## Abstract

Our main purpose is to introduce the theory of graph-skein modules of three-manifolds. This theory associates to each oriented three-manifold M an algebraic object (a module or an algebra) which is defined by considering the set of all ribbon graphs embedded in M modulo local linear skein relations. This idea is inspired by Przytycki's theory of skein modules which is also known as algebraic topology based on knots. Historically, this theory appeared as a generalization of the quantum invariants of links in the three-sphere to links in an arbitrary three-manifold. In this paper, we review the construction of the Kauffman bracket skein module and investigate its relationship with our graph-skein modules. We compute the graph-skein algebra in few cases. As an application we introduce new criteria for symmetries of spatial graphs which improve some results obtained earlier. The proof of these criteria is based on some easy calculation in the graph-skein module of the solid torus.

Original language | English |
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Title of host publication | Handbook of Material Science Research |

Publisher | Nova Science Publishers, Inc. |

Pages | 473-489 |

Number of pages | 17 |

ISBN (Print) | 9781607417989 |

Publication status | Published - 2010 |

## ASJC Scopus subject areas

- General Materials Science