Symplectic quandles and parabolic representations of 2-bridge knots and links
| Autor: | Hyuk Kim, Kyeonghee Jo |
|---|---|
| Rok vydání: | 2020 |
| Předmět: |
Pure mathematics
General Mathematics 010102 general mathematics Diagram Geometric Topology (math.GT) 0102 computer and information sciences 01 natural sciences Bridge (interpersonal) Arc (geometry) Mathematics - Geometric Topology 010201 computation theory & mathematics 57M25 57M27 FOS: Mathematics 0101 mathematics Mathematics Symplectic geometry |
| Zdroj: | International Journal of Mathematics. 31:2050081 |
| ISSN: | 1793-6519 0129-167X |
| DOI: | 10.1142/s0129167x20500810 |
| Popis: | In this paper, we study the parabolic representations of 2-bridge links by finiding arc coloring vectors on the Conway diagram. The method we use is to convert the system of conjugation quandle equations to that of symplectic quandle equations. In this approach, we have an integer coefficient monic polynomial [Formula: see text] for each 2-bridge link [Formula: see text], and each zero of this polynomial gives a set of arc coloring vectors on the diagram of [Formula: see text] satisfying the system of symplectic quandle equations, which gives an explicit formula for a parabolic representation of [Formula: see text]. We then explain how these arc coloring vectors give us the closed form formulas of the complex volume and the cusp shape of the representation. As other applications of this method, we show some interesting arithmetic properties of the Riley polynomial and of the trace field, and also describe a necessary and sufficient condition for the existence of epimorphisms between 2-bridge link groups in terms of divisibility of the corresponding Riley polynomials. |
| Databáze: | OpenAIRE |
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