Graphical virtual links and a polynomial of signed cyclic graphs

Qingying Deng; Xian'an Jin; Louis H Kauffman

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arXiv · 1712.06711

Graphical virtual links and a polynomial of signed cyclic graphs

Qingying Deng·Xian'an Jin·Louis H Kauffman

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Abstract

For a signed cyclic graph G, we can construct a unique virtual link L by taking the medial construction and convert 4-valent vertices of the medial graph to crossings according to the signs. If a virtual link can occur in this way then we say that the virtual link is graphical. In the article we shall prove that a virtual link L is graphical if and only if it is checkerboard colorable. On the other hand, we introduce a polynomial F[G] for signed cyclic graphs, which is defined via a deletion-marking recursion. We shall establish the relationship between F[G] of a signed cyclic graph G and the bracket polynomial of one of the virtual link diagrams associated with G. Finally we give a spanning subgraph expansion for F[G].

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BibTeX RIS

Qingying Deng, Xian'an Jin, Louis H Kauffman. 2018-05-31. Graphical virtual links and a polynomial of signed cyclic graphs. https://arxiv.org/abs/1712.06711

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