On standard forms of 1--dominations between knots with same Gromov volumes

2008/01/13 by Michel Boileau, Boileau, Michel, Yi Ni +3
Biochemistry, Genetics and Molecular Biology · Mathematics · #55M20 #57M25 #57M50 #Connective tissue disorders research #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #math.GT #msc:55M20 #msc:57M25 #msc:57M50

paper · pdf · doi:10.48550/arxiv.0801.1943

15 pages

arxiv created 2008/01/13 · openalex publication_date 2008/01/13 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let k and k' be two knots in 3-sphere. Say k 1--dominates k', if there is a proper degree 1 map f\co E(k)→ E(k'), between knot exterior of k_i. Theorem: Suppose that any companion of k is prime. If k 1--dominates k' with the same Gromov volume, then k' can be obtained from k by finitely many de-satellizations. The condition of "same Gromov volume" clearly can not be removed. We also give a new construction of 1-domination between knots with same Gromov volume to show that the condition "any companion of k is prime" can not be removed.

Citations

• Non-zero degree maps between 2n-manifolds

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