Advances in Low-Dimensional Topology and Related Topics

Dear Colleagues,

Algebraic topology has a long history and provides basic studies and many branches to analyse topological spaces in terms of something algebraic. Typical examples of the study are (co)-homology (with local system), algebraic structure from cohomology rings, fundamental groups, linking forms, and on.

Even in low-dimensional topology, such studies have provided many applications. For example, in knot theory, the Alexander module (polynomial) is a classical but powerful method to explore basic properties of knots and is summarized as the homology in abelian coefficients. In addition, R. C. Blanchfield [Blanchfield1957] made the corresponding generalization for linking forms, as well as some applications to concordance problems in knot theory.

Thus, it is natural to generalize the Blanchfield pairing. For example, as in Cochran–Orr–Teichner theory, the pairing has been defined even in solvable coefficients. Other topologists propose other twisted approaches from homology in local coefficient.

In this Special Issue, after providing a background for Blanchfield pairing, another approach to generalizing pairings is suggested: set theoretical Yang–Baxter equations. Moreover, some computations of the modified pairings and some relationships with classical Blanchfield pairing are discussed.

Dr. Takefumi Nosaka
Guest Editor

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• knot
• Blanchfield pairing
• twisted cohomology
• Yang-Baxter equation