### Abstract

For every polynomial f of degree n with no double roots, there is an associated family C(f) of harmonic algebraic curves, fibred over the circle, with at most n-1 singular fibres. We study the combinatorial topology of C(f) in the generic case when there are exactly n - 1 singular fibres. In this case, the topology of C(f) is determined by the data of an n-tuple of noncrossing matchings on the set {0,1,..., 2n - 1} with certain extra properties. We prove that there are 2(2n)^{n-2} such n-tuples, and that all of them arise from the topology of C(f) for some polynomial f.

Original language | English (US) |
---|---|

Pages (from-to) | 3083-3107 |

Number of pages | 25 |

Journal | Transactions of the American Mathematical Society |

Volume | 361 |

Issue number | 6 |

DOIs | |

State | Published - Jun 2009 |

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### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

**Polynomials, meanders, and paths in the lattice of noncrossing partitions.** / Savitt, David L.

Research output: Contribution to journal › Article

*Transactions of the American Mathematical Society*, vol. 361, no. 6, pp. 3083-3107. https://doi.org/10.1090/S0002-9947-08-04579-0

}

TY - JOUR

T1 - Polynomials, meanders, and paths in the lattice of noncrossing partitions

AU - Savitt, David L

PY - 2009/6

Y1 - 2009/6

N2 - For every polynomial f of degree n with no double roots, there is an associated family C(f) of harmonic algebraic curves, fibred over the circle, with at most n-1 singular fibres. We study the combinatorial topology of C(f) in the generic case when there are exactly n - 1 singular fibres. In this case, the topology of C(f) is determined by the data of an n-tuple of noncrossing matchings on the set {0,1,..., 2n - 1} with certain extra properties. We prove that there are 2(2n)n-2 such n-tuples, and that all of them arise from the topology of C(f) for some polynomial f.

AB - For every polynomial f of degree n with no double roots, there is an associated family C(f) of harmonic algebraic curves, fibred over the circle, with at most n-1 singular fibres. We study the combinatorial topology of C(f) in the generic case when there are exactly n - 1 singular fibres. In this case, the topology of C(f) is determined by the data of an n-tuple of noncrossing matchings on the set {0,1,..., 2n - 1} with certain extra properties. We prove that there are 2(2n)n-2 such n-tuples, and that all of them arise from the topology of C(f) for some polynomial f.

UR - http://www.scopus.com/inward/record.url?scp=77950538008&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77950538008&partnerID=8YFLogxK

U2 - 10.1090/S0002-9947-08-04579-0

DO - 10.1090/S0002-9947-08-04579-0

M3 - Article

AN - SCOPUS:77950538008

VL - 361

SP - 3083

EP - 3107

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 6

ER -