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Negative amphichiral knots and the half-Conway polynomial.
- Source :
-
Revista Mathematica Iberoamericana . 2024, Vol. 40 Issue 2, p581-622. 42p. - Publication Year :
- 2024
-
Abstract
- In 1979, Hartley and Kawauchi proved that the Conway polynomial of a strongly negative amphichiral knot factors as f (z)f (-z). In this paper, we normalize the factor f (z) to define the half-Conway polynomial. First, we prove that the half-Conway polynomial satisfies an equivariant skein relation, giving the first feasible computational method, which we use to compute the half-Conway polynomial for knots with 12 or fewer crossings. This skein relation also leads to a diagrammatic interpretation of the degree-one coefficient, from which we obtain a lower bound on the equivariant unknotting number. Second, we completely characterize polynomials arising as half-Conway polynomials of knots in S3, answering a problem of Hartley-Kawauchi. As a special case, we construct the first examples of non-slice strongly negative amphichiral knots with determinant one, answering a question of Manolescu. The double branched covers of these knots provide potentially non-trivial torsion elements in the homology cobordism group. [ABSTRACT FROM AUTHOR]
- Subjects :
- *POLYNOMIALS
*TORSION
*DIFFERENTIAL topology
*KNOT theory
Subjects
Details
- Language :
- English
- ISSN :
- 02132230
- Volume :
- 40
- Issue :
- 2
- Database :
- Academic Search Index
- Journal :
- Revista Mathematica Iberoamericana
- Publication Type :
- Academic Journal
- Accession number :
- 176026590
- Full Text :
- https://doi.org/10.4171/RMI/1442