Origins and Immunity Networking Functions of EDS1 Family Proteins

The EDS1 family of structurally unique lipase-like proteins EDS1, SAG101, and PAD4 evolved in seed plants, on top of existing phytohormone and nucleotide-binding–leucine-rich-repeat (NLR) networks, to regulate immunity pathways against host-adapted biotrophic pathogens. Exclusive heterodimers between EDS1 and SAG101 or PAD4 create essential surfaces for resistance signaling. Phylogenomic information, together with functional studies in Arabidopsis and tobacco, identify a coevolved module between the EDS1–SAG101 heterodimer and coiled-coil (CC) HET-S and LOP-B (CCHELO) domain helper NLRs that is recruited by intracellular Toll-interleukin1-receptor (TIR) domain NLR receptors to confer host cell death and pathogen immunity. EDS1–PAD4 heterodimers have a different and broader activity in basal immunity that transcriptionally reinforces local and systemic defenses triggered by various NLRs. Here, we consider EDS1 family protein functions across seed plant lineages in the context of networking with receptor and helper NLRs and downstream resistance machineries. The different modes of action and pathway connectivities of EDS1 family members go some way to explaining their central role in biotic stress resilience.

The slope conjecture for Montesinos knots

The slope conjecture relates the degree of the colored Jones polynomial of a knot to boundary slopes of essential surfaces. We develop a general approach that matches a state-sum formula for the colored Jones polynomial with the parameters that describe surfaces in the complement. We apply this to Montesinos knots proving the slope conjecture for Montesinos knots, with some restrictions.

Crosscap numbers of a family of Montesinos knots

This paper is concerned with the crosscap numbers of Montesinos knots. For a family of Montesinos knots, we give a lower bound of [Formula: see text] among all essential surfaces [Formula: see text] embedded in the exterior, where [Formula: see text] denotes the ratio of negative Euler characteristic of the surface and the number of sheets. From this result, we determine the crosscap numbers of a family of Montesinos knots. Our method relies on the algorithm of enumerating all essential surfaces for Montesinos knots given by Hatcher and Oertel.

A result on the Slope conjectures for 3-string Montesinos knots

The Slope Conjecture and the Strong Slope Conjecture predict that the degree of the colored Jones polynomial of a knot is matched by the boundary slope and the Euler characteristic of some essential surfaces in the knot complement. By solving a problem of quadratic integer programming to find the maximal degree and using the Hatcher–Oertel edgepath system to find the corresponding essential surface, we verify the Slope Conjectures for a family of 3-string Montesinos knots satisfying certain conditions.