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There are infinitely many pretzel links with the same Alexander polynomial (actually with trivial Alexander polynomial). By contrast, in this note we revisit the Jones polynomial of pretzel links and prove that, given a natural number S, there is only a finite number of pretzel links whose Jones polynomials have span S. More concretely, we provide an algorithm useful for deciding whether or not a given knot is pretzel. As an application we identify all the pretzel knots up to nine crossings, proving in particular that $8_{12}$ is the first non-pretzel knot., Comment: 31 pages, 6 figures, 2 tables, 1 appendix

The groups $G_n^k$ were defined by V. O. Manturov in order to describe dynamical systems in configuration systems. In the paper we consider two applications of this theory: we define a biquandle structure on the groups $G_n^k$, and construct a homomorphism from the surface singular braid monoid to the group $G_n^2$.

We study a local twist move on welded knots that is an analog of the virtualization move on virtual knots. Since this move is an unknotting operation we define an invariant, unknotting twist number, for welded knots. We relate the unknotting twist number with warping degree and welded unknotting number, and establish a lower bound on the twist number using Alexander quandle coloring. We also study the Gordian distance between welded knots by twist move and define the corresponding Gordian complex., Comment: 18 pages, 25 figures

The spaces of triangulations of a given manifold have been widely studied. The celebrated theorem of Pachner~\cite{Pachner} says that any two triangulations of a given manifold can be connected by a sequence of bistellar moves, or Pachner moves, see also~\cite{GKZ,Nabutovsky}. In the present paper we consider groups which naturally appear when considering the set of triangulations with fixed number of simplices of maximal dimension. There are three ways of introducing this groups: the geometrical one, which depends on the metric, the topological one, and the combinatorial one. The second one can be thought of as a ``braid group'' of the manifold and, by definition, is an invariant of the topological type of manifold; in a similar way, one can construct the smooth version. We construct a series of groups $\Gamma_{n}^{k}$ corresponding to Pachner moves of $(k-2)$-dimensional manifolds and construct a canonical map from the braid group of any $k$-dimensional manifold to $\Gamma_{n}^{k}$ thus getting topological/smooth invariants of these manifolds., Comment: arXiv admin note: substantial text overlap with arXiv:1905.08049 authors' note: arXiv:1905.08049 is a survey paper which includes the results of the present paper

We define the Wirtinger width of a knot. Then we prove the Wirtinger width of a knot equals its Gabai width. The algorithmic nature of the Wirtinger width leads to an efficient technique for establishing upper bounds on Gabai width. As an application, we use this technique to calculate the Gabai width of approximately 50000 tabulated knots., 17 pages, 9 Figures

The non-orientable 4-genus of a knot in the 3-sphere is defined as the smallest first Betti number of any non-orientable surface smoothly and properly embedded in the 4-ball, with boundary the given knot. We compute the non-orientable 4-genus for all knots with crossing number 8 or 9. As applications we prove a conjecture of Murakami's and Yasuhara's, and give a new lower bound for the slicing number of knot., Fixed two values of $\gamma_4$ arising from a typographical error. Updated references. Added Journal Ref

We introduce colorings of oriented surface-links by biquasiles using marked graph diagrams. We use these colorings to define counting invariants and Boltzmann enhancements of the biquasile counting invariants for oriented surface-links. We provide examples to show that the invariants can distinguish both closed surface-links and cobordisms and are sensitive to orientation., 12 pages.To appear in Topology and its Applications

In this paper, we compute the slice genus for many low-crossing virtual knots. For instance, we show that 1295 out of 92800 virtual knots with 6 or fewer crossings are slice, and that all but 248 of the rest are not slice. Key to these results are computations of Turaev's graded genus, which we show extends to give an invariant of virtual knot concordance. The graded genus is remarkably effective as a slice obstruction, and we develop an algorithm that applies virtual unknotting operations to determine the slice genus of many virtual knots with 6 or fewer crossings., 26 pages, many figures, supplementary dataset available online at https://micah46.wixsite.com/micahknots/slicegenus

We calculate the alternating number of torus knots with braid index 4 and less. For the lower bound, we use the upsilon-invariant recently introduced by Ozsv\'ath, Stipsicz, and Szab\'o. For the upper bound, we use a known bound for braid index $3$ and a new bound for braid index $4$. Both bounds coincide, so that we obtain a sharp result., Comment: 6 color figures

We consider the question of which Dehn surgeries along a given knot bound rational homology balls. We use Ozsv\'ath and Szab\'o's correction terms in Heegaard Floer homology to obtain general constraints on the surgery coefficients. We then turn our attention to the case of integral surgeries, with particular emphasis on positive torus knots. Finally, combining these results with a lattice-theoretic obstruction based on Donaldon's theorem, we classify which integral surgeries along torus knots of the form $T_{kq\pm 1,q}$ bound rational homology balls., Comment: 32 pages, many figures. Several minor improvements and corrections. Comments are welcome!

We define a group-valued invariant of virtual knots and relate it to various other group-valued invariants of virtual knots, including the extended group of Silver-Williams and the quandle group of Manturov and Bardakov-Bellingeri. A virtual knot is called almost classical if it admits a diagram with an Alexander numbering, and in that case we show that the group factors as a free product of the usual knot group and Z. We establish a similar formula for mod p almost classical knots, and we use these results to derive obstructions to a virtual knot K being mod p almost classical. Viewed as knots in thickened surfaces, almost classical knots correspond to those that are homologically trivial. We show they admit Seifert surfaces and relate their Alexander invariants to the homology of the associated infinite cyclic cover. We prove the first Alexander ideal is principal, recovering a result first proved by Nakamura et al. using different methods. The resulting Alexander polynomial is shown to satisfy a skein relation, and its degree gives a lower bound for the Seifert genus. We tabulate almost classical knots up to 6 crossings and determine their Alexander polynomials and virtual genus., Comment: 44 pages

Region crossing change is a local operation on link diagrams. The behavior of region crossing change on $S^2$ is well understood. In this paper, we study the behavior of (modified) region crossing change on higher genus surfaces., 20 pages, 13 figures

We consider diagrams of links in $S^2$ obtained by projection from $S^3$ with the Hopf map and the minimal crossing number for such diagrams. Knots admitting diagrams with at most one crossing are classified. Some properties of these knots are exhibited. In particular, we establish which of these knots are algebraic and, for such knots, give an answer to a problem posed by Fiedler., 25 pages, 9 figures, accepted to be published in the Osaka Journal of Mathematics

The $\nu^+$-equivalence is an equivalence relation on the knot concordance group. This relation can be seen as a certain stable equivalence on knot Floer complexes $CFK^{\infty}$, and many concordance invariants derived from Heegaard Floer theory are invariant under the equivalence. In this paper, we show that any genus one knot is $\nu^+$-equivalent to one of the trefoil, its mirror and the unknot., Comment: 46 pages, 8 figures;(v2)typos corrected

In links with two components there are three different types of crossings: self-crossings in the first component, self crossings in the second component, and crossings between components. In this paper we examine the minimum number of crossing changes needed to unlink without changing the crossings between components. We restrict our attention to unlinking two component links with linking number zero and both components unknotted. We provide data for links with no more than ten crossings and general results about asymmetry of unlinking between components., 10 pages, 12 figures

If a knot $K$ in $S^3$ admits a pair of truly cosmetic surgeries, we show that the surgery slopes are either $\pm 2$ or $\pm 1/q$ for some value of $q$ that is explicitly determined by the knot Floer homology of $K$. Moreover, in the former case the genus of $K$ must be two, and in the latter case there is bound relating $q$ to the genus and the Heegaard Floer thickness of $K$. As a consequence, we show that the cosmetic crossing conjecture holds for alternating knots (or more generally, Heegaard Floer thin knots) with genus not equal to two. We also show that the conjecture holds for any knot $K$ for which each prime summand of $K$ has at most 16 crossings; our techniques rule out cosmetic surgeries in this setting except for slopes $\pm 1$ and $\pm 2$ on a small number of knots, and these remaining examples can be checked by comparing hyperbolic invariants. These results make use of the surgery formula for Heegaard Floer homology, which has already proved to be a powerful tool for obstructing cosmetic surgeries; we get stronger obstructions than previously known by considering the full graded theory. We make use of a new graphical interpretation of knot Floer homology and the surgery formula in terms of immersed curves, which makes the grading information we need easier to access., v2: The main theorem (Theorem 2) is enhanced by including an explicit formula for q, which previously appeared later in the paper (Proposition 30 in v1). Other minor changes are implemented following referee feedback, the example in section 5 has been expanded for clarity, and a few figures have been added

Let K be a an alternating prime knot in the 3-sphere. We investigate the category of flypes between reduced alternating diagrams for K. As a consequence, we show that any odd prime order action on K is isotopic through maps of pairs to a single flype. This implies that for any odd prime order action on K there is either a reduced alternating periodic diagram or a reduced alternating free periodic diagram. Finally, we deduce that the quotient of an odd periodic alternating knot is also alternating.

We give a formula of the connected component decomposition of the Alexander quandle: $\mathbb{Z}[t^{\pm1}]/(f_1(t),\ldots, f_k(t))=\bigsqcup^{a-1}_{i=0}\mathrm{Orb}(i)$, where $a=\gcd (f_1(1),\ldots, f_k(1))$. We show that the connected component $\mathrm{Orb}(i)$ is isomorphic to $\mathbb{Z}[t^{\pm1}]/J$ with an explicit ideal $J$. By using this, we see how a quandle is decomposed into connected components for some Alexander quandles. We introduce a decomposition of a quandle into the disjoint union of maximal connected subquandles. In some cases, this decomposition is obtained by iterating a connected component decomposition. We also discuss the maximal connected sub-multiple conjugation quandle decomposition.

We show that a ribbon concordance between two links induces an injective map on Khovanov homology., Published version; errors corrected from previous version. 5 pages, 1 figure

We construct a group $\Gamma_{n}^{4}$ corresponding to the motion of points in $\mathbb{R}^{3}$ from the point of view of Delaunay triangulations. We study homomorphisms from pure braids on $n$ strands to the product of copies of $\Gamma_{n}^{4}$. We will also study the group of pure braids in $\mathbb{R}^{3}$, which is described by a fundamental group of the restricted configuration space of $\mathbb{R}^{3}$, and define the group homomorphism from the group of pure braids in $\mathbb{R}^{3}$ to $\Gamma_{n}^{4}$. In the end of this paper we give some comments about relations between the restricted configuration space of $\mathbb{R}^{3}$ and triangulations of the 3-dimensional ball and Pachner moves.

In a previous paper, the authors proved that Milnor link-homotopy invariants modulo $n$ classify classical string links up to $2n$-move and link-homotopy. As analogues to the welded case, in terms of Milnor invariants, we give here two classifications of welded string links up to $2n$-move and self-crossing virtualization, and up to $V^{n}$-move and self-crossing virtualization, respectively., 16 pages, 18 figures. arXiv admin note: text overlap with arXiv:1902.06079

We show that a band-connected sum of knots $K_0$ and $K_1$ along a band $b$ is equal to the connected sum $K_0\# K_1$ if and only if $b$ is a trivial band., Comment: 5 pages, 4 figures

We consider the question, asked by Friedl, Livingston and Zentner, of which sums of torus knots are concordant to alternating knots. After a brief analysis of the problem in its full generality, we focus on sums of two torus knots. We describe some effective obstructions based on Heegaard Floer homology., Comment: 19 pages, comments are welcome. Final version to appear in Bulletin of the London Mathematical Society

We give asymptotically sharp upper bounds for the Khovanov width and the dealternation number of positive braid links, in terms of their crossing number. The same braid-theoretic technique, combined with Ozsv\'ath, Stipsicz, and Szab\'o's Upsilon invariant, allows us to determine the exact cobordism distance between torus knots with braid index two and six., Comment: 11 pages, 6 figures, comments welcome! V2: minor changes and corrections. This version corresponds to the published article

We prove a Kauffman-Murasugi-Thistlethwaite theorem for alternating links in thickened surfaces. It states that any reduced alternating diagram of a link in a thickened surface has minimal crossing number, and any two reduced alternating diagrams of the same link have the same writhe. This result is proved more generally for link diagrams that are adequate, and the proof involves a two-variable generalization of the Jones polynomial for surface links defined by Krushkal. The main result is used to establish the first and second Tait conjectures for links in thickened surfaces and for virtual links., Comment: 32 pages, 20 figures, and 1 table

Gordian complex of knots was defined by Hirasawa and Uchida as the simplicial complex whose vertices are knot isotopy classes in $\mathbb{S}^3$. Later Horiuchi and Ohyama defined Gordian complex of virtual knots using $v$-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 that the constructed virtual knots have the same affine index polynomial., Comment: 23 pages, 35 figures, 1 table

In this paper, we analyze L-space surgeries on two component L-space links. We show that if one surgery coefficient is negative for the L-space surgery, then the corresponding link component is an unknot. If the link admits very negative (i.e. $d_{1}, d_{2}\ll0$) L-space surgeries, it is the Hopf link. We also give a way to characterize the torus link $T(2, 2l)$ by observing an L-space surgery $S^{3}_{d_{1}, d_{2}}(\mathcal{L})$ with $d_{1}d_{2}, 28 pages

We use the Bar-Natan Zh-correspondence to identify the generalized Alexander polynomial of a virtual knot with the Alexander polynomial of a two component welded link. We show that the Zh-map is functorial under concordance, and also that Satoh's Tube map (from welded links to ribbon knotted tori in $S^4$) is functorial under concordance. In addition, we extend classical results of Chen, Milnor, and Hillman on the lower central series of link groups to links in thickened surfaces. Our main result is that the generalized Alexander polynomial vanishes on any knot in a thickened surface which is concordant to a homologically trivial knot. In particular, this shows that it vanishes on virtually slice knots. We apply it to complete the calculation of the slice genus for virtual knots with four crossings and to determine non-sliceness for a number of 5-crossing and 6-crossing virtual knots., Comment: 32 pages, 24 figures

We show that the head and tail functions of the colored Jones polynomial of adequate links are the product of head and tail functions of the colored Jones polynomial of alternating links that can be read-off an adequate diagram of the link. We apply this to strengthen a theorem of Kalfagianni, Futer and Purcell on the fiberedness of adequate links., 11 pages, 9 figures. arXiv admin note: text overlap with arXiv:1112.3995

In this paper we look for closed expressions to calculate the number of colourings of prime knots for given linear Alexander quandles. For this purpose the colouring matrices are simplified to a triangular form, when possible. The operations used to perform this triangularization preserve the property that the entries in each row add up to zero, thereby simplifying the solution of the equations giving the number of colourings. When the colouring matrices (of prime knots up to ten crossings) can be triangularized, closed expressions giving the number of colourings can be obtained in a straightforward way. We use these results to show that there are colouring matrices that cannot be triangularized. In the case of knots with triangularizable colouring matrices we present a way to find linear Alexander quandles that distinguish by colourings knots with different Alexander polynomials. The colourings of knots with the same Alexander polynomial are also studied as regards when they can and cannot be distinguished by colourings., 51 pages

For every odd integer $c\ge 21$, we raise an example of a prime component-preservingly amphicheiral link with the minimal crossing number $c$. The link has two components, and consists of an unknot and a knot which is $(-)$-amphicheiral with odd minimal crossing number. We call the latter knot a {\it Stoimenow knot}. We also show that the Stoimenow knot is not invertible by the Alexander polynomials., 25 pages, 14 figures

Let $K_1, K_2$ be two knots with $t(K_1)+t(K_2)>2$ and $t(K_1 # K_2)=2$. Then, in the present paper, we will show that any genus three Heegaard splittings of $E(K_1 # K_2)$ is strongly irreducible and that $E(K_1 # K_2)$ has at most four genus three Heegaard splittings up to homeomorphism. Moreover, we will give a complete classification of those four genus three Heegaard splittings and show unknotting tunnel systems of knots $K_1 # K_2$ corresponding to those Heegaard splittings., 13 pages, 12 figures

In this paper we define a set of numerical criteria for a handlebody link to be irreducible. It provides an effective, easy-to-implement method to determine the irreducibility of handlebody links; particularly, it recognizes the irreducibility of all handlebody knots in the Ishii-Kishimoto-Moriuchi-Suzuki knot table and most handlebody links in the Bellettini-Paolini-Paolini-Wang link table., Comment: 13 pages, 3 figures

In this paper, we study the parabolic representations of 2-bridge links by finiding arc coloring vectors on the Conway diagram. The method we use is to convert the system of conjugation quandle equations to that of symplectic quandle equations. In this approach, we have an integer coefficient monic polynomial [Formula: see text] for each 2-bridge link [Formula: see text], and each zero of this polynomial gives a set of arc coloring vectors on the diagram of [Formula: see text] satisfying the system of symplectic quandle equations, which gives an explicit formula for a parabolic representation of [Formula: see text]. We then explain how these arc coloring vectors give us the closed form formulas of the complex volume and the cusp shape of the representation. As other applications of this method, we show some interesting arithmetic properties of the Riley polynomial and of the trace field, and also describe a necessary and sufficient condition for the existence of epimorphisms between 2-bridge link groups in terms of divisibility of the corresponding Riley polynomials.

The Slope Conjecture proposed by Garoufalidis asserts that the degree of the colored Jones polynomial determines a boundary slope, and its refinement, the Strong Slope Conjecture proposed by Kalfagianni and Tran asserts that the linear term in the degree determines the topology of an essential surface that satisfies the Slope Conjecture. Under certain hypotheses, we show that twisted, generalized Whitehead doubles of a knot satisfies the Slope Conjecture and the Strong Slope Conjecture if the original knot does. Additionally, we provide a proof that there are Whitehead doubles which are not adequate., This version has been accepted for publication in Quantum Topology. We restructured results and presentation to focus on the maximum degree of colored Jones polynomial for improved exposition and clarity. A cancellation error is also corrected. The changes mainly impact sections 1 and 2

For a virtual knot $K$ and an integer $r\geq 0$, the $r$-covering $K^{(r)}$ is defined by using the indices of chords on a Gauss diagram of $K$. In this paper, we prove that for any finite set of virtual knots $J_0,J_2,J_3,\dots,J_m$, there is a virtual knot $K$ such that $K^{(r)}=J_r$ $(r=0\mbox{ and }2\leq r\leq m)$, $K^{(1)}=K$, and otherwise $K^{(r)}=J_0$., 9 pages, 6 figures

We introduce a new link invariant called the algebraic genus, which gives an upper bound for the topological slice genus of links. In fact, the algebraic genus is an upper bound for another version of the slice genus proposed here: the minimal genus of a surface in the four-ball whose complement has infinite cyclic fundamental group. We characterize the algebraic genus in terms of cobordisms in three-space, and explore the connections to other knot invariants related to the Seifert form, the Blanchfield form, knot genera and unknotting. Employing Casson-Gordon invariants, we discuss the algebraic genus as a candidate for the optimal upper bound for the topological slice genus that is determined by the S-equivalence class of Seifert matrices., Comment: 30 pages, 5 figures, comments welcome! V3: Minor changes and corrections; added a footnote sketching a proof for the folklore fact that Taylor's invariant is a lower bound for the topological slice genus

For an oriented diagram of a link $L$ in the 3-sphere, Cho and Murakami defined the potential function whose critical point, slightly different from the usual sense, corresponds to a boundary parabolic $\mathrm{PSL}(2,\mathbb{C})$-representation of $\pi_1(S^3 \setminus L)$. They also showed that the volume and Chern-Simons invariant of such a representation can be computed from the potential function with its partial derivatives. In this paper, we extend the potential function to a $\mathrm{PSL}(2,\mathbb{C})$-representation that is not necessarily boundary parabolic. Under a mild assumption, it leads us to a combinatorial formula for computing the volume and Chern-Simons invariant of a $\mathrm{PSL}(2,\mathbb{C})$-representation of a closed 3-manifold., Comment: 22 pages

We incorporate quandle cocycle information into the quandle coloring quivers we defined in arXiv:1807.10465 to define weighted directed graph-valued invariants of oriented links we call \textit{quandle cocycle quivers}. This construction turns the quandle cocycle invariant into a small category, yielding a categorification of the quandle cocycleinvariant. From these graphs we define several new link invariants including a 2-variable polynomial which specializes to the usual quandle cocycle invariant. Examples and computations are provided., 9 pages

Ribbon tangles are proper embeddings of tori and cylinders in the 4-ball $B^4$, “bounding” 3-manifolds with only ribbon disks as singularities. We construct an Alexander invariant $\mathbf{A}$ of ribbon tangles equipped with a representation of the fundamental group of their exterior in a free abelian group $G$. This invariant induces a functor in a certain category $\mathbf{R}ib_G$ of tangles, which restricts to the exterior powers of Burau–Gassner representation for ribbon braids, that are analogous to usual braids in this context. We define a circuit algebra $\mathbf{C}ob_G$ over the operad of smooth cobordisms, inspired by diagrammatic planar algebras introduced by Jones [Jon99], and prove that the invariant $\mathbf{A}$ commutes with the compositions in this algebra. On the other hand, ribbon tangles admit diagrammatic representations, through welded diagrams. We give a simple combinatorial description of $\mathbf{A}$ and of the algebra $\mathbf{C}ob_G$, and observe that our construction is a topological incarnation of the Alexander invariant of Archibald [Arc10]. When restricted to diagrams without virtual crossings, $\mathbf{A}$ provides a purely local description of the usual Alexander poynomial of links, and extends the construction by Bigelow, Cattabriga and the second author [BCF15].

The lattice stick number of knots is defined to be the minimal number of straight sticks in the cubic lattice required to construct a lattice stick presentation of the knot. We similarly define the lattice stick number [Formula: see text] of spatial graphs [Formula: see text] with vertices of degree at most six (necessary for embedding into the cubic lattice), and present an upper bound in terms of the crossing number [Formula: see text] [Formula: see text] where [Formula: see text] has [Formula: see text] edges, [Formula: see text] vertices, [Formula: see text] cut-components, [Formula: see text] bouquet cut-components, and [Formula: see text] knot components.

For a nontrivial knot [Formula: see text], Negami found an upper bound on the stick number [Formula: see text] in terms of its crossing number [Formula: see text] which is [Formula: see text]. Later, Huh and Oh utilized the arc index [Formula: see text] to present a more precise upper bound [Formula: see text]. Furthermore, Kim, No and Oh found an upper bound on the equilateral stick number [Formula: see text] as follows; [Formula: see text]. As a sequel to this research program, we similarly define the stick number [Formula: see text] and the equilateral stick number [Formula: see text] of a spatial graph [Formula: see text], and present their upper bounds as follows; [Formula: see text] [Formula: see text] where [Formula: see text] and [Formula: see text] are the number of edges and vertices of [Formula: see text], respectively, [Formula: see text] is the number of bouquet cut-components, and [Formula: see text] is the number of non-splittable components.

We prove that every $\mathbb{Z}_2$H-thin link has no $2^k$-torsion for $k>1$ in its Khovanov homology. Together with previous results by Eun Soo Lee and the author, this implies that integer Khovanov homology of non-split alternating links is completely determined by the Jones polynomial and signature. Our proof is based on establishing an algebraic relation between Bockstein and Turner differentials on Khovanov homology over $\mathbb{Z}_2$. We conjecture that a similar relation exists between the corresponding spectral sequences., 12 pages, 5 figures

We give criteria for oriented links to be periodic of prime order using the quantum $\mathrm{SL}(N)$-invariant. The criteria are based upon an observation on the linking number between a periodic knot and its axis of the rotation.

We characterize the (1, 1) knots in the three-sphere and lens spaces that admit non-trivial L-space surgeries. As a corollary, 1-bridge braids in these manifolds admit non- trivial L-space surgeries. We also recover a characterization of the Berge manifold amongst 1-bridge braid exteriors., 17 pages, 5 figures; v2: corrects minor errors, has a new figure (Figure 2). This is the version to appear in Compositio Mathematica

We present two families of knots which have straight number higher than crossing number. In the case of the second family, we have computed the straight number explicitly. We also give a general theorem about alternating knots that states adding an even number of crossings to a twist region will not change whether the knots are perfectly straight or not perfectly straight., 10 pages, 6 figures, Comments and suggestions welcome. Minor grammatical corrections, added two questions

In this paper, we prove that $w(K) =4w(J)$, where $w(.)$ is the width of a knot and $K$ is the Whitehead double of a nontrivial knot $J$., 10 pages, 4 figures. In the new version, we re-draw all the figures and update the acknowledgement

Every knot has a plat projection, obtained by closing up a braid with bridges. The plat projection is determined by the number of strands and the number of rows of twist regions in the braid, and an integer number of crossings in each twist region. In recent work, we showed that under certain restrictions, including that the number of rows is odd, a minimal width plat projection is unique. In this paper we extend the results to even plats. Using new arguments, we show that if each of their twist regions contains at least three crossings, and their length is sufficiently long with respect to their width, then the projection is unique. This essentially "doubles" the set of knots for which such diagrams classify the links., This paper is being withdrawn because there is a gap in Section 3

Jablan and Radovi\'c originally defined two invariants called the Meander number and OGC number of knots for certain classes of knots. We generalize these definitions to all knots and name the straight number and contained straight number of a knot, respectively, and prove they are well defined. We answer two questions and prove a generalization of a conjecture of Jablan and Radovi\'c. We also give some relations to crossing number and petal number. Then we compute the straight numbers for all the knots in the standard knot table and present some interesting questions and the complete table of knots with 10 or fewer crossing and their straight number and contained straight number., Comment: 16 page, 7 figures, 1 table. Questions and comments are welcome -- Added reference to a paper that is related and answered questions they have asked. Made two corrections in the table

The aim of the present note is to enhance groups $G_{n}^{3}$ and to construct new invariants of classical braids. In particular, we construct invariants valued in $G_{N}^{2}$ groups. In groups $G_{n}^{2}$, the identity problem is solved, besides, their structure is much simpler than that of $G_{n}^{3}$. I am grateful to Huyue Yan for pointing out a small mistake in the previous version., Comment: 9 pages, 2 figures; 1 small mistake corrected