[Paper Review] Alexander polynomials and hyperbolic volume of arborescent links

A. Stoimenow|ArXiv.org|Dec 6, 2007

Geometric and Algebraic Topology39 references3 citations

TL;DR

This paper constructs hyperbolic arborescent links with prescribed monic Alexander polynomials, proving that any such polynomial can be realized by fibered hyperbolic links of any number of components (including infinitely many 4-component links). It establishes an upper bound on minimal hyperbolic volume for knots with a given Alexander polynomial and constructs knots of arbitrarily large volume with fixed free genus and Alexander polynomial, extending results on volume and polynomial invariants in low-dimensional topology.

ABSTRACT

We realize a given (monic) Alexander polynomial by a (fibered) hyperbolic arborescent knot and link of any number of components, and by infinitely many such links of at least 4 components. As a consequence, a Mahler measure minimizing polynomial, if it exists, is realized as the Alexander polynomial of a fibered hyperbolic link of at least 2 components. For given polynomial, we give also an upper bound on the minimal hyperbolic volume of knots/links, and contrarily, construct knots of arbitrarily large volume, which are arborescent, or have given free genus at least 2.

Motivation & Objective

• To realize any monic Alexander polynomial as the Alexander polynomial of a fibered hyperbolic arborescent link with any number of components.
• To establish an upper bound on the minimal hyperbolic volume of knots with a given Alexander polynomial, depending only on its degree.
• To construct knots and links of arbitrarily large hyperbolic volume while preserving a fixed Alexander polynomial and free genus at least 2.
• To extend infinite realizability of fibered links to arborescent links with four or more components, including canonical fiber surfaces.
• To confirm a conjecture by Silver and Williams that a Mahler measure minimizing polynomial, if it exists, is realized as the Alexander polynomial of a fibered hyperbolic 2-component link.

Proposed method

• Using Conway notation and tangle surgery techniques to systematically construct arborescent links with desired Alexander polynomials via Seifert matrices and skein relations.
• Applying Stallings twists and tangle surgeries to generate infinite families of links with fixed Alexander polynomial and increasing hyperbolic volume.
• Employing cut-and-paste arguments and results from sutured manifold theory (Gabai) to prove fibering and hyperbolicity of constructed links.
• Leveraging Oertel and Wu’s results on hyperbolicity of Montesinos links to verify hyperbolicity for multi-component arborescent links.
• Using canonical surfaces and fibered surface constructions to ensure minimal genus and fibered structure in the knots and links.
• Analyzing coefficient constraints in Alexander polynomials of pretzel knots to show limitations on realizability via symmetric polynomials and log-concavity.

Experimental results

Research questions

• RQ1 Can every monic Alexander polynomial be realized as the Alexander polynomial of a fibered hyperbolic arborescent link with any number of components?
• RQ2 Is there a universal upper bound on the minimal hyperbolic volume of knots with a given Alexander polynomial, depending only on its degree?
• RQ3 Can hyperbolic volume be made arbitrarily large while preserving a fixed Alexander polynomial and free genus at least 2?
• RQ4 Can infinite families of arborescent links with canonical fiber surfaces be constructed for links with four or more components?
• RQ5 Are there Alexander polynomials that cannot be realized by any generalized pretzel knot with an odd number of odd parameters?

Key findings

• Every monic Alexander polynomial is realized by a fibered hyperbolic arborescent link with any number of components, including infinitely many such 4-component links.
• An upper bound on the minimal hyperbolic volume of a knot with a given Alexander polynomial is established, depending only on the degree of the polynomial.
• Knots of arbitrarily large hyperbolic volume are constructed with fixed Alexander polynomial and free genus at least 2, using tangle surgeries and Stallings twists.
• The construction extends infinite realizability of fibered arborescent links to 4-component and higher links, even with canonical fiber surfaces.
• It is shown that certain coefficient patterns in Alexander polynomials (e.g., $0 < ho_4 < ho_2 < ho_6$) cannot be realized by generalized pretzel knots with odd parameters, due to log-concavity of elementary symmetric polynomials.
• The paper confirms a claim by Silver and Williams: if a Mahler measure minimizing polynomial exists, it is realized as the Alexander polynomial of a fibered hyperbolic 2-component link.