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Quotients of the Gordian and H(2)-Gordian graphs
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Abstract
The Gordian graph and H(2)-Gordian graphs of knots are abstract graphs whose vertex sets represent isotopy classes of unoriented knots, and whose edge sets record whether pairs of knots are related by crossing changes or H(2)-moves, respectively. We investigate quotients of these graphs under equivalence relations defined by several knot invariants including the determinant, the span of the Jones polynomial, and an invariant related to tricolorability. We show, in all cases considered, that the quotient graphs are Gromov hyperbolic. We then prove a collection of results about the graph isomorphism type of the quotient graphs. In particular, we find that the H(2)-Gordian graph of links modulo the relation induced by the span of the Jones polynomial is isomorphic with the complete graph on infinitely many vertices.
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We define a metric filtration of the Gordian graph by an infinite family of 1-dense subgraphs. The n-th subgraph of this family is generated by all knots whose fundamental groups surject to a symmetric group with parameter at least n, where all meridians are mapped to transpositions. Incidentally, we verify the Meridional Rank Conjecture for a family of knots with unknotting number one yet arbitrarily high bridge number.
M.Hirasawa and Y.Uchida defined the Gordian complex of knots which is a simplicial complex whose vertices consist of all knot types in S3 by using "a crossing change". In this paper, we define the Ck-Gordian complex of knots which is an extension of the Gordian complex of knots. Let k be a natural number more than 2 and we show that for any knot K0 and any given natural number n, there exists a family of knots {K0,K1,...,Kn} such that for any pair (Ki,Kj) of distinct elements of the family, the Ck-distance dCk(Ki,Kj) = 1.
After the works of Kauffman-Banchoff and Yamasaki, it is known that a local move
called the pass move is strongly related to the Arf invariant, which is
equivalent to the parity of the coefficient of the degree two term in the Conway
polynomial. Our main result is the following: There exists a pair of knots such
that their Conway polynomials coincide, and that the sets of Conway polynomials
of knots obtained from them by a single pass move do not coincide.
A braid defines a link which has a signature. This defines a map from the braid group to the integers which is not a bomomorphism. We relate the homornorphism defect of this map to Meyer cocycle and Maslov class. We give some information about the global geometry of the gordian metric space.
We introduce new simplicial complexes by using various invariants and local moves for knots, which give generalizations of the Gordian complex defined by Hirasawa and Uchida. In particular, we focus on the simplicial complex defined by using the Alexander-Conway polynomial and the Delta-move, and show that the simplicial complex is Gromov hyperbolic and quasi-isometric to the real line. Comment: 9 pages, 6 figures
In this paper, we build a bridge between Conway–Coxeter friezes (CCFs) and rational tangles through the Kauffman bracket polynomials. One can compute a Kauffman bracket polynomial attached to rational links by using CCFs. As an application, one can give a complete invariant on CCFs of zigzag-type.
In this paper, the [Formula: see text]-move is defined. We show that for any knot [Formula: see text], there exists an infinite family of knots [Formula: see text] such that the Gordian distance [Formula: see text] and pass-move-Gordian distance [Formula: see text] for any [Formula: see text]. We also show that there is another infinite family of knots [Formula: see text] (where [Formula: see text]) such that the [Formula: see text]-move-Gordian distance [Formula: see text] and [Formula: see text]-Gordian distance [Formula: see text] for any [Formula: see text] and all [Formula: see text].
In this paper, the (Formula presented.)-Gordian complex of knots is defined. The authors show that for any knot (Formula presented.) and any positive integer (Formula presented.), there is a finite set of knots of size (Formula presented.) containing (Formula presented.), such that the (Formula presented.)-Gordian distance between any two knots in this set is 1 for all (Formula presented.).
We analyze various local moves on knot diagrams to show that Jones polynomials must have certain algebraic properties. In particular, we show that the Jones polynomial of a knot cannot be a nontrivial monomial.
Some evaluations of the Gordian distance from a knot to another knot are given by using three polynomial invariants called the HOMFLY polynomial, the Jones polynomial and the Q-polynomial. Furthermore, Gordian distances between a lot of pairs of knots are determined.
We define the Gordian complex of knots, which is a simplicial complex whose vertices consist of all oriented knot types in the 3-sphere. We show that for any knot K, there exists an infinite family of distinct knots containing K such that for any pair (K i ,K j ) of members of the family, the Gordian distance d G (K i ,K j )=1.
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A NEW combinatorial formulation of the Jones polynomial of a link is used to establish some basic properties of this polynomial. A striking consequence of these properties is the result that a link admitting an alternating diagram with m crossings and with no “nugatory” crossing cannot be projected with fewer than m crossings.
For each k⩾2, we exhibit infinite families of prime k-component links with Jones polynomial equal to that of the k-component unlink.
We consider a relation between two kinds of unknotting numbers defined by
using a band surgery on unoriented knots; the band-unknotting number and
H(2)-unknotting number, which we may characterize in terms of the first Betti
number of surfaces in S^3 spanning the knot and the trivial knot. We also give
several examples for these numbers.
My research area is applications of topology to biology, especially DNA topology. DNA topology studies the shape and path of DNA in three dimensional space. My thesis relates to the study of DNA topology in a protein-DNA complex by solving tangle equations and calculating distances between DNA knots.
- Stanislav Jabuka
- Beibei Liu
- Allison H Moore
Stanislav Jabuka, Beibei Liu, and Allison H. Moore. Knot graphs and Gromov hyperbolicity.
arXiv:1912.03766v1 [math.GT], 2020.
The Jones polynomial. Unpublished note
- F R Vaughan
- Jones
Vaughan F. R. Jones. The Jones polynomial. Unpublished note, 8 2005.
Knotinfo: Table of knot invariants. URL: knotinfo.math
- Charles Livingston
- Allison H Moore
Charles Livingston and Allison H. Moore. Knotinfo:
Table of knot invariants. URL:
knotinfo.math.indiana.edu, June 2020.