On T-positive links

TL;DR

This paper characterizes T-positive links as strongly quasipositive links that are closures of T-homogeneous braids, exploring their properties, behaviors under operations, and their relation to other positivity notions.

Contribution

It provides a new characterization of T-positive links via T-homogeneous braids and analyzes their behavior and classification among strongly quasipositive links.

Findings

All strongly quasipositive, fibered knots with ≤12 crossings are T-positive.

T-positive links are closed under cabling and connected sums.

Comparison with other positivity notions and open questions are discussed.

Abstract

T-positive links form a subset of strongly quasipositive links that strictly contains the set of all non-split braid positive links. Analogous to Baader's characterisation of positive links as precisely the strongly quasipositive and homogeneous links, we show that T-positive links are precisely the strongly quasipositive links that are the closures of T-homogeneous braids. This complements previous characterizations of T-positive links by Rudolph and Banfield as links arising as boundaries of positive Hopf-plumbed baskets, or closures of staircase braids. We examine the behavior of T-positive links under cabling operations and connected sums, and demonstrate that all strongly quasipositive, fibered knots with at most 12 crossings are T-positive. Additionally, we compare T-positivity with other positivity notions for links and compile open questions.