ar X iv : 0 80 2 . 10 72 v 2 [ m at h . G T ] 1 4 M ay 2 00 8 A note on closed 3 - braids ∗
نویسندگان
چکیده
This is a review article about knots and links of braid index 3. Its goal is to gather together, in one place, some of the tools that are special to knots and links of braid index 3, in a form that could be useful for those who have a need to calculate, and need to know precisely all the exceptional cases. Knots and links which are closed 3-braids are a very special class. Like 2-bridge knots and links, they are simple enough to admit a complete classification. At the same time they are rich enough to serve as a source of examples on which, with luck, a researcher may be able to test various conjectures. As an example, non-invertibility is a property of knots which is fairly subtle, yet we know (see Theorem 2 below) precisely which links of braid index 3 are and are not invertible. Therefore 3-braids could be very useful for testing potential invariants that might detect non-invertibility. The purpose of this note is to gather together, in one place, some of the tools that are special to knots and links of braid index 3, in a form that could be useful for those who have a need to calculate, and need to know precisely all the exceptional cases. It includes the work in an earlier preprint [4] which was restricted to 3-braid representatives of transversal knots. The preprint [4] was written when [26] was in preparation, and the authors of [26] asked us about low crossing examples to test their then-new invariants of transversal knots. 1 Three basic theorems In this section we present the basic lemmas and theorems about links that are closed three-braids. Initially, we will use the classical presentation for the braid group B3: < σ1, σ2;σ1σ2σ1 = σ2σ1σ2 > (1) this article is an expanded version of the preprint “A note on transversal knots which are closed 3-braids”, arXiv math/0703669. partially supported by the US National Science Foundation Grant DMS 0405586. partially supported by the US National Science Foundation Grant DMS 0306062
منابع مشابه
ar X iv : 0 80 5 . 32 59 v 1 [ m at h . A G ] 2 1 M ay 2 00 8 SELF - DUAL PROJECTIVE TORIC VARIETIES
Let T be a torus over an algebraically closed field k of characteristic 0, and consider a projective T -module P(V ). We determine when a projective toric subvariety X ⊂ P(V ) is self-dual, in terms of the configuration of weights of V .
متن کاملar X iv : 0 80 5 . 24 44 v 1 [ m at h . A G ] 1 6 M ay 2 00 8 Symmetry and holomorphy of the second member of the second Painlevé hierarchy
We study symmetry and holomorphy of the second member of the second Painlevé hierarchy P (2) II .
متن کاملar X iv : 0 80 4 . 14 18 v 2 [ m at h . D G ] 2 7 M ay 2 00 8 SMOOTH YAMABE INVARIANT AND SURGERY
We prove a surgery formula for the smooth Yamabe invariant σ(M) of a compact manifold M . Assume that N is obtained from M by surgery of codimension at least 3. We prove the existence of a positive constant Λn, depending only on the dimension n of M , such that σ(N) ≥ min{σ(M),Λn}.
متن کاملar X iv : m at h / 98 05 07 8 v 1 [ m at h . G T ] 1 8 M ay 1 99 8 POSITIVE KNOTS , CLOSED BRAIDS AND THE JONES POLYNOMIAL
We prove that there are only finitely many positive knots with the same Alexander polynomial and the (properly normalized) Vassiliev invariants of degrees 2 and 3 of positive knots are not less than their genera. We prove, that the minimal degree of the Jones polynomial of a closed positive braid is at least a quarter of its crossing number and there are only finitely many positive knots with t...
متن کاملar X iv : 0 80 5 . 37 89 v 1 [ m at h . A P ] 2 4 M ay 2 00 8 The balance between diffusion and absorption in semilinear parabolic equations ∗
Let h : [0,∞) 7→ [0,∞) be continuous and nondecreasing, h(t) > 0 if t > 0, and m,q be positive real numbers. We investigate the behavior when k → ∞ of the fundamental solutions u = uk of ∂tu − ∆u m + h(t)u = 0 in Ω × (0, T ) satisfying uk(x, 0) = kδ0. The main question is wether the limit is still a solution of the above equation with an isolated singularity at (0, 0), or a solution of the asso...
متن کامل