### Abstract

We show that there exist infinitely many examples of pairs of knots, K_{1} and K_{2}, that have no epimorphism π_{1}(S^{3} \ K_{1}) → π_{1}(S^{3} \ K_{2}) 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 |

### Fingerprint

### Keywords

- A polynomial
- Knot group
- Tangle sum

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*New York Journal of Mathematics*,

*21*, 823-835.

**Tangle sums and factorization of a-polynomials.** / Ishikawa, Masaharu; Mattman, Thomas W.; Shimokawa, Koya.

Research output: Contribution to journal › Article

*New York Journal of Mathematics*, vol. 21, pp. 823-835.

}

TY - JOUR

T1 - Tangle sums and factorization of a-polynomials

AU - Ishikawa, Masaharu

AU - Mattman, Thomas W.

AU - Shimokawa, Koya

PY - 2015/1/1

Y1 - 2015/1/1

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

UR - http://www.scopus.com/inward/record.url?scp=84940471464&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84940471464&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:84940471464

VL - 21

SP - 823

EP - 835

JO - New York Journal of Mathematics

JF - New York Journal of Mathematics

SN - 1076-9803

ER -