[Paper Review] Braid ordering and knot genus

Tetsuya Ito|ArXiv.org|May 14, 2008

Geometric and Algebraic Topology4 references3 citations

TL;DR

This paper establishes a new lower bound for the genus of a knot using the Dehornoy floor, a measure of braid complexity derived from the Dehornoy ordering. By analyzing braid foliations on Seifert surfaces and applying properties of the Dehornoy floor, the authors prove that the Dehornoy floor of a braid is strictly bounded by a function of the knot's Euler characteristic, yielding a topological lower bound on knot genus.

ABSTRACT

The genus of knots is a one of the fundamental invariant and can be seen as a complexity of knots. In this paper, we give a lower bound of genus using Dehornoy floor, which is a measure of complexity of braids in terms of braid ordering.

Motivation & Objective

• To relate the algebraic complexity of braids, measured by the Dehornoy floor, to the topological complexity of knots, measured by genus.
• To establish a quantitative link between the Dehornoy ordering—a left-invariant total order on braid groups—and classical knot invariants.
• To provide a new lower bound for the genus of a knot using the Dehornoy floor of its braid representative.
• To extend the applicability of the Dehornoy floor as a complexity measure by connecting it to geometric topology via Seifert surfaces.

Proposed method

• Define the Dehornoy floor $[\beta]_D$ as the minimal non-negative integer $m$ such that $\Delta^{-2m-2} <_D \beta <_D \Delta^{2m+2}$, using the Dehornoy ordering on braid groups.
• Apply Birman-Menasco’s braid foliation theory to isotopically simplify a Seifert surface $F$ of a closed braid $\widehat{\beta}$ to satisfy braid-foliation conditions.
• Analyze singularities (aa, bb, ab) in the foliation of $F$ to estimate the number of $\sigma_1^\pm$ generators in a word representative of the braid $\beta$.
• Use the conjugacy invariance and subadditivity properties of the Dehornoy floor (from Proposition 1) to bound $[\beta]_D$ in terms of the number of singularities and band generators.
• Derive the inequality $[\beta]_D < \frac{3}{2} - \frac{2\chi(\widehat{\beta})}{n+2}$ using the Euler characteristic formula and vertex type classification in the foliation.
• Generalize the result to other Thurston-type orderings by noting that analogous properties of the Dehornoy floor hold for the Thurston floor.

Experimental results

Research questions

• RQ1 Can the Dehornoy floor of a braid be used to bound the genus of its closure?
• RQ2 How does the algebraic complexity of a braid in the Dehornoy ordering relate to the topological complexity of the corresponding knot?
• RQ3 What is the minimal number of $\sigma_1^\pm$ generators required in a word representative of a braid, given the foliation structure of a maximal Euler characteristic Seifert surface?
• RQ4 Does the inequality $[\beta]_D < \frac{3}{2} - \frac{2\chi(\widehat{\beta})}{n+2}$ hold for all closed braids?
• RQ5 Can the result be extended beyond the Dehornoy ordering to other left-invariant orderings of braid groups?

Key findings

• The Dehornoy floor $[\beta]_D$ of a braid $\beta$ is strictly bounded above by $\frac{3}{2} - \frac{2\chi(\widehat{\beta})}{n+2}$, where $\chi(\widehat{\beta})$ is the maximal Euler characteristic of an orientable spanning surface of the closure $\widehat{\beta}$.
• For an oriented knot $K$ represented as the closure of an $n$-braid $\beta$, the inequality $[\beta]_D < \frac{4g(K)}{n+2} - \frac{2}{n+2} + \frac{3}{2} \leq g(K) + 1$ holds, providing a new lower bound on the knot genus.
• The bound is sharp in the sense that a braid with high Dehornoy floor must correspond to a knot of high genus, confirming that Dehornoy ordering complexity aligns with topological complexity.
• The proof relies on classifying vertex types in braid foliations and bounding the number of $\sigma_1^\pm$ generators in word representatives via ab-singularity and type B interval analysis.
• The result extends to other Thurston-type orderings: the analogous inequality holds when replacing the Dehornoy floor with the Thurston floor.
• The bound implies that the closure of a braid with high Dehornoy floor cannot be a simple knot, reinforcing the topological significance of the ordering.