Explicit formulas for Vasil'ev invariants of the fourth order for knots

Allenov, S. V.

doi:10.1007/s10958-009-9322-5

J Math Sci

2009

DOI: 10.1007/s10958-009-9322-5

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Explicit formulas for Vasil'ev invariants of the fourth order for knots

S. V. Allenov

Abstract: In this work, the integer-valued basic Vasil'ev invariants of the fourth order for the oriented knots V 1 4 (K) and V 2 4 (K) are presented. The explicit diagram-arrow formulas with rational coefficients defined according to the Gaussian diagram of a knot are obtained for them.

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2020

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Twin Nicomachean q-identities and conjectures for the associated discriminants, polynomials, and inequalities

Kim

Stolarsky

2020

Int. J. Number Theory

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We insert additional variables into Warnaar’s [Formula: see text]-analogue of Nicomachus’ identity and other related identities, and compute discriminants with respect to [Formula: see text]. Factorization of these discriminants reveals pairs of partitions that conjecturally relate in the manner of Wheatstone. The factorization also yields, conjecturally, families of polynomials with relations to various Molien series, remarkable rational generating functions, and notable root distributions. For a [Formula: see text]-analogue of Nicomachus’ identity produced by Cigler, we provide proofs of the partition properties. We also state and in part prove tight inequalities for the elements of two interlacing sequences that led us to the “twin” of Warnaar’s [Formula: see text]-analogue.

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Twin Nicomachean q-identities and conjectures for the associated discriminants, polynomials, and inequalities

Kim

Stolarsky

2020

Int. J. Number Theory

View full text Add to dashboard Cite

show abstract

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Explicit formulas for Vasil'ev invariants of the fourth order for knots

Abstract: In this work, the integer-valued basic Vasil'ev invariants of the fourth order for the oriented knots V 1 4 (K) and V 2 4 (K) are presented. The explicit diagram-arrow formulas with rational coefficients defined according to the Gaussian diagram of a knot are obtained for them.

Cited by 1 publication

References 6 publications

Twin Nicomachean q-identities and conjectures for the associated discriminants, polynomials, and inequalities

Twin Nicomachean q-identities and conjectures for the associated discriminants, polynomials, and inequalities

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