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September 10, 2025Mathematical Proceedings of the Cambridge Philosophical Society0 citationsOpen Access

A partial resolution of Hedden’s conjecture on satellite homomorphisms

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RJRandall JohanningsmeierHKHillary KimAMAllison N. Miller

Key Points

  • The result shows that specific patterns failing to exhibit homomorphism advances our understanding of knot theory.
  • Proving that maps defined by even winding numbers do not form a homomorphism deepens the implications for smooth knot concordance.
  • The exploration focuses on self-maps within the smooth knot concordance group to clarify Hedden's conjecture further.
  • This finding may necessitate reevaluation of criteria for homomorphic behavior in satellite constructions.

Abstract

Abstract A pattern knot in a solid torus defines a self-map of the smooth knot concordance group. We prove that if the winding number of a pattern is even but not divisible by 8, then the corresponding map is not a homomorphism, thus partially establishing a conjecture of Hedden.

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Cite This Study

Johanningsmeier et al. (2025) studied this question.

synapsesocial.com/papers/68c1d60654b1d3bfb60f961ehttps://doi.org/10.1017/s0305004125101485
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