### Abstract

ConwayGordon proved that for every spatial complete graph on six vertices, the sum of the linking numbers over all of the constituent 2-component links is congruent to 1 modulo 2, and for every spatial complete graph on seven vertices, the sum of the Arf invariants over all of the Hamiltonian knots is also congruent to 1 modulo 2. In this paper, we give a ConwayGordon type theorem for any graph which is obtained from the complete graph on six or seven vertices by a finite sequence of △Y-exchanges.

Original language | English |
---|---|

Article number | 1250067 |

Journal | Journal of Knot Theory and its Ramifications |

Volume | 21 |

Issue number | 7 |

DOIs | |

Publication status | Published - 2012 Jun |

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### Keywords

- δY-exchange
- Intrinsic knottedness
- Intrinsic linkedness
- Spatial graph

### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

*Journal of Knot Theory and its Ramifications*,

*21*(7), [1250067]. https://doi.org/10.1142/S0218216512500678

**δy-exchanges and the conwaygordon theorems.** / Nikkuni, Ryo; Taniyama, Kouki.

Research output: Contribution to journal › Article

*Journal of Knot Theory and its Ramifications*, vol. 21, no. 7, 1250067. https://doi.org/10.1142/S0218216512500678

}

TY - JOUR

T1 - δy-exchanges and the conwaygordon theorems

AU - Nikkuni, Ryo

AU - Taniyama, Kouki

PY - 2012/6

Y1 - 2012/6

N2 - ConwayGordon proved that for every spatial complete graph on six vertices, the sum of the linking numbers over all of the constituent 2-component links is congruent to 1 modulo 2, and for every spatial complete graph on seven vertices, the sum of the Arf invariants over all of the Hamiltonian knots is also congruent to 1 modulo 2. In this paper, we give a ConwayGordon type theorem for any graph which is obtained from the complete graph on six or seven vertices by a finite sequence of △Y-exchanges.

AB - ConwayGordon proved that for every spatial complete graph on six vertices, the sum of the linking numbers over all of the constituent 2-component links is congruent to 1 modulo 2, and for every spatial complete graph on seven vertices, the sum of the Arf invariants over all of the Hamiltonian knots is also congruent to 1 modulo 2. In this paper, we give a ConwayGordon type theorem for any graph which is obtained from the complete graph on six or seven vertices by a finite sequence of △Y-exchanges.

KW - δY-exchange

KW - Intrinsic knottedness

KW - Intrinsic linkedness

KW - Spatial graph

UR - http://www.scopus.com/inward/record.url?scp=84859406231&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84859406231&partnerID=8YFLogxK

U2 - 10.1142/S0218216512500678

DO - 10.1142/S0218216512500678

M3 - Article

VL - 21

JO - Journal of Knot Theory and its Ramifications

JF - Journal of Knot Theory and its Ramifications

SN - 0218-2165

IS - 7

M1 - 1250067

ER -