Positive Knots and Ribbon Concordance

TL;DR

The paper investigates whether positive knots are minimal in the ribbon-concordance partial order and whether concordance classes admit more than one positive knot. It combines geometric analysis of band sums and incompressible Seifert surfaces with algebraic tools from knot Floer homology and rational Alexander modules to derive rigidity results. It proves that positive knots are band-prime, establishes a broad minimality criterion under a leading-coefficient condition on the Alexander polynomial, and shows concordant positive knots with a signature-degree bound have isomorphic rational Alexander modules, with extensions to almost positive knots. Collectively, these results advance understanding of the concordance landscape for positive knots and provide evidence toward the conjecture that each concordance class contains at most one positive knot.

Abstract

Ribbon concordances between knots generalize the notion of ribbon knots. Agol, building on work of Gordon, proved ribbon concordance gives a partial order on knots in $S^3$. In previous work, the author and Greene conjectured that positive knots are minimal in this ordering. In this note we prove this conjecture for a large class of positive knots, and show that a positive knot cannot be expressed as a non-trivial band sum -- both results extend earlier theorems of Greene and the author for special alternating knots. In a related direction, we prove that if positive knots $K$ and $K'$ are concordant and $|σ(K)| \geq 2g(K) - 2$, then $K$ and $K'$ have isomorphic rational Alexander modules. This strengthens a result of Stoimenow, and gives evidence toward a conjecture that any concordance class contains at most one positive knot.

Figures (1)

• Figure 1:

Theorems & Definitions (31)

• Conjecture 1.1: bogr24,tag23
• Theorem 1.2
• Theorem 1.3
• Conjecture 1.5: sto15
• Theorem 1.6
• Corollary 1.7
• Conjecture 1.8
• Lemma 2.1
• proof
• Theorem 2.2: oza02