Tangle sums and factorization of a-polynomials

Masaharu Ishikawa; Thomas W. Mattman; Koya Shimokawa

Tangle sums and factorization of a-polynomials

Masaharu Ishikawa, Thomas W. Mattman, Koya Shimokawa

Research output: Contribution to journal › Article › peer-review

Abstract

We show that there exist infinitely many examples of pairs of knots, K₁ and K₂, that have no epimorphism π₁(S³ \ K₁) → π₁(S³ \ K₂) preserving peripheral structure although their A-polynomials have the factorization A_K2 (L, M) ∣ A_K1 (L, M). Our construction accounts for most of the known factorizations of this form for knots with 10 or fewer crossings. In particular, we conclude that while an epimorphism will lead to a factorization of A-polynomials, the converse generally fails.

Original language	English
Pages (from-to)	823-835
Number of pages	13
Journal	New York Journal of Mathematics
Volume	21
Publication status	Published - 2015 Jan 1
Externally published	Yes

Keywords

A polynomial
Knot group
Tangle sum

ASJC Scopus subject areas

Mathematics(all)

Cite this

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N2 - We show that there exist infinitely many examples of pairs of knots, K1 and K2, that have no epimorphism π1(S3 \ K1) → π1(S3 \ K2) preserving peripheral structure although their A-polynomials have the factorization AK2 (L, M) ∣ AK1 (L, M). Our construction accounts for most of the known factorizations of this form for knots with 10 or fewer crossings. In particular, we conclude that while an epimorphism will lead to a factorization of A-polynomials, the converse generally fails.

AB - We show that there exist infinitely many examples of pairs of knots, K1 and K2, that have no epimorphism π1(S3 \ K1) → π1(S3 \ K2) preserving peripheral structure although their A-polynomials have the factorization AK2 (L, M) ∣ AK1 (L, M). Our construction accounts for most of the known factorizations of this form for knots with 10 or fewer crossings. In particular, we conclude that while an epimorphism will lead to a factorization of A-polynomials, the converse generally fails.

KW - A polynomial

KW - Knot group

KW - Tangle sum

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JO - New York Journal of Mathematics

JF - New York Journal of Mathematics

ER -

Tangle sums and factorization of a-polynomials

Abstract

Keywords

ASJC Scopus subject areas

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