Connected sum of virtual knots and F-polynomials

2021 ◽  
Author(s):  
Maxim Ivanov
Keyword(s):  
Infinite Family ◽  
Connected Sum ◽  
Connected Sums ◽  
Virtual Knots ◽  
Knot Diagrams

It is known that connected sum of two virtual knots is not uniquely determined and depends on knot diagrams and choosing the points to be connected. But different connected sums of the same virtual knots cannot be distinguished by Kauffman’s affine index polynomial. For any pair of virtual knots [Formula: see text] and [Formula: see text] with [Formula: see text]-dwrithe [Formula: see text] we construct an infinite family of different connected sums of [Formula: see text] and [Formula: see text] which can be distinguished by [Formula: see text]-polynomials.

scholarly journals Minimal diagrams of free knots

2014 ◽  
Vol 23 (06) ◽  
pp. 1450032
Author(s):  
Tomas Boothby ◽  
Allison Henrich ◽  
Alexander Leaf
Keyword(s):  
Equivalence Class ◽  
Virtual Knots ◽  
Knot Diagrams

Manturov recently introduced the idea of a free knot, i.e. an equivalence class of virtual knots where equivalence is generated by crossing change and virtualization moves. He showed that if a free knot diagram is associated to a graph that is irreducibly odd, then it is minimal with respect to the number of classical crossings. Not all minimal diagrams of free knots are associated to irreducibly odd graphs, however. We introduce a family of free knot diagrams that arise from certain permutations that are minimal but not irreducibly odd.

scholarly journals The SL(2, C) Casson Invariant for Knots and the Â-polynomial

2016 ◽  
Vol 68 (1) ◽  
pp. 3-23 ◽  
Author(s):  
Hans U. Boden ◽  
Cynthia L. Curtis
Keyword(s):  
Large Class ◽  
Product Formula ◽  
Integral Homology ◽  
Connected Sum ◽  
Connected Sums ◽  
Knot Invariant ◽  
Definition Of ◽  
Arbitrary Knots

AbstractIn this paper, we extend the definition of the SL(2,ℂ) Casson invariant to arbitrary knots K in integral homology 3-spheres and relate it to the m-degree of the Â-polynomial of K. We prove a product formula for the Â-polynomial of the connected sum K1#K2 of two knots in S3 and deduce additivity of the SL(2,ℂ) Casson knot invariant under connected sums for a large class of knots in S3. We also present an example of a nontrivial knot K in S3 with trivial Â-polynomial and trivial SL(2,ℂ) Casson knot invariant, showing that neither of these invariants detect the unknot.

Picture-valued parity-biquandle bracket

2020 ◽  
Vol 29 (02) ◽  
pp. 2040004 ◽  
Author(s):  
Denis P. Ilyutko ◽  
Vassily O. Manturov
Keyword(s):  
Classical Case ◽  
Linear Combinations ◽  
Virtual Knot ◽  
Virtual Knots ◽  
Underlying Graph ◽  
Kauffman Bracket ◽  
Knot Diagrams

In V. O. Manturov, On free knots, preprint (2009), arXiv:math.GT/0901.2214], the second named author constructed the bracket invariant [Formula: see text] of virtual knots valued in pictures (linear combinations of virtual knot diagrams with some crossing information omitted), such that for many diagrams [Formula: see text], the following formula holds: [Formula: see text], where [Formula: see text] is the underlying graph of the diagram, i.e. the value of the invariant on a diagram equals the diagram itself with some crossing information omitted. This phenomenon allows one to reduce many questions about virtual knots to questions about their diagrams. In [S. Nelson, M. E. Orrison and V. Rivera, Quantum enhancements and biquandle brackets, preprint (2015), arXiv:math.GT/1508.06573], the authors discovered the following phenomenon: having a biquandle coloring of a certain knot, one can enhance various state-sum invariants (say, Kauffman bracket) by using various coefficients depending on colors. Taking into account that the parity can be treated in terms of biquandles, we bring together the two ideas from these papers and construct the picture-valued parity-biquandle bracket for classical and virtual knots. This is an invariant of virtual knots valued in pictures. Both the parity bracket and Nelson–Orrison–Rivera invariants are partial cases of this invariant, hence this invariant enjoys many properties of various kinds. Recently, the authors together with E. Horvat and S. Kim have found that the picture-valued phenomenon works in the classical case.

Geodesic graphs for a homotopy class of virtual knots

2017 ◽  
Vol 26 (13) ◽  
pp. 1750090
Author(s):  
Sumiko Horiuchi ◽  
Yoshiyuki Ohyama
Keyword(s):  
Homotopy Class ◽  
Finite Sequence ◽  
Dimensional Lattice ◽  
Virtual Knot ◽  
Virtual Knots ◽  
Minimum Number ◽  
Vertex Set ◽  
Number Formula ◽  
Lattice Graph ◽  
Knot Diagrams

We consider a local move, denoted by [Formula: see text], on knot diagrams or virtual knot diagrams.If two (virtual) knots [Formula: see text] and [Formula: see text] are transformed into each other by a finite sequence of [Formula: see text] moves, the [Formula: see text] distance between [Formula: see text] and [Formula: see text] is the minimum number of times of [Formula: see text] moves needed to transform [Formula: see text] into [Formula: see text]. By [Formula: see text], we denote the set of all (virtual) knots which can be transformed into a (virtual) knot [Formula: see text] by [Formula: see text] moves. A geodesic graph for [Formula: see text] is the graph which satisfies the following: The vertex set consists of (virtual) knots in [Formula: see text] and for any two vertices [Formula: see text] and [Formula: see text], the distance on the graph from [Formula: see text] to [Formula: see text] coincides with the [Formula: see text] distance between [Formula: see text] and [Formula: see text]. When we consider virtual knots and a crossing change as a local move [Formula: see text], we show that the [Formula: see text]-dimensional lattice graph for any given natural number [Formula: see text] and any tree are geodesic graphs for [Formula: see text].

scholarly journals Gordian complexes of knots and virtual knots given by region crossing changes and arc shift moves

2020 ◽  
Vol 29 (10) ◽  
pp. 2042008
Author(s):  
Amrendra Gill ◽  
Madeti Prabhakar ◽  
Andrei Vesnin
Keyword(s):  
Simplicial Complex ◽  
Infinite Family ◽  
High Dimensional ◽  
Dimensional Simplex ◽  
Virtual Knot ◽  
Virtual Knots ◽  
The Family

Gordian complex of knots was defined by Hirasawa and Uchida as the simplicial complex whose vertices are knot isotopy classes in [Formula: see text]. Later Horiuchi and Ohyama defined Gordian complex of virtual knots using [Formula: see text]-move and forbidden moves. In this paper, we discuss Gordian complex of knots by region crossing change and Gordian complex of virtual knots by arc shift move. Arc shift move is a local move in the virtual knot diagram which results in reversing orientation locally between two consecutive crossings. We show the existence of an arbitrarily high-dimensional simplex in both the Gordian complexes, i.e. by region crossing change and by the arc shift move. For any given knot (respectively, virtual knot) diagram we construct an infinite family of knots (respectively, virtual knots) such that any two distinct members of the family have distance one by region crossing change (respectively, arc shift move). We show that the constructed virtual knots have the same affine index polynomial.

A PARITY AND A MULTI-VARIABLE POLYNOMIAL INVARIANT FOR VIRTUAL LINKS

2013 ◽  
Vol 22 (13) ◽  
pp. 1350073 ◽  
Author(s):  
YOUNG HO IM ◽  
KYOUNG IL PARK
Keyword(s):  
Knot Theory ◽  
Virtual Link ◽  
Polynomial Invariant ◽  
Polynomial Invariants ◽  
Virtual Knot ◽  
Virtual Knots ◽  
Link Diagrams ◽  
Virtual Links ◽  
Knot Diagrams

We introduce a parity of classical crossings of virtual link diagrams which extends the Gaussian parity of virtual knot diagrams and the odd writhe of virtual links that extends that of virtual knots introduced by Kauffman [A self-linking invariants of virtual knots, Fund. Math.184 (2004) 135–158]. Also, we introduce a multi-variable polynomial invariant for virtual links by using the parity of classical crossings, which refines the index polynomial introduced in [Index polynomial invariants of virtual links, J. Knot Theory Ramifications19(5) (2010) 709–725]. As consequences, we give some properties of our invariant, and raise some examples.

scholarly journals FILAMENTATIONS FOR VIRTUAL LINKS

2006 ◽  
Vol 15 (03) ◽  
pp. 327-338 ◽  
Author(s):  
WILLIAM J. SCHELLHORN
Keyword(s):  
Virtual Knot ◽  
Virtual Knots ◽  
Chord Diagrams ◽  
Virtual Links ◽  
Knot Diagrams

In 2002, Hrencecin and Kauffman defined a filamentation invariant on oriented chord diagrams that may determine whether the corresponding flat virtual knot diagrams are non-trivial. A virtual knot diagram is non-classical if its related flat virtual knot diagram is non-trivial. Hence filamentations can be used to detect non-classical virtual knots. We extend these filamentation techniques to virtual links with more than one component. We also give examples of virtual links that they can detect as non-classical.

Classification of genus 1 virtual knots having at most five classical crossings

2014 ◽  
Vol 23 (06) ◽  
pp. 1450031 ◽  
Author(s):  
Akimova Alena Andreevna ◽  
Sergei Vladimirovich Matveev
Keyword(s):  
Virtual Knot ◽  
Virtual Knots ◽  
Kauffman Polynomial ◽  
Thickened Torus ◽  
Knot Diagrams ◽  
Genus One

The goal of this paper is to tabulate all genus one prime virtual knots having diagrams with ≤ 5 classical crossings. First, we construct all nonlocal prime knots in the thickened torus T × I which have diagrams with ≤ 5 crossings and admit no destabilizations. Then we use a generalized version of the Kauffman polynomial to prove that all those knots are different. Finally, we convert the knot diagrams in T thus obtained into virtual knot diagrams in the plane.

scholarly journals Kauffman–Harary conjecture for alternating virtual knots

2015 ◽  
Vol 24 (08) ◽  
pp. 1550046
Author(s):  
Zhiyun Cheng
Keyword(s):  
Affirmative Answer ◽  
Virtual Knot ◽  
Virtual Knots ◽  
Prime Determinant ◽  
Knot Diagrams

In 1999, Kauffman–Harary conjectured that every non-trivial Fox p-coloring of a reduced, alternating knot diagram with prime determinant p is heterogeneous. Ten years later this conjecture was proved by Mattman and Solis. Williamson generalized this conjecture to alternating virtual knots and proved it for certain families of alternating virtual knots. In this paper, by studying the coloring matrices and the determinants of alternating virtual knot diagrams we give an affirmative answer to the Kauffman–Harary conjecture for alternating virtual knots.

Characterization of signed Gauss paragraphs and skew-symmetric graded matrices

2018 ◽  
Vol 27 (01) ◽  
pp. 1850002 ◽  
Author(s):  
José Gregorio Rodríguez-Nieto
Keyword(s):  
Embedded Graphs ◽  
Intersection Pairing ◽  
Virtual Knot ◽  
Virtual Knots ◽  
Virtual Links ◽  
The One ◽  
Minimal Realizations ◽  
Knot Diagrams ◽  
Bracket Polynomials

In this paper, we use theory of embedded graphs on oriented and compact [Formula: see text]-surfaces to construct minimal realizations of signed Gauss paragraphs. We prove that the genus of the ambient surface of these minimal realizations can be seen as a function of the maximum number of Carter’s circles. For the case of signed Gauss words, we use a generating set of [Formula: see text], given in [G. Cairns and D. Elton, The Planarity problem for signed Gauss world, J. Knots Theor. Ramif. 2(4) (1993) 359–367], and the intersection pairing of immersed [Formula: see text]-normal curves to present a short solution of the signed Gauss word problem. We relate this solution with the one given by Cairns and Elton. Moreover, we define the join operation on signed Gauss paragraphs to produce signed Gauss words such that both can be realized on the same minimal genus [Formula: see text]-surface. We connect the characterization of signed Gauss paragraph with the recognition virtual links problem. Also we present a combinatorial algorithm to compute, in an easier way, skew-symmetric graded matrices [V. Turaev, Cobordism of knots on surfaces, J. Topol. 1(2) (2008) 285–305] for virtual knots through the concept of triplets [M. Toro and J. Rodríguez, Triplets associated to virtual knot diagrams, Rev. Integración (2011)]. Therefore, we can prove that the Kishino’s knot is not classical, moreover, we prove that the virtual knots of the family [Formula: see text] given in [H. A. Dye, Virtual knots undetected by [Formula: see text] and [Formula: see text]-strand bracket polynomials, Topol. Appl. 153 (2005) 141–160] are not classical knots.

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