Abstract
This note concerns knots which are positive in the sense that they are closures of positive words in the classical braid generators. Such knots are known to be fibred, non-slice and non-amphicheiral. Recursive knot-theoretic methods of J.H. Conway yield criteria sufficiently restrictive to settle the question of positivity for all save four of the 249 knots on ten or fewer crossings and provide alternate proofs for classical and recent results on positive knots.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Alexander, J.W.: Topological invariants of knots and links. Trans. Amer. Math. Soc. 30 (1928) 275–306.
Birman, J.: Braids, Links and Mapping Class Groups. Princeton University Press, Princeton, N.J. (1974).
Birman, J. and Williams, R.: Knotted periodic orbits in dynamical systems I: Lorenz’s equation. Topology 22 (1983) 47–82.
Conway, J.H.: An enumeration of knots and links and some of their algebraic properties. Computational Problems in Abstract Algebra. Pergamon Press, New York, N.Y. (1970) 329–358.
Fox, R.H.: A quick trip through knot theory, Topology of 3-Manifolds, Prentice-Hall, Englewood Cliffs, N.J. (1962) 120–167.
Giller, C: A family of links and the Conway calculus. Trans. Amer. Math. Soc. 270 (1982) 75–109.
Kauffman, L.: The Conway Polynomial. Topology 20 (1980) 101–108.
Milnor, J.: Singular Points of Complex Hypersurfaces. Princeton University Press, Princeton, N.J. (1968).
Murasugi, K.: On Closed 3-Braids. Memoirs Amer. Math. Soc. No. 151 (1974).
Rolfsen, D.: Knots and Links. Publish or Perish Inc., Wilmington, Del. (1976).
Rudolph, L.: Non-trivial positive braids have positive signature. Topology 21 (1982) 325–327.
Stallings, J.: Constructions of fibred knots and links. Proc. Sympos. Pure Math. 32 (1978).
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1985 Springer-Verlag
About this paper
Cite this paper
Van Buskirk, J.M. (1985). Positive knots have positive Conway polynomials. In: Rolfsen, D. (eds) Knot Theory and Manifolds. Lecture Notes in Mathematics, vol 1144. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0075018
Download citation
DOI: https://doi.org/10.1007/BFb0075018
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-15680-2
Online ISBN: 978-3-540-39616-1
eBook Packages: Springer Book Archive