scholarly journals Higher-order finite type invariants of classical and virtual knots and unknotting operations

2019 ◽  
Vol 264 ◽  
pp. 210-222
Author(s):  
Noboru Ito ◽  
Migiwa Sakurai
Keyword(s):  
Finite Type ◽  
Higher Order ◽  
Virtual Knots ◽  
Finite Type Invariants

A LOWER BOUND FOR THE NUMBER OF FORBIDDEN MOVES TO UNKNOT A LONG VIRTUAL KNOT

2013 ◽  
Vol 22 (06) ◽  
pp. 1350024 ◽  
Author(s):  
MYEONG-JU JEONG
Keyword(s):  
Lower Bound ◽  
Finite Type ◽  
Finite Sequence ◽  
Virtual Knot ◽  
Virtual Knots ◽  
Finite Type Invariants ◽  
Combinatorial Representations ◽  
Congruent Modulo

Nelson and Kanenobu showed that forbidden moves unknot any virtual knot. Similarly a long virtual knot can be unknotted by a finite sequence of forbidden moves. Goussarov, Polyak and Viro introduced finite type invariants of virtual knots and long virtual knots and gave combinatorial representations of finite type invariants. We introduce Fn-moves which generalize the forbidden moves. Assume that two long virtual knots K and K′ are related by a finite sequence of Fn-moves. We show that the values of the finite type invariants of degree 2 of K and K′ are congruent modulo n and give a lower bound for the number of Fn-moves needed to transform K to K′.

2- AND 3-VARIATIONS AND FINITE TYPE INVARIANTS OF DEGREE 2 AND 3

2013 ◽  
Vol 22 (08) ◽  
pp. 1350042 ◽  
Author(s):  
MIGIWA SAKURAI
Keyword(s):  
Finite Type ◽  
Lower Bounds ◽  
Virtual Knots ◽  
Finite Type Invariants ◽  
Type Invariant

Goussarov, Polyak and Viro defined a finite type invariant and a local move called an n-variation for virtual knots. In this paper, we give the differences of the values of the finite type invariants of degree 2 and 3 between two virtual knots which can be transformed into each other by a 2- and 3-variation, respectively. As a result, we obtain lower bounds of the distance between long virtual knots by 2-variations and the distance between virtual knots by 3-variations by using the values of the finite type invariants of degree 2 and 3, respectively.

scholarly journals Finite-type invariants of classical and virtual knots

Topology ◽  
2000 ◽  
Vol 39 (5) ◽  
pp. 1045-1068 ◽  
Author(s):  
Mikhail Goussarov ◽  
Michael Polyak ◽  
Oleg Viro
Keyword(s):  
Finite Type ◽  
Virtual Knots ◽  
Finite Type Invariants

scholarly journals PARITY AND EXOTIC COMBINATORIAL FORMULAE FOR FINITE-TYPE INVARIANTS OF VIRTUAL KNOTS

2012 ◽  
Vol 21 (13) ◽  
pp. 1240001 ◽  
Author(s):  
MICAH WHITNEY CHRISMAN ◽  
VASSILY OLEGOVICH MANTUROV
Keyword(s):  
Finite Type ◽  
Combinatorial Formula ◽  
Knot Invariants ◽  
Virtual Knot ◽  
Additional Information ◽  
Virtual Knots ◽  
Finite Type Invariants ◽  
Gauss Diagram

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.

Finite-type invariants of order one of long and framed virtual knots

2019 ◽  
Vol 28 (10) ◽  
pp. 1950064 ◽  
Author(s):  
Nicolas Petit
Keyword(s):  
Finite Type ◽  
Virtual Knot ◽  
Virtual Knots ◽  
Finite Type Invariants

We generalize three invariants, first discovered by Henrich, to the long and/or framed virtual knot case. These invariants are all finite-type invariants of order one, and include a universal invariant. The generalization will require us to extend the notion of a based matrix of a virtual string, first introduced by Turaev and later generalized by Henrich, to the long and framed cases.

scholarly journals FINITE TYPE LINK HOMOTOPY INVARIANTS II: Milnor's $\bar\mu$-Invariants

2000 ◽  
Vol 09 (06) ◽  
pp. 735-758 ◽  
Author(s):  
BLAKE MELLOR
Keyword(s):  
Finite Type ◽  
Higher Order ◽  
Finite Type Invariants ◽  
Homotopy Invariant ◽  
Type Invariant ◽  
Link Homotopy ◽  
Homotopy Invariants

We define a notion of finite type invariants for links with a fixed linking matrix. We show that Milnor's link homotopy invariant [Formula: see text] is a finite type invariant, of type 1, in this sense. We also generalize this approach to Milnor's higher order [Formula: see text] invariants and show that they are also, in a sense, of finite type. Finally, we compare our approach to another approach for defining finite type invariants within linking classes.

scholarly journals Some Dimensions of Spaces of Finite Type Invariants of Virtual Knots

2011 ◽  
Vol 20 (3) ◽  
pp. 282-287 ◽  
Author(s):  
Dror Bar-Natan ◽  
Iva Halacheva ◽  
Louis Leung ◽  
Fionntan Roukema
Keyword(s):  
Finite Type ◽  
Virtual Knots ◽  
Finite Type Invariants

scholarly journals Finite Type Invariants and the Kauffman Bracket Polynomials of Virtual Knots

2014 ◽  
Vol 54 (4) ◽  
pp. 639-653 ◽  
Author(s):  
Myeong-Ju Jeong ◽  
Chan-Young Park ◽  
Soon Tae Yeo
Keyword(s):  
Finite Type ◽  
Virtual Knots ◽  
Finite Type Invariants ◽  
Kauffman Bracket ◽  
Bracket Polynomials
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