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abstract: 'We introduce invariants of graphs embedded in $S^3$ which are related to the Wu invariant and the Simon invariant. Then we use our invariants to prove that certain graphs are intrinsically chiral, and to obtain lower bounds for the minimal crossing number of particular embeddings of graphs in $S^3$.'
author:
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    ERICA FLAPAN\
    Department of MathematicsPomona CollegeClaremont, CA 91711, USA WILL FLETCHER\
    Biophysics ProgramStanford UniversityStanford, CA 94305, USA
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     RYO NIKKUNI\
    Department of Mathematics, Tokyo Woman’s Christian University, 2-6-1 Zempukuji, Suginami-ku, Tokyo 167-8585, Japan
title: Reduced Wu and Generalized Simon Invariants for Spatial Graphs
---

Introduction
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While there are numerous invariants for embeddings of graphs in $3$-manifolds, most have limited applications either because they are hard to compute or because they are only defined for particular types of graphs. For example, Thompson [@Tho] defined a powerful polynomial invariant for graphs embedded in arbitrary 3-manifolds, which can detect whether an embedding of a graph in $S^3$ is planar. However, computing Thompson’s invariant requires identifying topological features of a sequence of 3-manifolds, such as whether each manifold is compressible.

Yamada [@Ya] and Yokota [@Yo] introduced polynomial invariants for spatial graphs (i.e., graphs embedded in $S^3$). The Yamada 