LINEAR INDEPENDENCE OF CABLES IN THE KNOT CONCORDANCE GROUP

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We produce infinite families of knots {K-i}(i >= 1) for which the set of cables {K-p,1(i)}(i,p >= 1) is linearly independent in the knot concordance group, C. We arrange that these examples lie arbitrarily deep in the solvable and bipolar filtrations of C, denoted by {F-n} and {B-n} respectively. As a consequence, this result cannot be reached by any combination of algebraic concordance invariants, Casson-Gordon invariants, and Heegaard-Floer invariants such as tau, epsilon, and Y. We give two applications of this result. First, for any n >= 0, there exists an infinite family {K-i}(i >= 1) such that for each fixed i, {K-2j,1(i)}(j >= 0) is a basis for an infinite rank summand of F-n and {K-p,1(i)}(i,p >= 1) is linearly independent in F-n/F-n.5. Second, for any n >= 1, we give filtered counterexamples to Kauffman's conjecture on slice knots by constructing smoothly slice knots with genus one Seifert surfaces where one derivative curve has nontrivial Arf invariant and the other is nontrivial in both F-n/F-n.5 and Bn-1/Bn+1. We also give examples of smoothly slice knots with genus one Seifert surfaces such that one derivative has nontrivial Arf invariant and the other is topologically slice but not smoothly slice.
Publisher
AMER MATHEMATICAL SOC
Issue Date
2021-06
Language
English
Article Type
Article
Citation

TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, v.374, no.6, pp.4449 - 4479

ISSN
0002-9947
DOI
10.1090/tran/8336
URI
http://hdl.handle.net/10203/285386
Appears in Collection
MA-Journal Papers(저널논문)
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