Braid Index Research Articles

On the Semi-threading of Knot Diagrams with Minimal Overpasses

Given a knot diagram D, we construct a semi-threading circle of it which can be an axis of D as a closed braid depending on knot diagrams. In particular, we consider semi-threading circles of minimal diagrams of a knot with respect to overpasses which give us some information related to the braid index. By this notion, we try to give another proof of the fact that, for every nontrivial knot K, the braid index b(K) of K is not less than the minimum number l(K) of overpasses of diagrams. Also, they are the same for a torus knot.

Volume bounds for generalized twisted torus links

Twisted torus knots and links are given by twisting adjacent strands of a torus link. They are geometrically simple and contain many examples of the smallest volume hyperbolic knots. Many are also Lorenz links. We study the geometry of twisted torus links and related generalizations. We determine upper bounds on their hyperbolic volumes that depend only on the number of strands being twisted. We exhibit a family of twisted torus knots for which this upper bound is sharp, and another family with volumes approaching infinity. Consequently, we show there exist twisted torus knots with arbitrarily large braid index and yet bounded volume.

Braid Presentation of Spatial Graphs

We define braid presentation of edge-oriented spatial graphs as a natural generalization of braid presentation of oriented links. We show that every spatial graph has a braid presentation. For an oriented link, it is known that the braid index is equal to the minimal number of Seifert circles. We show that an analogy does not hold for spatial graphs.

Conjugacy Classes of the 3-Braid Group

In this article, we describe the summit sets in the 3-braid group, the smallest element in a summit set, and we compute the Hilbert series corresponding to conjugacy classes. The results are related to the Birman–Menasco classification of knots with braid index three or less than three.

S-graphs and braid index of links

We study the relationship between reduction operations on link diagrams and S-graphs associated with them. We are motivated by the problem of computing the braid index of a link and some well known conjectures concerning the braid index of a link and the writhe of its diagrams. Possible counterexamples are discussed in terms of both S-graphs and link diagrams. We also indicate the relation of S-graphs to singular links regarded up to an appropriate equivalence relation.

THE DEALTERNATING NUMBER AND THE ALTERNATION NUMBER OF A CLOSED 3-BRAID

We give an upper bound for the dealternating number of a closed 3-braid. As applications, we determine the dealternating numbers, the alternation numbers and the Turaev genera of some closed positive 3-braids. We also show that there exist infinitely many positive knots with any dealternating number (or any alternation number) and any braid index.

A NOTE ON THE CROSSING NUMBER AND THE BRAID INDEX FOR VIRTUAL LINKS

It is well known that any virtual link is described as the closure of a virtual braid. Therefore, we can define the virtual braid index. Ohyama proved an inequality for the crossing number and the braid index of a classical link. In this paper, we prove an analogous inequality for the (total) crossing number and the braid index of a virtual link.

Fibered Transverse Knots and the Bennequin Bound

We prove that a nicely fibered link (by which we mean the binding of an open book) in a tight contact manifold (M, ξ) with zero Giroux torsion has a transverse representative realizing the Bennequin bound if and only if the contact structure it supports (since it is also the binding of an open book) is ξ. This gives a geometric reason for the non-sharpness of the Bennequin bound for fibered links. We also give the classification, up to contactomorphism, of maximal self-linking number links in these knot types. Moreover, in the standard tight contact structure on S3 we classify, up to transverse isotopy, transverse knots with maximal self-linking number in the knot types given as closures of positive braids and given as fibered strongly quasi-positive knots. We derive several braid theoretic corollaries from this. In particular, we give conditions under which quasi-positive braids are related by positive Markov stabilizations and when a minimal braid index representative of a knot is quasi-positive. Finally, we observe that our main result can be used to show, and make rigorous the statement, that contact structures on a given manifold are in a strong sense classified by the transverse knot theory they support.

The spiral index of knots

AbstractIn this paper, we introduce two new invariants that are closely related to Milnor's curvature-torsion invariant. The first, a particularly natural invariant called the spiral index of a knot, captures the number of local maxima in a knot projection that is free of inflection points. This invariant is sandwiched between the bridge and braid index of a knot, and captures more subtle properties. The second invariant, the projective superbridge index, provides a method of counting the greatest number of local maxima that occur in a given projection. In addition to investigating the relationships among these invariants, we use them to classify all those knots for which Milnor's curvature-torsion invariant is 6π.

TABULATING AND DISTINGUISHING MUTANTS

We explain progress in computing the cabled Jones, HOMFLY and Kauffman polynomial. This is applied, first, in combination with some group theoretic considerations, to the tabulation of low-crossing mutants. Then we study the distinction of mutants, with particular regard to the symmetric mutants. We discuss the determination of braid index as another application of our computational methods.

The complexity of Platonic and Archimedean polyhedral links

A mathematical methodology for understanding the complexity of Platonic and Archimedean polyhedral links has been developed based on some topological invariants from knot theory. Knot invariants discussed here include crossing number, unknotting number, genus and braid index, which are considered significant in view of DNA nanotechnology. Our results demonstrate that the braid index provides the most structural information; hence, it can be used, among four knot invariants, as the most useful complexity measure. Using such an invariant, it indicates that the complexity of polyhedral links is directed by the number of their building blocks. The research introduces a simple but important concept in the theoretical characterization and analysis of DNA polyhedral catenanes.

Braiding a link with a fixed closed braid

We show that every oriented link diagram with a closed braid diagram as a sublink diagram can be deformed into a closed braid diagram by a deformation keeping the sublink diagram and, under a mild condition, the number of Seifert circles fixed. As an application, we give an upper bound for the braid index of the link obtained by reversing the orientation of its sublink by using only the information of an original link.

A formula for the braid index of links

Morton and Franks–Williams independently gave a lower bound for the braid index b ( L ) of a link L in S 3 in terms of the v-span of the Homfly-pt polynomial P L ( v , z ) of L: 1 2 span v P L ( v , z ) + 1 ⩽ b ( L ) . Up to now, many classes of knots and links satisfying the equality of this Morton–Franks–Williams's inequality have been founded. In this paper, we give a new such a class K of knots and links and make an explicit formula for determining the braid index of knots and links that belong to the class K . This gives simultaneously a new class of knots and links satisfying the Jones conjecture which says that the algebraic crossing number in a minimal braid representation is a link invariant. We also give an algorithm to find a minimal braid representative for a given knot or link in K .

Khovanov-Rozansky homology and the braid index of a knot

We construct a knot whose braid index is not detected by the Morton-Franks-Williams (MFW) inequality but is detected by a related KR-MFW inequality that comes from the Khovanov-Rozansky homology. We also construct infinitely many knots whose braid indices are not detected by the KR-MFW inequality.

A new twist on Lorenz links

Twisted torus links are given by twisting a subset of strands on a closed braid representative of a torus link. T--links are a natural generalization, given by repeated positive twisting. We establish a one-to-one correspondence between positive braid representatives of Lorenz links and T--links, so Lorenz links and T--links coincide. Using this correspondence, we identify over half of the simplest hyperbolic knots as Lorenz knots. We show that both hyperbolic volume and the Mahler measure of Jones polynomials are bounded for infinite collections of hyperbolic Lorenz links. The correspondence provides unexpected symmetries for both Lorenz links and T-links, and establishes many new results for T-links, including new braid index formulas.

Defining and measuring braiding intensity

AbstractGeomorphological studies of braided rivers still lack a consistent measurement of the complexity of the braided pattern. Several simple indices have been proposed and two (channel count and total sinuosity) are the most commonly applied. For none of these indices has there been an assessment of the sampling requirements and there has been no systematic study of the equivalence of the indices to each other and their sensitivity to river stage. Resolution of these issues is essential for progress in studies of braided morphology and dynamics at the scale of the channel network.A series of experiments was run using small‐scale physical models of braided rivers in a 3 m ∞ 20 m flume. Sampling criteria for braid indices and their comparability were assessed using constant‐discharge experiments. Sample hydrographs were run to assess the effect of flow variability.Reach lengths of at least 10 times the average wetted width are needed to measure braid indices with precision of the order of 20% of the mean. Inherent variability in channel pattern makes it difficult to achieve greater precision. Channel count indices need a minimum of 10 cross‐sections spaced no further apart than the average wetted width of the river. Several of the braid indices, including total sinuosity, give very similar numerical values but they differ substantially from channel‐count index values. Consequently, functional relationships between channel pattern and, for example, discharge, are sensitive to the choice of braid index. Braid indices are sensitive to river stage and the highest values typically occur below peak flows of a diurnal (melt‐water) hydrograph in pro‐glacial rivers. There is no general relationship with stage that would allow data from rivers at different relative stage to be compared. At present, channel count indices give the best combination of rapid measurement, precision, and range of sources from which measurements can be reliably made. They can also be related directly to bar theory for braided pattern development. Copyright © 2008 John Wiley & Sons, Ltd.

A NOTE ON CLOSED 3-BRAIDS

This is a review article about knots and links of braid index 3. Its goal is to gather together, in one place, some of the tools that are special to knots and links of braid index 3, in a form that could be useful for those who have a need to calculate, and need to know precisely all the exceptional cases.

Heegaard Floer homology and genus one, one-boundary component open books

We compute the Heegaard Floer homology of any rational homology 3-sphere with an open book decomposition of the form (T, ϕ), where T is a genus one surface with one-boundary component. In addition, we compute the Heegaard Floer homology of every T2-bundle over S1 with first Betti number equal to 1, and we compare our results with those of Lebow on the embedded contact homology of such torus bundles. We use these computations to place restrictions on Stein-fillings of the contact structures compatible with such open books, to narrow down somewhat the class of 3-braid knots with finite concordance order, and to identify all quasi-alternating links with braid index at most 3.

A property of the skein polynomial with an application to contact geometry

We prove a finiteness property of the values of the skein polynomial of homogeneous knots which allows to establish large classes of such knots to have arbitrarily unsharp Bennequin inequality (for the Thurston-Bennequin invariant of any of their Legendrian embeddings in the standard contact structure of R^3), and a give a short proof that there are only finitely many among these knots that have given genus and given braid index.

Invariants of knots, surfaces in R3, and foliations

We give a survey of some known results related to combinatorial and geometric properties of finite-order invariants of knots in a three-dimensional space. We study the relationship between Vassiliev invariants and some classical numerical invariants of knots and point out the role of surfaces in the investigation of these invariants. We also consider combinatorial and geometric properties of essential tori in standard position in closed braid complements by using the braid foliation technique developed by Birman, Menasco, and other authors. We study the reductions of link diagrams in the context of finding the braid index of links.