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Volume Conjecture for Knots pp 1–10Cite as

Preliminaries

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In this chapter we describe fundamental definitions and theorems. For details, see for example Burde et al. (Knots, extended ed., De Gruyter studies in mathematics, vol 5. De Gruyter, Berlin, 2014. MR 3156509), Lickorish (An introduction to knot theory. Graduate texts in mathematics, vol 175. Springer, New York, 1997. MR 98f:57015), and Rolfsen (Knots and links. Mathematics lecture series, vol 7. Publish or Perish, Inc., Houston, 1990; Corrected reprint of the 1976 original. MR 1277811 (95c:57018)).

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Notes

  1. 1.

    Precisely speaking, p counts how many times P intersects with the meridian (a circle on the torus that bounds a disk inside T) and q counts how many times P intersects with the longitude (a circle on the torus that is null-homologous outside T). The dotted circle in Fig. 1.11 shows the longitude of the trefoil. Observe that the linking number between the two circles is 0, since the dotted line goes under the solid line three times in the positive direction and three times in the negative direction.

    Fig. 1.11
    figure 11

    A longitude of the trefoil

    Full size image
  2. 2.

    A surface S in a three-manifold M is incompressible if the inclusion π 1(S) → π 1(M) is injective.

  3. 3.

    Two tori are parallel if they bound a thickened torus (S 1 × S 1 × [0, 1]) and a torus in a knot complement is called boundary-parallel if it is parallel to the boundary of a tubular neighborhood of the knot in S 3.

  4. 4.

    A closed three-manifold is called prime if it cannot be a connected-sum of two three-manifolds, none of which is the three-sphere.

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Authors and Affiliations

  1. Graduate School of Information Sciences, Tohoku University, Sendai, Japan

    Hitoshi Murakami

  2. Department of Mathematics and Information Science, Tokyo Metropolitan University, Tokyo, Japan

    Yoshiyuki Yokota

Authors
  1. Hitoshi Murakami
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  2. Yoshiyuki Yokota
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Murakami, H., Yokota, Y. (2018). Preliminaries. In: Volume Conjecture for Knots. SpringerBriefs in Mathematical Physics, vol 30. Springer, Singapore. https://doi.org/10.1007/978-981-13-1150-5_1

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