Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-28T14:09:01.987Z Has data issue: false hasContentIssue false
  • English
  • Français

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: [email protected], [email protected], [email protected]
RACHEL HUDSON
Affiliation:
Department of Mathematics, Bronfman Science Center, Williams College, Williamstown, MA 01267, U.S.A. e-mail: [email protected], [email protected], [email protected]
RALPH MORRISON
Affiliation:
Department of Mathematics, Bronfman Science Center, Williams College, Williamstown, MA 01267, U.S.A. e-mail: [email protected], [email protected], [email protected]
WILLIAM GEORGE
Affiliation:
Department of Mathematics, University of Toronto, Toronto, Ontario, Canada M5S 2E4. e-mail: [email protected]
LAURA STARKSTON
Affiliation:
Department of Mathematics, Harvard University, 1 Oxford St., Cambridge, MA 02138, U.S.A. e-mail: [email protected]
SAMUEL TAYLOR
Affiliation:
Department of Mathematics, University of Texas, 1 University Station C1200, Austin, TX 78712, U.S.A. e-mail: [email protected]
OLGA TURANOVA
Affiliation:
Department of Mathematics, University of Chicago, 5734 S. University Ave., Chicago, IL 60637, U.S.A. e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Alexander, J. W.A lemma on systems of knotted curves. Proc. Nat. Acad. Sci. USA 9 (3) (1923), 93–95.Google Scholar
[2]Brennan, B., Adams, C. and Greilsheimer, D.Stick numbers and compositions of knots and links. J. Knot Theory Ramifications 6 (2) (1997), 149–161.Google Scholar
[3]Franks, J. and Williams, R. F.Braids and the Jones polynomial. Trans. Amer. Math. Soc. 303 (1) (1987), 97–108.CrossRefGoogle Scholar
[4]Honma, N. and Saeki, O.On Milnor's curvature-torsion invariant for knots and links. Kobe J. Math 11 (2) (1994), 225–239.Google Scholar
[5]Kuiper, N.A new knot invariant. Math. Ann. 278 (1987), 193–209.CrossRefGoogle Scholar
[6]Menasco, W.Closed incompressible surfaces in alternating knots and link complements. Topology 23 (1) (1984), 37–44.CrossRefGoogle Scholar
[7]Menasco, W. and Thistlethwaite, M.The classification of alternating links. Ann. Math. 138 (1) (1993), 113–171.CrossRefGoogle Scholar
[8]Milnor, J.On total curvatures of closed space curves. Math. Scand. 1 (1953), 289–296.Google Scholar
[9]Morton, H. R.Seifert circles and knot polynomials. Math. Proc. Camb. Phil. Soc. 99 (1) (1986), 107–109.Google Scholar
[10]Yamada, S.The minimal number of seifert circles equals the braid index of a link. Inven. Math. 89 (2) (1987), 347–356.Google Scholar