On the fundamental 3-classes of knot group representations

  • Takefumi NosakaEmail author
Original Paper


We discuss the fundamental (relative) 3-classes of knots (or hyperbolic links), and provide diagrammatic descriptions of the push-forwards with respect to link-group representations. The point is an observation of a bridge between the relative group homology and quandle homology from the viewpoints of Inoue–Kabaya map (Geom Dedicata 171(1):265–292, 2014). Furthermore, we give an algorithm to algebraically describe the fundamental 3-class of any hyperbolic knot.


Knot Relative group homology Hyperbolicity Malnormal subgroups Quandle 



The author is greatly indebted to Tetsuya Ito and Yuichi Kabaya for useful discussions on quandle, malnormality, and hyperbolicity. He also expresses his gratitude to the referee for useful comments This work is partially supported by JSPS KAKENHI Grant Number 00646903.


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Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of MathematicsTokyo Institute of TechnologyMeguro-KuJapan

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