Akimova and Matveev classified the prime virtual knots of genus 1 which admit diagrams with at most 5 classicalcrossings 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 provethat the $ L $- and $ F $-polynomials for these knots coincide with the affine indexpolynomial. Also, we establish that all Akimova–Matveev knots are totally flat-trivialand calculate their affine index polynomials.

Original languageEnglish
Pages (from-to)994-1001
Number of pages8
JournalSiberian Mathematical Journal
Issue number6
Publication statusPublished - Nov 2020


  • 515.162.8
  • affine index polynomial
  • knot in a thickened torus
  • virtual knot


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