Abstract
Akimova and Matveev classified the prime virtual knots of genus 1 which admit diagrams with at most 5 classical crossings in 2017. In 2018, Kaur, Prabhakar, and Vesnin introduced the families of the \( L \)- and \( F \)-polynomials of virtual knots generalizing the Kauffman affine index polynomial. We introduce the notion of a totally flat-trivial virtual knot. We prove that the \( L \)- and \( F \)-polynomials for these knots coincide with the affine index polynomial. Also, we establish that all Akimova–Matveev knots are totally flat-trivial and calculate their affine index polynomials.
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Funding
The authors were supported by the Laboratory of Topology and Dynamics of Novosibirsk State University (Grant 14.Y26.31.0025 of the Ministry of Education and Science of the Russian Federation).
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Vesnin, A.Y., Ivanov, M.E. The Polynomials of Prime Virtual Knots of Genus 1 and Complexity at Most 5. Sib Math J 61, 994–1001 (2020). https://doi.org/10.1134/S003744662006004X
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DOI: https://doi.org/10.1134/S003744662006004X