An algorithm for checking property P for knots with complements of Heegaard genus

Author:
R. P. Osborne

Journal:
Proc. Amer. Math. Soc. **88** (1983), 357-362

MSC:
Primary 57M25

DOI:
https://doi.org/10.1090/S0002-9939-1983-0695275-8

MathSciNet review:
695275

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: One of the most fundamental questions about knots is: If we know the topological type of the complement of a knot, is the knot determined? In this paper we give an algorithm for deciding if certain knots called tunnel number one knots are determined by their complements. This algorithm turns out to be practical and efficient in that it can be used on knots with ten crossings without the aid of a computer and one can expect to be able to handle knots with, say, twenty crossings with the aid of a desk-top computer.

**[B&M]**R. H. Bing and J. Martin,*Cubes with knotted holes*, Trans. Amer. Math. Soc.**155**(1971), 217-231. MR**0278287 (43:4018a)****[Fox]**R. H. Fox,*A quick trip through knot theory*, Topology of -Manifolds and Related Topics (M. K. Fort, Ed.), Prentice-Hall, Englewood Cliffs, N. J., 1962. MR**0140099 (25:3522)****[H.O.T.]**T. Homma, M. Ochiai and M. Takahasi,*An algorithm for recognizing**in**-manifolds with Heegaard splittings of genus two*, Osaka J. Math.**17**(1980), 625-648. MR**591141 (82i:57013)****[Ka]**A. Kawauchi,*The invertibility problem for amphieheiral excellent knots*, Proc. Japan Acad.**10**(1979), 399-402. MR**559040 (81b:57003)****[M.K.S.]**W. Magnus, A. Karrass and D. Solitar,*Combinatorial group theory*, Interscience, New York, 1966.**[Os]**R. P. Osborne,*Knots with Heegaard genus 2 complements are invertible*, Proc. Amer. Math. Soc.**81**(1981), 501-502. MR**597671 (82c:57004)****[O&SI]**R. P. Osborne and R. S. Stevens,*Group presentations corresponding to spines of**-manifolds*. I, Amer. J. Math.**96**(1974), 454-471. MR**0356058 (50:8529)****[O&SII]**-,*Group presentations corresponding to spines of**-manifolds*. II, Trans. Amer. Math. Soc.**234**(1977), 213-243. MR**0488062 (58:7634a)****[Rolf]**D. Rolfsen,*Knots and links*, Publish or Perish, Berkeley, Calif., 1976. MR**0515288 (58:24236)****[Sim]**J. Simon,*Some classes of knots with property*, Topology of Manifolds (Cantrell and Edwards, Eds.), Markham, Chicago, Ill., 1970. MR**0278288 (43:4018b)****[Tak]**Moto-o-Takahashi,*Two bridge knots have property*, Mem. Amer. Math. Soc. No. 239 (1981). MR**597092 (82f:57010)****[Thurs]**W. Thurston,*The Smith conjecture*, notes.**[Wh]**J. H. C. Whitehead,*On certain sets of elements in a free group*, Proc. London Math. Soc.**41**(1936), 48-56.**[Will]**Mark Willis, Masters Paper, Colorado State Univ., 1982.**[Zie]**H. Zieschang,*On simple systems of paths on complete pretzels*, Trans. Amer. Math. Soc.**92**(1970), 127-137.

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
57M25

Retrieve articles in all journals with MSC: 57M25

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1983-0695275-8

Article copyright:
© Copyright 1983
American Mathematical Society