Narrow Results

This work examines the flexural and impact behavior of self-reinforced poly(ethylene terephthalate) (srPET) composites, which were produced by film stacking from fabrics composed of braiding commingled yarns with high-tenacity PET (serving as the reinforcements) and copolymerized PET (mPET) (serving as the matrix). The influence of the hybrid yarns, fabric architectures, and processing conditions on the mechanical properties of srPETs were studied.

In this paper we prove certain Hurwitz equivalence properties of B_n. In particular we prove that for n = 3 every two Artin factorizations of of the form H_i1 ⋯ H_i6, F_j1 ⋯ F_j6 (with i_k, j_k ∈ {1, 2}) where {H₁, H₂}, {F₁, F₂} are frames, are Hurwitz equivalent. The proof provided here is geometric, based on a newly defined frame type.

The results will be applied to the classification of algebraic surfaces up to deformation. It is already known that there exist surfaces that are diffeomorphic to each other but are not deformations of each other (Manetti's example). We construct a new invariant based on Hurwitz equivalence of factorizations, to distinguish among diffeomorphic surfaces which are not deformation of each other. The main result of this paper will help us to compute the new invariant.

A double-torus knot is a knot embedded in a genus two Heegaard surface in S³. We consider double-torus knots L such that is connected, and consider fibred knots in various classes.

In this paper we give new presentations of the braid groups and the pure braid groups of a closed surface. We also give an algorithm to solve the word problem in these groups, using the given presentations.

In this paper, we give a description of the equivariant signature of knots from the symplectic topology point of view. For certain knots K in S³, we define a symplectic Floer homology for the representation space of the knot group π₁ (S³\ K) into SU(2) with trace along all meridians (p is an odd prime and 0<k<p). The symplectic Floer homology of knots is a new invariant of knots and its Euler number is half of the equivariant signature of knots.

Using the new approach of braiding sequences we give a proof of the Lin-Wing conjecture, stating that a Vassiliev invariant ν of degree k has a value O_ν (c(K)^k) on a knot K, where c(K) is the crossing number of K and O_ν depends on ν only. We extend our method to give a quadratic upper bound in k for the crossing number of alternating/positive knots, the values on which suffice to determine uniquely a Vassiliev invariant of degree k. This also makes orientation and mutation sensitivity of Vassiliev invariants decidable by testing them on alternating/positive knots/mutants only.

We give an exponential upper bound for the number of Vassiliev invariants on a special class of closed braids.

A simple complete combinatorial invariant for elements of the braid group is found. It admits some generalizations, e.g. a complete invariant of spherical braids and a complete invariant of cylindrical braids. Values of the invariants are well recognizable, i.e., they provide the complete algorithmic classification of elements in the named braid groups.

We propose a conjecture concerning 3-manifolds (which implies the ℚ conjecture of Myers') and construct a countable sequence of examples which supports it.

We apply the Murasugi–Tristram inequality to real algebraic curves of odd degree in RP² with a deep nest, i.e. a nest of the depth k - 1 where 2k + 1 is the degree. For such curves, the ingredients of the Murasugi–Tristram inequality can be computed (or estimated) inductively using the computations for iterated torus links due to Eisenbud and Neumann as the base case of the induction and Conway's skein relation as the induction step. As an example of applications, we prove that some isotopy types are not realizable by M-curves of degree 9. In Appendix B, we give some generalization of the skein relation for Conway polynomial.

A generalization of the topological fundamental group is developed in order to construct a completion of Artin's braid group on infinitely many strands with respect to the following notion of convergence: b_n → id iff for each M > 0, eventually the first M strands of b_n are trivial.

A left orderable completely metrizable topological group is exhibited containing Artin's braid group on infinitely many strands. The group is the mapping class group (rel boundary) of the closed unit disk with a sequence of interior punctures converging to the boundary. This resolves an issue suggested by work of Dehornoy.

The first author defined a well-order in the set of links by embedding it into a canonical well-ordered set of (integral) lattice points. He also gave elementary transformations among lattice points to enumerate the prime links in terms of lattice points under this order. In this paper, we add some new elementary transformations and explain how to enumerate the prime links. We show a table of the first 443 prime links arising from the lattice points of lengths up to 10 under this order. Our argument enables us to find 7 omissions and 1 overlap in Conway's table of prime links of 10 crossings.

The inverse braid monoid consists of n-string braids from which some subset of the strings has been deleted. Such partial braids may be composed by concatenation followed by the deletion of any incomplete strings, and this operation gives the structure of an inverse monoid. We derive presentations of by first using a groupoid presentation, and then deriving equivalent inverse monoid presentations. In particular, we offer a new derivation of the presentation for that has been given by Easdown and Lavers.

Dehornoy showed that the Artin braid groups B_n are left-orderable. This ordering is discrete, but we show that, for n > 2 the Dehornoy ordering, when restricted to certain natural subgroups, becomes a dense ordering. Among subgroups which arise are the commutator subgroup and the kernel of the Burau representation (for those n for which the kernel is nontrivial). These results follow from a characterization of least positive elements of any normal subgroup of B_n which is discretely ordered by the Dehornoy ordering.

Fix a straight line L in Euclidean 3-space and consider the fibration of the complement of L by half-planes. A generic knot K in the complement of L has neither fiber quadrisecants nor fiber extreme secants such that K touches the corresponding half-plane at 2 points. Both types of secants occur in generic isotopies of knots. We give lower bounds for the number of these fiber secants in all isotopies connecting given isotopic knots. The bounds are expressed in terms of invariants calculable in linear time with respect to the number of crossings.

For any knot K represented as an n-braid (n ≥ 3), we construct an infinite sequence of pairwise non-conjugate (n + 1)-braids {b_m, m ∈ ℕ} representing K.

The braid group B_n, endowed with Artin's presentation, admits two distinguished involutions. One is the anti-automorphism rev : B_n →B_n, , defined by reading braids in the reverse order (from right to left instead of left to right). Another one is the conjugation τ : x ↦ Δ^-1xΔ by the generalized half-twist (Garside element).

More generally, the involution rev is defined for all Artin groups (equipped with Artin's presentation) and the involution τ is defined for all Artin groups of finite type. A palindrome is an element invariant under rev. We study palindromes and palindromes invariant under τ in Artin groups of finite type. Our main results are the injectivity of the map in all finite-type Artin groups, the existence of a left-order compatible with rev for Artin groups of type A, B, D, and the existence of a decomposition for general palindromes. The uniqueness of the latter decomposition requires that the Artin groups carry a left-order.

Braid groups and mapping class groups have many features in common. Similarly to the notion of inverse braid monoid, inverse mapping class monoid is defined. It concerns surfaces with punctures, but among given n punctures, several can be omitted. This corresponds to braids where the number of strings is not fixed. In the paper we give the analogue of the Dehn–Nilsen–Baer theorem, propose a presentation of the inverse mapping class monoid for a punctured sphere and study the word problem. This shows that certain properties and objects based on mapping class groups may be extended to the inverse mapping class monoids. We also give analogues of Artin presentation with two generators.

In this paper we examine the relationship between various types of positivity for knots and the concordance invariant τ discovered by Ozsváth and Szabó and independently by Rasmussen. The main result shows that, for fibered knots, τ characterizes strong quasipositivity. This is quantified by the statement that for K fibered, τ(K) = g(K) if and only if K is strongly quasipositive. A corollary is that any knot admitting a lens space surgery or, more generally, an L-space surgery, is strongly quasipositive. In addition, we survey existing results regarding τ and forms of positivity and highlight several consequences concerning the types of knots which are (strongly) (quasi) positive.

We consider the problem of classification of links up to (2, 2)-moves. Our motivation comes from the theory of skein modules, more specifically from the skein module of S³ based on the deformation of (2, 2)-move. As it was proved in D-P-2, not every link can be reduced to a trivial link by (2, 2)-moves, for instance, the closure of (σ₁σ₂)⁶. In this paper, we classify 3-braids up to (2, 2)-moves and, we show how the Harikae–Nakanishi–Uchida conjecture can be modified to hold for closed 3-braids. As an important step in the classification we prove the conjecture for 2-algebraic links and classify (2, 2)-equivalence classes for links up to nine crossings. We also analyze an action of (2, 2)-move on Kei (involutive quandle) associated to a link diagram. We define Burnside Kei, Q(m, n), and ask for which values of m and n, is Q(m, n) finite. This question is motivated by classical Burnside problem.