Geodesic
The generalized Kauffman–Harary conjecture is true
Bakshi, Rhea Palak1 ; Guo, Huizheng2 ; Montoya-Vega, Gabriel3 ; Mukherjee, Sujoy4 ; Przytycki, Józef H5
1Institute for Theoretical Studies, ETH Zurich, Zurich, Switzerland, Department of Mathematics, University of California, Santa Barbara, Santa Barbara, CA, United States
2Department of Mathematics, The George Washington University, Washington, DC, United States
3Department of Mathematics, The Graduate Center CUNY, New York, NY, United States, Department of Mathematics, University of Puerto Rico at Río Piedras, San Juan, PR, United States
4Department of Mathematics, University of Denver, Denver, CO, United States
5Department of Mathematics, The George Washington University, Washington, DC, United States, Department of Mathematics, University of Gdańsk, Gdańsk, Poland
Algebraic and Geometric Topology, Tome 25 (2025) no. 4, pp. 2067-2081
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For a reduced alternating diagram of a knot with a prime determinant p, the Kauffman–Harary conjecture states that every nontrivial Fox p-coloring of the knot assigns different colors to its arcs. We prove a generalization of the conjecture, stated nineteen years ago by Asaeda, Przytycki and Sikora: for every pair of distinct arcs in the reduced alternating diagram of a prime link with determinant δ, there exists a Fox δ-coloring that distinguishes them.

DOI : 10.2140/agt.2025.25.2067
Keywords: determinants of links, double branched cover, Fox colorings, Kauffman–Harary conjecture, knots and links, pseudocolorings

Bakshi, Rhea Palak  1   ; Guo, Huizheng  2   ; Montoya-Vega, Gabriel  3   ; Mukherjee, Sujoy  4   ; Przytycki, Józef H  5

1 Institute for Theoretical Studies, ETH Zurich, Zurich, Switzerland, Department of Mathematics, University of California, Santa Barbara, Santa Barbara, CA, United States
2 Department of Mathematics, The George Washington University, Washington, DC, United States
3 Department of Mathematics, The Graduate Center CUNY, New York, NY, United States, Department of Mathematics, University of Puerto Rico at Río Piedras, San Juan, PR, United States
4 Department of Mathematics, University of Denver, Denver, CO, United States
5 Department of Mathematics, The George Washington University, Washington, DC, United States, Department of Mathematics, University of Gdańsk, Gdańsk, Poland 
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Bakshi, Rhea Palak; Guo, Huizheng; Montoya-Vega, Gabriel; Mukherjee, Sujoy; Przytycki, Józef H. The generalized Kauffman–Harary conjecture is true. Algebraic and Geometric Topology, Tome 25 (2025) no. 4, pp. 2067-2081. doi: 10.2140/agt.2025.25.2067

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