A result on the Slope conjectures for 3-string Montesinos knots

Abstract

The (Strong) Slope Conjecture relates the degree of the colored Jones polynomial of a knot to certain essential surfaces in the knot complement. We verify the Slope Conjecture and the Strong Slope Conjecture for 3-string Montesinos knots satisfying certain conditions. 2010 Mathematics Subject Classification. 57N10, 57M25 ., and [s 0 , · · · , s p ] and [t 0 , · · · , t q ] are defined similarly. Note that our conventions for Montesinos knots coincide with those of [22]. This article is a succeeding work of [4] and [23] (however, the results of this article is not strictly the generalization of that of [4] or [23] because we can not loosen the restriction m, p, q ≥ 1 in Lemma 3.4), and the goal of these articles is to provide more data and evidences to the (Strong) Slope Conjecture for Montesinos knots. The reason to choose the family of Montesinos knots is that as a generalization of 2-bridge knots, it is large and representative, and meanwhile well parameterized. Moreover, C. R. S. Lee and R. van der Veen's method [4] to deal with the colored Jones polynomial and its degree and Hatcher and Oertel's algorithm [15] to determine the incompressible surfaces of Montesinos knots pave the way for the proof. The strategy of the proof is straight-forward: we first find out the maximal degree of the colored Jones polynomial and then choose the essential surface which matches the degree by the boundary slope and the Euler characteristic provided by the Hatcher-Oertel algorithm.As we will see, for 3-string Montesinos knots M([r 0 , · · · , r m ], [s 0 , · · · , s p ], [t 0 , · · · , t q ]), the increasing of m, p, q does not cause much complexity, and like the cases in [4] and [23], r 0 , s 0 and s 0 , particularly the discriminant ∆ (see Theorem 2.4 and its proof in Section 3) still dominate the maximal degree of the colored Jones polynomial (Theorem 2.4) as well as the selection of the essential surface (Theorem 2.5). More specifically, when ∆ < 0, the degree of colored Jones polynomial is matched by a typical type I essential surface; when ∆ ≥ 0, it is matched by a type II essential surface, but this type II surface generally (when at least one of m,p and q is greater than 1) does not correspond to a Seifert surface while it does in [4] and [23]. The Slope ConjecturesLet K denote a knot in S 3 and N(K) denote its tubular neighbourhood. A surface S properly embedded in the knot exterior E(K) = S 3 − N(K) is

Cancellations in the Degree of the Colored Jones Polynomial

“…In order for a choice of a skein element in the Fig. 15 The middle pair of projectors are shown with dashed boxes Fig. 16 All the possibilities for a skein element in the expansion of the bottom projector after choosing a skein element in the expansion for the top projector.…”

“…For non-adequate knots, it is natural to ask the extent to which similar results hold. Results extending relationships observed for adequate knots exist [2,8,9,[13][14][15]18]. However, it is difficult to study the colored Jones polynomial in complete generality, since the state sum which may be used to define the polynomial often has cancellations that are difficult to control.…”