Knots with Heegaard genus $2$ complements are invertible
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- by Richard P. Osborne PDF
- Proc. Amer. Math. Soc. 81 (1981), 501-502 Request permission
Abstract:
Let $K$ be a polyhedral oriented knot in ${S^3}$ and $N(K)$ be a regular neighborhood of $K$. If ${S^3} \sim \mathring {N} (K)$ can be constructed by attaching a single $2$-handle to a genus two handlebody, then there is a homeomorphism of ${S^3}$ onto itself mapping $K$ onto itself and reversing the orientation of $K$.References
- R. P. Osborne and R. S. Stevens, Group presentations corresponding to spines of $3$-manifolds. II, Trans. Amer. Math. Soc. 234 (1977), no. 1, 213–243. MR 488062, DOI 10.1090/S0002-9947-1977-0488062-9
- Joan S. Birman and Hugh M. Hilden, Heegaard splittings of branched coverings of $S^{3}$, Trans. Amer. Math. Soc. 213 (1975), 315–352. MR 380765, DOI 10.1090/S0002-9947-1975-0380765-8
- Dale Rolfsen, Knots and links, Mathematics Lecture Series, No. 7, Publish or Perish, Inc., Berkeley, Calif., 1976. MR 0515288
- R. H. Fox, A quick trip through knot theory, Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961) Prentice-Hall, Englewood Cliffs, N.J., 1962, pp. 120–167. MR 0140099
- H. F. Trotter, Non-invertible knots exist, Topology 2 (1963), 275–280. MR 158395, DOI 10.1016/0040-9383(63)90011-9
Additional Information
- © Copyright 1981 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 81 (1981), 501-502
- MSC: Primary 57M25
- DOI: https://doi.org/10.1090/S0002-9939-1981-0597671-4
- MathSciNet review: 597671