[Paper Review] Braids, knots and contact structures

Joan S. Birman|ArXiv.org|Mar 28, 2004

Geometric and Algebraic Topology12 references3 citations

TL;DR

This paper establishes the existence of distinct transversal knot types in R³ with the same topological knot type and identical Bennequin invariant, using a topological obstruction based on braiding and flype templates. By analyzing closed 3-braids via negative flype templates and applying Eliashberg's isotopy extension theorem, the authors prove that certain braids cannot be related by transverse isotopy despite sharing key invariants, thus demonstrating that the transverse Markov theorem does not hold in full generality.

ABSTRACT

These notes were prepared to supplement the talk that I gave on Feb 19, 2004, at the First East Asian School of Knots and Related Topics, Seoul, South Korea. In this article I review aspects of the interconnections between braids, knots and contact structures on Euclidean 3-space. I discuss my recent work with William Menasco (arXiv math.GT/0310279)} and (arXiv math.GT/0310280). In the latter we prove that there are distinct transversal knot types in contact 3-space having the same topological knot type and the same Bennequin invariant.

Motivation & Objective

• To resolve whether transverse isotopy is determined solely by topological type and Bennequin invariant.
• To investigate the limitations of the transverse Markov theorem in classifying transverse knots.
• To construct explicit examples of transverse knots that are topologically equivalent but not transversally isotopic.
• To establish a topological obstruction using braiding assignments and flype templates to rule out transverse isotopy.

Proposed method

• Analyzes closed 3-braids using block-strand diagrams and flype templates to study isotopy relations.
• Applies the negative flype template to show that certain braids cannot be connected by transverse isotopy.
• Uses braiding assignments to blocks in the template to define 2-component links with distinct β-invariants.
• Applies Eliashberg’s isotopy extension theorem to show that transverse isotopy must preserve β-invariants of components.
• Employs a contradiction argument: if a transverse isotopy existed, it would preserve β-invariants, but the flype changes them, so no such isotopy can exist.
• Relies on a specialized version of the Markov theorem for transverse knots, restricting to 3-braids and excluding destabilization or exchange moves.

Experimental results

Research questions

• RQ1 Can two transverse knots with the same topological type and Bennequin invariant fail to be transversally isotopic?
• RQ2 Is the transverse Markov theorem equivalent to the classical Markov theorem in braid theory?
• RQ3 What topological or dynamical obstructions prevent transverse isotopy between closed braids with identical invariants?
• RQ4 Do flype templates in 3-braids admit both positive and negative flypes under transverse isotopy?
• RQ5 Can β-invariants of components in a link be preserved under transverse isotopy, and how does this constrain isotopy existence?

Key findings

• The paper constructs explicit examples of transverse knots with the same topological type and Bennequin invariant (14 − 3 = 11) that are not transversally isotopic.
• The examples are closed 3-braids: σ₁⁵σ₂⁴σ₁⁶σ₂⁻¹ and σ₁⁵σ₂⁻¹σ₁⁶σ₂⁴, which are topologically equivalent via a negative flype.
• A transverse isotopy between these knots would require a template that supports both braids, but the only viable template is the negative flype template.
• The isotopy cannot exist because it would need to preserve β-invariants of components, but the flype changes β(L₁) and β(L₂) from (−1, −3) to (−3, −1), violating invariance.
• The contradiction arises from Eliashberg’s theorem, which ensures that any transverse isotopy extends to the ambient 3-sphere and thus preserves component invariants.
• The result implies that the transverse Markov theorem does not hold in full generality, as not all transverse representatives of a given topological type are related by positive stabilization/destabilization alone.