arXiv math/99110851999-11math.GT

The Culler-Shalen seminorms of the (-2,3,n) pretzel knot

Thomas W. Mattman

备注：: 34 pages, 11 figures; Results unchanged but major revision of logic and more detailed arguments

英文摘要：

We show that the SL(2,C)-character variety of the (-2,3,n) pretzel knot consists of two (respectively three) algebraic curves when 3 does not divide n (respectively 3 divides n) and give an explicit calculation of the Culler-Shalen seminorms of these curves. Using this calculation, we describe the fundamental polygon and Newton polygon for these knots and give a list of Dehn surgeries yielding a manifold with finite or cyclic fundamental group. This constitutes a new proof of property P for these knots.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

Modules, fields of definition, and the Culler-Shalen norm

Culler-Shalen theory uses the algebraic geometry of the SL(2,C)-character variety of a 3-manifold to construct essential surfaces in the manifold. There are module structures associated to the coordinate rings of the irreducible components of character varieties that are intimately related to essential surface construction. When these modules are finitely generated, we derive a formula for their rank that incorporates the irreducible component's field of definition and the Culler-Shalen norm.

2018-05-11math.GT

欧几里得长度与斜率的Culler-Shalen范数

Euclidean lengths and the Culler-Shalen norms of slopes

College of Humanities and Sciences, Nihon University（日本大学人文科学部）

研究例外Dehn填充中两个斜率函数（horotorus上的欧几里得长度和Culler-Shalen范数）的关系，建立两个不等式，并给出边界斜率直径的界。

2026-06-06math.GT

Culler-Shalen seminorms of fillings of the Whitehead link exterior

We determine the total Culler-Shalen seminorms for the 3-manifolds W_{p/q}:=W(p/q,-) obtained by Dehn filling with slope p/q on one boundary component of the Whitehead link exterior W when p is odd. As part of the proof, we use an explicit parametrization of the eigenvalue variety of W to find a one-variable polynomial whose roots characterize characters of p-reps of π_{1}(W_{p/q}), i.e. representations with values in SL_2(C) which are parabolic on the peripheral subgroup.

2020-03-17math.GT

The Culler-Shalen seminorms of the (-3,3,4) pretzel knot

We describe a method to compute the Culler-Shalen seminorms of a knot, using the (-3,3,4) pretzel knot as an illustrative example. We deduce that the SL2(C)-character variety of this knot consists of three algebraic curves and that it admits no non-trivial cyclic or finite surgeries. We also summarize similar results for other (-3,3,n) pretzel knots including the observation that the Seifert surgeries for these knots are precisely those integral slopes lying between two of the boundary slopes.

2001-02-06math.GT

任意特征下的Culler-Shalen理论导引

An invitation to Culler-Shalen theory in arbitrary characteristic

本文将Culler-Shalen理论推广到任意代数闭域的任意特征情形，奠定通用理论基础，并给出特征2下A-多项式变化、两类闭哈肯双曲3维流形本质曲面检测差异等实例。

2024-12-12math.GT

On Culler-Shalen seminorms and Dehn filling

Consider the exterior M of a hyperbolic knot lying in a closed, connected, orientable 3-manifold. Culler and Shalen defined norm on H_1(dM;R) using the SL(2,C) character variety of pi_1(M). The Culler-Shalen norm encodes many topological properties of M; in particular it provides information about Dehn fillings of M. Their construction may be applied to arbitrary curves in the SL_2(C)-character variety of a connected, compact, orientable, irreducible 3-manifold whose boundary is a torus, though in this generality one can only guarantee that it will define a seminorm. The first half of this pap…

1998-11-01math.GT

更多相关论文收起更多论文

Cyclic and finite surgeries on pretzel knots

We classify Dehn surgeries on (p,q,r) pretzel knots resulting in a manifold M(s) having cyclic fundamental group and analyze those leading to a finite fundamental group. The proof uses the theory of cyclic and finite surgeries developed by Culler, Shalen, Boyer, and Zhang. In particular, Culler-Shalen seminorms play a central role.

2001-12-13math.GT

Exceptional surgery and boundary slopes

Let X be a norm curve in the SL(2,C)-character variety of a knot exterior M. Let t = || b || / || a || be the ratio of the Culler-Shalen norms of two distinct non-zero classes a, b in H_1(\partial M, Z). We demonstrate that either X has exactly two associated strict boundary slopes \pm t, or else there are strict boundary slopes r_1 and r_2 with |r_1| > t and |r_2| < t. As a consequence, we show that there are strict boundary slopes near cyclic, finite, and Seifert slopes. We also prove that the diameter of the set of strict boundary slopes can be bounded below using the Culler-Shalen norm of …

2003-12-09math.GT

Finite surgeries on three-tangle pretzel knots

We classify Dehn surgeries on (p,q,r) pretzel knots that result in a manifold of finite fundamental group. The only hyperbolic pretzel knots that admit non-trivial finite surgeries are (-2,3,7) and (-2,3,9). Agol and Lackenby's 6-theorem reduces the argument to knots with small indices p,q,r. We treat these using the Culler-Shalen norm of the SL(2,C)-character variety. In particular, we introduce new techniques for demonstrating that boundary slopes are detected by the character variety.

2009-03-27math.GT

Hyperbolic 3-manifolds with k-free fundamental group

The results of Culler and Shalen for 2,3 or 4-free hyperbolic 3-manifolds are contingent on properties specific to and special about rank two subgroups of a free group. Here we determine what construction and algebraic information is required in order to make a geometric statement about $M$, a closed, orientable hyperbolic 3-manifold with $k$-free fundamental group, for any value of $k$ greater than four. Main results are both to show what the formulation of the general statement should be, for which Culler and Shalen's result is a special case, and that it is true modulo a group-theoretic con…

2012-05-02math.GT

SU(2)-Cyclic Surgeries on Knots

A surgery on a knot in 3-sphere is called SU(2)-cyclic if it gives a manifold whose fundamental group has no non-cyclic SU(2) representations. Using holonomy perturbations on the Chern-Simons functional, we prove that the distance of two SU(2)-cyclic surgery coefficients is bounded by the sum of the absolute values of their numerators. This is an analog of Culler-Gordon-Luecke-Shalen's cyclic surgery theorem.

2013-07-02math.GT

Finding a system of essential 2-suborbifolds

We make an analogy of Culler-Morgan-Shalen theory. Our main goal is to show that there exists a non-empty system of essential 2-suborbifolds respecting a given splitting of the orbifold fundamental group.

2012-06-09math.ATmath.GT

Cyclic surgery, degrees of maps of character curves, and volume rigidity for hyperbolic manifolds

This paper proves a theorem about Dehn surgery using a new theorem about PSL(2, C) character varieties. Confirming a conjecture of Boyer and Zhang, this paper shows that a small hyperbolic knot in a homotopy sphere having a non-trivial cyclic slope r has an incompressible surface with non-integer boundary slope strictly between r-1 and r+1. A corollary is that any small knot which has only integer boundary slopes has Property P. The proof uses connections between the topology of the complement of the knot, M, and the PSL(2, C) character variety of M that were discovered by Culler and Shalen. T…

1998-10-25math.GT