Special Issue for Slavic Jablan; Guest Editors: L. H. Kauffman, R. Sazdanovic and S. LambropoulouNo Access

Determinant of links, spanning trees, and a theorem of Shank

Jun Ge

Department of Mathematics, Sichuan Normal University, Chengdu 610066, Sichuan, P. R. China

E-mail Address: mathsgejun@163.com

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and

Lianzhu Zhang

School of Mathematical Sciences, Xiamen University, Xiamen 361005, Fujian, P. R. China

E-mail Address: zhanglz@xmu.edu.cn

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https://doi.org/10.1142/S0218216516410054Cited by:0 (Source: Crossref)

Abstract

In this note, we first give an alternative elementary proof of the relation between the determinant of a link and the spanning trees of the corresponding Tait graph. Then, we use this relation to give an extremely short, knot theoretical proof of a theorem due to Shank stating that a link has component number one if and only if the number of spanning trees of its Tait graph is odd.

In memory of Professor Slavik Jablan

Keywords:

AMSC: 57M25, 57M15

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