scholarly journals Knot theory and matrix integrals

2018 ◽  
Author(s):  
Jeremie Bouttier
Keyword(s):  
Knot Theory ◽  
Virtual Knots ◽  
Quartic Potential ◽  
Link Diagrams ◽  
Large Size ◽  
Virtual Links ◽  
Matrix Integrals ◽  
Higher Genus ◽  
Enumeration Problems ◽  
Alternating Links

This article considers some enumeration problems in knot theory, with a focus on the application of matrix integral techniques. It first reviews the basic definitions of knot theory, paying special attention to links and tangles, especially 2-tangles, before discussing virtual knots and coloured links as well as the bare matrix model that describes coloured link diagrams. It shows how the large size limit of matrix integrals with quartic potential may be used to count alternating links and tangles. The removal of redundancies amounts to renormalization of the potential. This extends into two directions: first, higher genus and the counting of ‘virtual’ links and tangles, and second, the counting of ‘coloured’ alternating links and tangles. The article analyses the asymptotic behaviour of the number of tangles as the number of crossings goes to infinity

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.

Parities and polynomial invariants for virtual links

2014 ◽  
Vol 23 (12) ◽  
pp. 1450066 ◽  
Author(s):  
Young Ho Im ◽  
Kyoung Il Park ◽  
Mi Hwa Shin
Keyword(s):  
Knot Theory ◽  
Virtual Link ◽  
Polynomial Invariant ◽  
Polynomial Invariants ◽  
Virtual Knots ◽  
Link Diagrams ◽  
Virtual Links ◽  
Kauffman Polynomial

We introduce the odd Jones–Kauffman polynomial and odd Miyazawa polynomials of virtual link diagrams by using the parity of virtual link diagrams given in [Y. H. Im and K. I. Park, A parity and a multi-variable polynomial invariant for virtual links, J. Knot Theory Ramifications22(13) (2013), Article ID: 1350073, 18pp.], which are different from the original Jones–Kauffman and Miyazawa polynomials. Also, we give a family of parities and odd polynomials for virtual knots so that many virtual knots can be distinguished.

GENERALIZED INDEX POLYNOMIALS FOR VIRTUAL LINKS

2012 ◽  
Vol 21 (14) ◽  
pp. 1250128
Author(s):  
KYEONGHUI LEE ◽  
YOUNG HO IM
Keyword(s):  
Knot Theory ◽  
Recursive Method ◽  
Polynomial Invariant ◽  
Polynomial Invariants ◽  
Virtual Knots ◽  
Virtual Links

We construct some polynomial invariants for virtual links by the recursive method, which are different from the index polynomial invariant defined in [Y. H. Im, K. Lee and S. Y. Lee, Index polynomial invariant of virtual links, J. Knot Theory Ramifications19(5) (2010) 709–725]. We show that these polynomials can distinguish whether virtual knots can be invertible or not although the index polynomial cannot distinguish the invertibility of virtual knots.

scholarly journals A POLYNOMIAL INVARIANT OF VIRTUAL LINKS

2013 ◽  
Vol 22 (12) ◽  
pp. 1341002 ◽  
Author(s):  
ZHIYUN CHENG ◽  
HONGZHU GAO
Keyword(s):  
Knot Theory ◽  
Polynomial Invariant ◽  
Polynomial Invariants ◽  
Virtual Knots ◽  
The Third ◽  
Linking Number ◽  
Virtual Links ◽  
Definition Of ◽  
Knots And Links

In this paper, we define some polynomial invariants for virtual knots and links. In the first part we use Manturov's parity axioms [Parity in knot theory, Sb. Math.201 (2010) 693–733] to obtain a new polynomial invariant of virtual knots. This invariant can be regarded as a generalization of the odd writhe polynomial defined by the first author in [A polynomial invariant of virtual knots, preprint (2012), arXiv:math.GT/1202.3850v1]. The relation between this new polynomial invariant and the affine index polynomial [An affine index polynomial invariant of virtual knots, J. Knot Theory Ramification22 (2013) 1340007; A linking number definition of the affine index polynomial and applications, preprint (2012), arXiv:1211.1747v1] is discussed. In the second part we introduce a polynomial invariant for long flat virtual knots. In the third part we define a polynomial invariant for 2-component virtual links. This polynomial invariant can be regarded as a generalization of the linking number.

scholarly journals A twisted link invariant derived from a virtual link invariant

2017 ◽  
Vol 26 (09) ◽  
pp. 1743007
Author(s):  
Naoko Kamada
Keyword(s):  
Knot Theory ◽  
Orientable Surface ◽  
Link Diagram ◽  
Link Invariant ◽  
Virtual Knot ◽  
Chord Diagrams ◽  
Link Diagrams ◽  
Virtual Links ◽  
Virtual Knot Theory ◽  
Double Coverings

Virtual knot theory is a generalization of knot theory which is based on Gauss chord diagrams and link diagrams on closed oriented surfaces. A twisted knot is a generalization of a virtual knot, which corresponds to a link diagram on a possibly non-orientable surface. In this paper, we discuss an invariant of twisted links which is obtained from the JKSS invariant of virtual links by use of double coverings. We also discuss some properties of double covering diagrams.

scholarly journals The Alexander polynomial for virtual twist knots

2017 ◽  
Vol 26 (01) ◽  
pp. 1750007
Author(s):  
Isaac Benioff ◽  
Blake Mellor
Keyword(s):  
Knot Theory ◽  
Recursive Formula ◽  
Alexander Polynomial ◽  
Polynomial Invariants ◽  
Virtual Knot ◽  
Virtual Knots ◽  
Virtual Links ◽  
Polynomial Formula

We define a family of virtual knots generalizing the classical twist knots. We develop a recursive formula for the Alexander polynomial [Formula: see text] (as defined by Silver and Williams [Polynomial invariants of virtual links, J. Knot Theory Ramifications 12 (2003) 987–1000]) of these virtual twist knots. These results are applied to provide evidence for a conjecture that the odd writhe of a virtual knot can be obtained from [Formula: see text].

scholarly journals GRAPH-VALUED INVARIANTS OF VIRTUAL AND CLASSICAL LINKS AND MINIMALITY PROBLEM

2013 ◽  
Vol 22 (12) ◽  
pp. 1341006 ◽  
Author(s):  
VLADIMIR ALEKSANDROVICH KRASNOV ◽  
VASSILY OLEGOVICH MANTUROV
Keyword(s):  
Knot Theory ◽  
Classical Case ◽  
Strong Sense ◽  
Virtual Knot ◽  
Virtual Knots ◽  
Crucial Difference ◽  
Virtual Links ◽  
Two Component ◽  
The Rich ◽  
Virtual Knot Theory

The Kuperberg bracket is a well-known invariant of classical links. Recently, the second named author and Kauffman constructed the graph-valued generalization of the Kuperberg bracket for the case of virtual links: unlike the classical case, the invariant in the virtual case is valued in graphs which carry a significant amount of information about the virtual knot. The crucial difference between virtual knot theory and classical knot theory is the rich topology of the ambient space for virtual knots. In a paper by Chrisman and the second named author, two-component classical links with one fibered component were considered; the complement to the fibered component allows one to get highly non-trivial ambient topology for the other component. In this paper, we combine the ideas of the above mentioned papers and construct the "virtual" Kuperberg bracket for two-component links L = J ⊔ K with one component (J) fibered. We consider a new geometrical complexity for such links and establish minimality of diagrams in a strong sense. Roughly speaking, every other "diagram" of the knot in question contains the initial diagram as a subdiagram. We prove a sufficient condition for minimality in a strong sense where minimality cannot be established as introduced in the paper by Chrisman and the second named author.

scholarly journals MATRIX INTEGRALS AND THE GENERATION AND COUNTING OF VIRTUAL TANGLES AND LINKS

2004 ◽  
Vol 13 (03) ◽  
pp. 325-355 ◽  
Author(s):  
PAUL ZINN-JUSTIN ◽  
JEAN-BERNARD ZUBER
Keyword(s):  
Asymptotic Behavior ◽  
Riemann Surface ◽  
Generating Functions ◽  
Equivalence Classes ◽  
Feynman Diagrams ◽  
Complex Matrices ◽  
Virtual Links ◽  
Matrix Integrals ◽  
Thickened Surface ◽  
Alternating Links

Virtual links are generalizations of classical links that can be represented by links embedded in a "thickened" surface Σ×I, product of a Riemann surface of genus h with an interval. In this paper, we show that virtual alternating links and tangles are naturally associated with the 1/N2 expansion of an integral over N×N complex matrices. We suggest that it is sufficient to count the equivalence classes of these diagrams modulo ordinary (planar) flypes. To test this hypothesis, we use an algorithm coding the corresponding Feynman diagrams by means of permutations that generates virtual diagrams up to 6 crossings and computes various invariants. Under this hypothesis, we use known results on matrix integrals to get the generating functions of virtual alternating tangles of genus 1 to 5 up to order 10 (i.e. 10 real crossings). The asymptotic behavior for n large of the numbers of links and tangles of genus h and with n crossings is also computed for h=1,2,3 and conjectured for general h.

scholarly journals Alexander and writhe polynomials for virtual knots

2016 ◽  
Vol 25 (08) ◽  
pp. 1650050 ◽  
Author(s):  
Blake Mellor
Keyword(s):  
Knot Theory ◽  
Alexander Polynomial ◽  
Polynomial Invariant ◽  
Polynomial Invariants ◽  
Virtual Knot ◽  
Virtual Knots ◽  
Virtual Links ◽  
New Interpretation ◽  
Polynomial Formula ◽  
Knots And Links

We give a new interpretation of the Alexander polynomial [Formula: see text] for virtual knots due to Sawollek [On Alexander–Conway polynomials for virtual knots and Links, preprint (2001), arXiv:math/9912173] and Silver and Williams [Polynomial invariants of virtual links, J. Knot Theory Ramifications 12 (2003) 987–1000], and use it to show that, for any virtual knot, [Formula: see text] determines the writhe polynomial of Cheng and Gao [A polynomial invariant of virtual links, J. Knot Theory Ramifications 22(12) (2013), Article ID: 1341002, 33pp.] (equivalently, Kauffman’s affine index polynomial [An affine index polynomial invariant of virtual knots, J. Knot Theory Ramifications 22(4) (2013), Article ID: 1340007, 30pp.]). We also use it to define a second-order writhe polynomial, and give some applications.

ON EXTREME COEFFICIENTS OF THE JONES–KAUFFMAN POLYNOMIAL FOR VIRTUAL LINKS

2006 ◽  
Vol 15 (07) ◽  
pp. 853-868
Author(s):  
ROMAN S. AVDEEV
Keyword(s):  
Knot Theory ◽  
Classical Case ◽  
Coefficient Function ◽  
Planar Curves ◽  
Vassiliev Invariants ◽  
Virtual Knots ◽  
Chord Diagrams ◽  
Virtual Links ◽  
Kauffman Polynomial

An important problem of knot theory is to find or estimate the extreme coefficients of the Jones–Kauffman polynomial for (virtual) links with a given number of classical crossings. This problem has been studied by Morton and Bae [1] and Manchón [11] for the case of classical links. It turns out that the general case can be reduced to the case when the extreme coefficient function is expressible in terms of chord diagrams (previous authors consider only d-diagrams which correspond to the classical case [9]). We find the maximal absolute values for generic chord diagrams, thus, for generic virtual knots. Also we consider the "next" coefficient of the Jones–Kauffman polynomial in terms of framed chord diagrams and find its maximal value for a given number of chords. These two functions on chord diagrams are of their own interest because there are related to the Vassiliev invariants of classical knots and J-invariants of planar curves, as mentioned in [10].

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