Thin presentation of knots and lens spaces

Abstract

This paper concerns thin presentations of knots K in closed 3-manifolds M 3 which produce S 3 by Dehn surgery, for some slope γ . If M does not have a lens space as a connected summand, we first prove that all such thin presentations, with respect to any spine of M have only local maxima. If M is a lens space and K has an essential thin presentation with respect to a given standard spine (of lens space M ) with only local maxima, then we show that K is a 0-bridge or 1-bridge braid in M ; furthermore, we prove the minimal intersection between K and such spines to be at least three, and finally, if the core of the surgery K γ yields S 3 by r -Dehn surgery, then we prove the following inequality: |r| ≤ 2g, where g is the genus of K γ .

Spinal Knots in Lens Spaces

“…In [9], Goda and Teragaito show that |r| ≤ 12g − 7, if K is hyperbolic, and conjecture that | r |≤ 4g − 1. This inequality has recently been improved by Ichihara [17] : |r| ≤ 3 · 2 7 4 g. In [4], the authors prove that if K is a non-spinal knot in L then |r| ≤ 2g. We consider a non-spinal knot K which is not in a 3-ball of L disjoint from the spine.…”

“…Recall the basic definitions discussed in [4]. A spine Σ of a lens space L = L(p, q) is standard if it corresponds to the usual, refering to Rolfsen [23, p. 236].…”

“…In [4], a spinal knot is said standardly spinal. The authors define a thin presentation of any knot with respect to Σ.…”

“…The authors define a thin presentation of any knot with respect to Σ. A local maximum in this thin presentation is defined to be inessential if one can isotope it to Σ; otherwise, it is essential (see [4,Sec. 4] for more details).…”

“…If the thin presentation of K begins by inessential local maxima, we can isotope them inside N (Σ). Since K is not a spinal knot, it is not a 0 or 1-bridge braid in L. Therefore, by [4, Proposition 1.3] s ≥ 3 and by [4,Theorem 4.1] its thin presentation begins by a local minimum (after the possibly inessential local maxima isotoped inside N (Σ)).…”