---
address: |
    Harvard University, Cambridge MA 02138\
    Massachusetts Institute of Technology, Cambridge MA 02139
author:
- 'P. B. Kronheimer and T. S. Mrowka [^1]'
bibliography:
- 'fcs.bib'
title: 'Exact triangles for $\SO(3)$ instanton homology of webs'
---

=1

Introduction
============

Let $K \subset \R^{3}$ be an unoriented web, i.e. an embedded trivalent graph whose local model at the vertices is that of three arcs meeting with distinct tangent directions. In a previous paper [@KM-jsharp], the authors defined an invariant $\Jsharp(K)$ for such webs, as an $\SO(3)$ instanton homology with coefficients in the field $\F=\Z/2$. This instanton homology is functorial for *foams*, which are singular cobordisms between webs. The construction of $\Jsharp(K)$ closely resembles an invariant $\Isharp(K)$ defined earlier for knots and links in [@KM-unknot]. Knots and links are webs without vertices; but even for these, $\Isharp(K)$ and $\Jsharp(K)$ are different, because $\Isharp(K)$ was defined using $\SU(2)$ representation varieties, while $\Jsharp(K)$ uses $\SO(3)$. Our conventions and definitions are briefly recalled in section \[sec:recap\].

This paper is a continuation of [@KM-jsharp] and establishes a type of skein relation (an exact triangle) for $\Jsharp$. The main result concerns three webs $L_{2}$, $L_{1}$, $L_{0}$ which differ only inside a ball, as shown: $$\label{eq:intro-L-pics}
        