# The 2-leg vertex in K-theoretic DT theory

## TL;DR

This paper derives a formula for a specific K-theoretic vertex in Donaldson-Thomas theory when two entries are nontrivial, advancing computational techniques in enumerative geometry.

## Contribution

It provides a new explicit formula for the 2-leg K-theoretic vertex in DT theory, expanding the computational toolkit for curve counting in toric threefolds.

## Key findings

- Derived a formula for the 2-leg K-theoretic vertex with two nontrivial entries.
- Discussed applications of the vertex formula in enumerative geometry.
- Enhanced computational methods for K-theoretic DT counts.

## Abstract

K-theoretic Donaldson-Thomas counts of curves in toric and many related threefolds can be computed in terms of a certain canonical 3-valent tensor, the K-theoretic equivariant vertex. In this paper we derive a formula for the vertex in the case when two out of three entries are nontrivial. We also discuss some applications of this result.