[Paper Review] The monodromies of homogeneous links

Mark C. Bell|arXiv (Cornell University)|Jun 30, 2012

Geometric and Algebraic Topology3 references3 citations

TL;DR

This paper proves that only finitely many homogeneous links exist with a Conway polynomial of any given degree, enabling the construction of an inhomogeneous fibred knot. It further provides an algorithm to compute the monodromy of a homogeneous link complement directly from a homogeneous braid word, offering a computational tool for studying fibred structures in link theory.

ABSTRACT

We show that there are only finitely many homogeneous links whose Conway polynomial has any given degree. Using this we give an example of an inhomogeneous, fibred knot. Secondly, we show how to compute the monodromy of a homogeneous link complement from a homogeneous braid word representative.

Motivation & Objective

• To establish a finiteness result for homogeneous links based on the degree of their Conway polynomial.
• To demonstrate the existence of an inhomogeneous fibred knot, challenging the assumption that fibred knots must be homogeneous.
• To develop a method for computing the monodromy of a homogeneous link complement using a homogeneous braid word representative.
• To provide a constructive algorithm that links braid word structure to monodromy data in homogeneous link theory.

Proposed method

• Leveraging properties of homogeneous braids and their closures to analyze the structure of the associated link complements.
• Using the Conway polynomial as a degree-based invariant to bound the number of homogeneous links for any fixed degree.
• Applying known results on fibred links and mapping torus structures to identify conditions under which a link is fibred.
• Deriving a systematic procedure to compute the monodromy from the braid word by analyzing the braid's homogeneous structure and its action on the fiber surface.
• Utilizing the fact that homogeneous braids yield fibred links, and exploiting the recursive structure of such braids to compute monodromy maps.
• Combining topological invariants with braid group techniques to verify the monodromy computation via the mapping torus construction.

Experimental results

Research questions

• RQ1 Are there only finitely many homogeneous links with a Conway polynomial of a given degree?
• RQ2 Can an inhomogeneous knot still be fibred, and if so, can such a knot be explicitly constructed?
• RQ3 Is there a direct algorithm to compute the monodromy of a homogeneous link from a homogeneous braid word?
• RQ4 How does the braid word structure of a homogeneous link relate to the monodromy of its complement?
• RQ5 What topological invariants can be used to distinguish homogeneous from inhomogeneous fibred links?

Key findings

• There are only finitely many homogeneous links whose Conway polynomial has any fixed degree, establishing a strong finiteness constraint on the class.
• An explicit example of an inhomogeneous fibred knot is constructed, demonstrating that fibredness does not imply homogeneity.
• The monodromy of a homogeneous link complement can be algorithmically computed from a homogeneous braid word representative.
• The method relies on the intrinsic structure of homogeneous braids, allowing the monodromy to be derived without prior knowledge of the fiber surface.
• The monodromy computation is consistent with the mapping torus model of fibred links, confirming topological coherence.
• The results provide a computational bridge between braid word representation and monodromy data in homogeneous link theory.