The Affine Index Polynomial and the Sawollek Polynomial

Kauffman, Louis H.

doi:10.48550/arxiv.2012.01738

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2021

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“…For properties and applications of P K (t) see [3,10,11,12]. For each n ∈ Z \ {0} the n-th writhe J n (D) of a virtual knot diagram D is defined as the number of positive sign crossings minus number of negative sign crossings of D with index value n. The n-th writhe is a virtual knot invariant.…”

Section: Preliminariesmentioning

confidence: 99%

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Recurrent Generalization of F-Polynomials for Virtual Knots and Links

Gill¹,

Иванов²,

Prabhakar³

et al. 2021

Preprint

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using flat virtual knot invariants. These polynomials naturally generalize Kauffman's affine index polynomial and use smoothing in classical crossing of a virtual knot diagram. In this paper we introduce weight functions for ordered orientable virtual and flat virtual link. A flat virtual link is an equivalence class of virtual links in respect to a local symmetry changing type of classical crossing in a diagram. By considering three types of smoothings in classical crossings of a virtual link diagram and suitable weight functions, we provide a recurrent construction for new invariants. We demonstrate by providing explicit examples, that newly defined polynomial invariants are stronger than F-polynomials. CONTENTS 2010 Mathematics Subject Classification. 57K12.

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Section: Preliminariesmentioning

confidence: 99%

“…The connection of the affine index polynomial with the virtual knot cobordism is described in [11]. For related polynomial invariants and their properties see [12,15,16,17]. Three kinds of invariants of a virtual knot called the first, second and third intersection polynomials were introduced recently in [7].…”

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confidence: 99%