Parity and exotic combinatorial formulae for finite-type invariants of virtual knots

The present paper produces examples of Gauss diagram formulae for virtual knot invariants which have no analogue in the classical knot case. These combinatorial formulae contain additional information about how a subdiagram is embedded in a virtual knot diagram. The additional information comes from the second author's recently discovered notion of parity. For a parity of flat virtual knots, the new combinatorial formulae are Kauffman finite-type invariants. However, many of the combinatorial formulae possess exotic properties. It is shown that there exists an integer-valued virtualization invariant combinatorial formula of order n for every n (i.e. it is stable under the map which changes the direction of one arrow but preserves the sign). Hence, it is not of Goussarov-Polyak-Viro finite-type. Moreover, every homogeneous Polyak-Viro combinatorial formula admits a decomposition into an "even" part and an "odd" part. For the Gaussian parity, neither part of the formula is of GPV finite-type when it is non-constant on the set of classical knots. In addition, eleven new non-trivial combinatorial formulae of order 2 are presented which are not of GPV finite-type. © 2012 World Scientific Publishing Company.

Авторы
Chrisman M.W.1 , Manturov V.O. 2
Редакторы
-
Издательство
-
Номер выпуска
13
Язык
Английский
Страницы
-
Статус
Опубликовано
Подразделение
-
Номер
1240001
Том
21
Год
2012
Организации
  • 1 Monmouth University, West Long Branch, NJ, United States
  • 2 People's Friendship University of Russia, Faculty of Sciences, 3 Ordjonikidze Street, Moscow 117923, Russian Federation
Ключевые слова
combinatorial formula; finite-type invariant; Knot; parity; virtual knot
Дата создания
19.10.2018
Дата изменения
19.10.2018
Постоянная ссылка
https://repository.rudn.ru/ru/records/article/record/2243/