bridge and braid index of a knot, and captures more subtle properties. The second invariant, the projective superbridge index, provides a method of counting the greatest number of local maxima that occur in a given projection. In addition to investigating the relationships among these invariants, we use them to classify all those knots for which Milnor's curvature-torsion invariant is 6π.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 149 , Issue 2 , September 2010 , pp. 297 - 315
Copyright
Copyright © Cambridge Philosophical Society 2010

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References

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