Search

In this paper, we obtain the necessary and sufficient condition that two knot projections are related by a finite sequence of the first and second flat Reidemeister moves (Theorem 1). We also consider an equivalence relation that is called weak (1, 3) homotopy. This equivalence relation occurs by the first flat Reidemeister move and one of the third flat Reidemeister moves. We introduce a map sending weak (1, 3) homotopy classes to knot isotopy classes (Sec. 3). Using the map, we determine which knot projections are trivialized under weak (1, 3) homotopy (Corollary 3)., 13 pages, 25 figures

We introduce a knot semigroup as a cancellative semigroup whose defining relations are produced from crossings on a knot diagram in a way similar to the Wirtinger presentation of the knot group; to be more precise, a knot semigroup as we define it is closely related to such tools of knot theory as the twofold branched cyclic cover space of a knot and the involutory quandle of a knot. We describe knot semigroups of several standard classes of knot diagrams, including torus knots and torus links T(2, n) and twist knots. The description includes a solution of the word problem. To produce this description, we introduce alternating sum semigroups as certain naturally defined factor semigroups of free semigroups over cyclic groups. We formulate several conjectures for future research.

The traditional skein relation for the Alexander polynomial involves an oriented knot, K + , with a distinguished positive crossing; a knot K − , obtained by changing the distinguished positive crossing of K + to a negative crossing; and a link K 0 , the orientation preserving resolution of the distinguished crossing. We refer to ( K + , K − , K 0 ) as the oriented skein triple . A tangle is defined as a pair ( B, t ) of a 3-dimensional ball B and a collection of disjoint, simple, properly embedded arcs, denoted t . DeWitt Sumners and Claus Ernst developed the tangle model which uses the mathematics of tangles to model DNA-protein binding. The protein is seen as the 3-ball and the DNA bound by the protein as properly embedded curves in the 3-ball. Topoisomerases are proteins that break one segment of DNA allowing a DNA segment to pass through before resealing the break. Effectively, the action of these proteins can be modeled as K − ↔ K + . Recombinases are proteins that cut two segments of DNA and recombine them in some manner. While recombinase local action varies, most are mathematically equivalent to a resolution, i.e. K ± ↔ K 0 . The oriented triple is now viewed as K − = circular DNA substrate, K + = product of topoisomerase action, K 0 = product of recombinase action. The theorem stated in this dissertation gives a relationship between two 2-bridge knots, K + and K − , that differ by a crossing change and a link, K 0 created from the oriented resolution of that crossing. We apply this theorem to difference topology experiments using topoisomerase proteins to study SMC proteins. In recent years, link homology theories have become a popular invariant to develop and study. One such invariant knot Floer homology , was constructed by Peter Ozsvath, Zoltan Szabo, and independently Jacob Rasmussen, denoted by HFK . It is also a refinement of a classical invariant, the Alexander polynomial . The study of DNA knots and links are of great interest to molecular biologists as they are present in many cellular process. The variety of experimentally observed DNA knots and links makes separating and categorizing these molecules a critical issue. Thus, knowing the knot Floer homology will provide restrictions on knotted and linked products of protein action. We give a summary of the combinatorial version of knot Floer homology from known work, providing a worked out example. The thesis ends with reviewing knot Floer homology properties of three particular sub-families of biologically relevant links known as (2, p)- torus links, clasp knots and 3-strand pretzel…

A twisted torus knot is obtained from a torus knot by adding a number of full twists to some adjacent strands of the torus knot. In this paper, we show that if a twisted torus knot is a torus knot, then the number of added full twists is generically at most two in absolute value. We also show that this bound is the best possible by classifying twisted torus knots for which the upper bound is attained. 57N10

A knot shadow is a diagram with all crossing information missing. We cannot determine the original knot from a knot shadow in general. In this paper, we investigate properties (unknotting number, genus, braid index, etc.) of the original knot from a knot shadow.

It is conjectured that, in the asymptotic expansion of the Kashaev invariant of a hyperbolic knot, the first coefficient is represented by the complex volume of the knot complement, and the second coefficient is represented by a constant multiple of the square root of the twisted Reidemeister torsion associated with the holonomy representation of the hyperbolic structure of the knot complement. In particular, this conjecture has been rigorously proved for some simple hyperbolic knots, for which the second coefficient is presented by a modification of the square root of the Hessian of the potential function of the hyperbolic structure of the knot complement. In this paper, we define an invariant of a parametrized knot diagram as a modification of the Hessian of the potential function obtained from the parametrized knot diagram. Further, we show that this invariant is equal (up to sign) to a constant multiple of the twisted Reidemeister torsion for any two-bridge knot. 57M27

We present a new paradigm for three dimensional chaos, and specifically for the Lorenz equations. The main difficulty in these equations and for a generic flow in dimension three is the existence of singularities. We show how to use knot theory as a way to remove the singularities. Specifically, we claim: (1) For certain parameters, the Lorenz system has an invariant one dimensional curve, which is a trefoil knot. The knot is a union of invariant manifolds of the singular points. (2) The flow is topologically equivalent to an Anosov flow on the complement of this curve, and even to a geodesic flow. (3) When varying the parameters, the system exhibits topological phase transitions, i.e. for special parameter values, it will be topologically equivalent to an Anosov flow on a knot complement, and different knots appear for different parameter values. The steps of a mathematical proof of these statements are at different stages. Some have been proven, for some we present numerical evidence and some are still conjectural., 14 pages, 11 figures

Knot mosaic theory was introduced by Lomonaco and Kauffman in the paper on ‘Quantum knots and mosaics’ to give a precise and workable definition of quantum knots, intended to represent an actual physical quantum system. A knot [Formula: see text]-mosaic is an [Formula: see text] matrix whose entries are eleven mosaic tiles, representing a knot or a link by adjoining properly. In this paper, we introduce two variants of knot mosaics: period knot mosaics and toroidal knot mosaics, which are common features in physics and mathematics. We present an algorithm producing the exact enumeration of period knot [Formula: see text]-mosaics for any positive integers [Formula: see text] and [Formula: see text], toroidal knot [Formula: see text]-mosaics for co-prime integers [Formula: see text] and [Formula: see text], and furthermore toroidal knot [Formula: see text]-mosaics for a prime number [Formula: see text]. We also analyze the asymptotics of the growth rates of their cardinality.

Given a Morse function defined in the complement of a knot K ⊂ R 3 we obtain a lower bound for the number of its critical points, depending on a knot invariant t ( K ) known as the “tunnel number”. This lower bound is used to prove existence of many periodic solutions in a system of differential equations from celestial mechanics.

We introduce a way to color the regions of a classical knot diagram using ternary operations, so that the number of colorings is a knot invariant. By choosing appropriate substitutions in the algebras that we assign to diagrams, one obtains the relations from the knot group, and from the core group. Using the ternary operator approach, we generalize the Dehn presentation of the knot group to extra loops, and a similar presentation for the core group to the variety of Moufang loops., Comment: 19 pages

Seidel and Smith introduced the graded fixed-point symplectic Khovanov cohomology group Khsymp,inv(K) for a knot K ⊂ S3, as well as a spectral sequence converging to the Heegaard Floer homology group HFd (Σ(K)#(S2 × S1)) with E1-page isomorphic to a factor of Khsymp,inv(K) [26]. There the authors proved that Khsymp,inv is a knot invariant. We show here that the higher pages of their spectral sequence are knot invariants also, and further define a reduced version of the spectral sequence which directly computes HFd (Σ(K)). Under some degeneration conditions, one obtains a new absolute Maslov grading on the group HFd (Σ(K)). This occurs when K is a two-bridge knot, and we compute the grading in this case.

In the article we show that nondi eomorphic symplectic 4-manifolds whichadmit marked Lefschetz brations can share the same monodromy group. Explicitlywe prove that, for each integer g > 0, every knot surgery 4-manifold in a familyfE(2) K jK is a bered 2-bridge knot of genus g in S 3 g admits a marked Lefschetz bra-tion structure which has the same monodromy group. 1. IntroductionSeiberg-Witten invariants are one of the most powerful invariants in the classi- cation of smooth 4-manifolds and Fintushel-Stern’s knot surgery method is one ofthe most e ective methods to modify smooth structures on a given 4-manifold. ButSeiberg-Witten invariants are not complete invariants and there are known examplesof nondi eomorphic symplectic 4-manifolds which share the same Seiberg-Witteninvariants [3, 15].R. Fintushel and R. Stern showed that Seiberg-Witten invariants of knot surgery4-manifoldE(2) K = E(2)] F=m K S 1 (M K S 1 )can be computed by using the Alexander polynomial of the related knot K[2]. If werestrict our attention to a bered knot K, then E(2)

The virtual unknotting number of a virtual knot is the minimal number of crossing changes that makes the virtual knot to be the unknot, which is defined only for virtual knots virtually homotopic to the unknot. We focus on the virtual knot obtained from the standard (p,q)-torus knot diagram by replacing all crossings on one overstrand into virtual crossings and prove that its virtual unknotting number is equal to the unknotting number of the $(p,q)$-torus knot, i.e. it is (p-1)(q-1)/2., 14 pages, 8 figures

We compute the Heegaard–Floer homology of \(S^{3}_{1}(K)\) (the (+1) surgery on the torus knot Tp,q) in terms of the semigroup generated by p and q, and we find a compact formula (involving Dedekind sums) for the corresponding Ozsvath–Szabo d-invariant. We relate the result to known knot invariants of Tp,q as the genus and the Levine–Tristram signatures. Furthermore, we emphasize the striking resemblance between Heegaard–Floer homologies of (+1) and (−1) surgeries on torus knots. This relation is best seen at the level of τ functions.

It is known by few that a trivial knot can be transformed into a lattice knot whose three shadows are all trees (here, three shadows mean the projections of the lattice knot to the directions of ordinary orthogonal axes). Since a knot itself is a loop in the space, this fact may be rather astonishing. It will be interesting to ask whether a non-trivial knot has such a transformation or not. The purpose of this paper is to show that any two-bridge torus knot or link has a transformation into a lattice knot whose three shadows are all trees. The algorithm to construct such a position will be demonstrated.

If there are any 2-component counterexamples to the Generalized Property R Conjecture, a least genus component of all such counterexamples cannot be a fibered knot. Furthermore, the monodromy of a fibered component of any such counterexample has unexpected restrictions. The simplest plausible counterexample to the Generalized Property R Conjecture could be a 2-component link containing the square knot. We characterize all two-component links that contain the square knot and which surger to (S^1 x S^2) # (S^1 x S^2). We exhibit a family of such links that are probably counterexamples to Generalized Property R. These links can be used to generate slice knots that are not known to be ribbon., Comment: Combines and expands arXiv:0908.2795 and arXiv:0901.2319 into the version published in Geometry and Topology

A 2-universal knot is a universal knot having all branching indices 1 or 2. Here we construct the first example of a 2-universal knot which could be a hyperbolic knot (SnapPea). Hence, we could have an example of a universal group generated by 180° rotations.

Link-homotopy and self Δ-equivalence are equivalence relations on links. It was shown by J. Milnor (resp. the last author) that Milnor invariants determine whether or not a link is link-homotopic (resp. self Δ-equivalent) to a trivial link. We study link-homotopy and self Δ-equivalence on a certain component of a link with fixing the other components, in other words, homotopy and Δ-equivalence of knots in the complement of a certain link. We show that Milnor invariants determine whether a knot in the complement of a trivial link is null-homotopic, and give a sufficient condition for such a knot to be Δ-equivalent to the trivial knot. We also give a sufficient condition for knots in the complements of the trivial knot to be equivalent up to Δ-equivalence and concordance.

We will study boundaries of incompressible surfaces properly embedded in graph knot exteriors. We will first show that any bounded two-sided surface is meridional or longitudinal. In particular, iterated torus knot exteriors contain no bounded two-sided essential surface which is meridional or preferred longitudinal but Seifert surfaces. Even if the surface is possibly one-sided, the boundary is not of type (2p, 2q + 1) for any integers p and q.

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

We observe a behavior of a slice (an algebraic subset) S 0 ( K ) of the SL 2 ( C ) -character variety of a knot group under the connected sum of knots. It turns out that the number of 0-dimensional components of S 0 ( K ) is additive under the connected sum of knots.

We construct a class of knots with the CI ∗ {}^* property, that is, π 1 ( M ( n ) ∣ ∂ M ( n ) ) ≠ { e } \pi _1(M(n)\mid \partial M(n))\neq \{e\} for some n > 0 n>0 . It follows that the infinite cyclic covering of such a knot cannot be embedded in any compact 3-manifold.

We show that every bridge surface of certain types of (1,1) prime knot has the disjoint curve property. Also we determine when a bridge surface of a pretzel knot of type (i2,3,n) has the disjoint curve property.

We give a lower bound for the alternation number of a knot by using the Rasmussen s-invariant and the signature of a knot. Then, we determine the torus knots with alternation number one and show that many torus knots are "far" from the alternating knots. As an application, we determine the almost alternating torus knots, solving a conjecture due to Adams et al.

A knot diagram can be divided by a circle into two parts, such that each part can be coded by a planar tree with integer weights on its edges. A half of the number of intersection points of this circle with the knot diagram is called the girth. The girth of a knot is the minimal girth of all diagrams of this knot. The girth of a knot minus one is an upper bound of the Heegaard genus of the 2-fold branched covering of that knot. We will use Topological Quantum Field Theory (TQFT) coming from the Kauffman bracket to determine the girth of some knots. Consequently, our method can be used to determine the Heegaard genus of the 2-fold branched covering of some knots.

A partial order on the set of the prime knots can be defined by the existence of a surjective homomorphism between knot groups. In the previous paper, we determined the partial order in the knot table. In this paper, we prove that 31 and 41 are minimal elements. Further, we study which surjection a pair of a periodic knot and its quotient knot induces, and which surjection a degree one map can induce.

In the article, we study Fintushel-Stern's knot surgery four-manifold E(n) K and its monodromy factorization. For fibered knots we provide a smooth classification of knot surgery 4-manifolds up to twisted fiber sums. We then show that other constructions of 4-manifolds with the same Seiberg-Witten invariants are in fact diffeomorphic.

Ozsvath and Szabo conjectured that knot Floer homology detects fibred knots. We propose a strategy to approach this conjecture based on Gabai's theory of sutured manifold decomposition and contact topology. We implement this strategy for genus-one knots, obtaining as a corollary that, if rational surgery on a knot $K$ gives the Poincare homology sphere $\Sigma(2,3,5)$, then $K$ is the left-handed trefoil knot., Comment: The paper used to be known as "Knot Floer homology detects fibred links", however there was a mistake in the proof of Theorem 1.9 concerning links. The result about knots (Theorem 1.4) and the application to surgeries giving the Poincare' homology sphere (Corollary 1.7), as well as the applications by Ozsvath and Szabo (math.GT/0604079) and Yi Ni (math.GT/0607156) are uneffected by the mistake. This version is essentially the previous one, without Theorem 1.9 and Subsection 4.3. A more revised version is going to appear on Amer. J. Math. and is available on my personal webpage at http://www.labmath.uqam.ca/~ghiggini/papers.html

In the ordinary normal surface for a compact 3-manifold, any incompressible, ∂-incompressible, compact surface can be moved by an isotopy to a normal surface [9]. But in a non-compact 3-manifold with an ideal triangulation, the existence of a normal surface representing an incompressible surface cannot be guaranteed. The figure-8 knot complement is presented in a counterexample in [12]. In this paper, we show the existence of normal Seifert surface under some restriction for a given ideal triangulation of the knot complement.

Fox's shadow p-colorings of a knot K define two kinds of Laurent polynomials, Φp(K) and [Formula: see text], as invariants of K. We prove that the equality [Formula: see text] holds for any knot K. Also we prove that, if the p-coloring number of K is equal to p2, then [Formula: see text] has the form [Formula: see text] for some N ∈ ℤp.

Recently Stoimenow showed that for every knot K and any n ∈ N and u 0 ⩾ u ( K ) there is a prime knot K n , u o which is n-equivalent to the knot K and has unknotting number u ( K n , u o ) equal to u 0 . The similar result has been obtained for the 4-ball genus g s of a knot. Stoimenow also proved that any admissible value of the Tristram–Levine signature σ ξ can be realized by a knot with the given Vassiliev invariants of bounded order. In this paper, we show that for every knot K with genus g ( K ) and any n ∈ N and m ⩾ g ( K ) there exists a prime knot L which is n-equivalent to K and has genus g ( L ) equal to m.

We define some new numerical invariants for knots with zero winding number in the solid torus. These invariants explore some geometric features of knots embedded in the solid torus. We characterize these invariants and interpret them on the level of the knot projection. We also find some relations among some of these invariants. Moreover, we give lower bounds for some of these invariants using Vassiliev invariants of type one. We connect our invariants to the bridge number in the solid torus. We give a lower bound and an upper bound of the wrap of a knot in the solid torus in terms of our new invariants.

We say a knot $k$ in the 3-sphere $\mathbb S^3$ has {\it Property $IE$} if the infinite cyclic cover of the knot exterior embeds into $\mathbb S^3$. Clearly all fibred knots have Property $IE$. There are infinitely many non-fibred knots with Property $IE$ and infinitely many non-fibred knots without property $IE$. Both kinds of examples are established here for the first time. Indeed we show that if a genus 1 non-fibred knot has Property $IE$, then its Alexander polynomial $\Delta_k(t)$ must be either 1 or $2t^2-5t+2$, and we give two infinite families of non-fibred genus 1 knots with Property $IE$ and having $\Delta_k(t)=1$ and $2t^2-5t+2$ respectively. Hence among genus one non-fibred knots, no alternating knot has Property $IE$, and there is only one knot with Property $IE$ up to ten crossings. We also give an obstruction to embedding infinite cyclic covers of a compact 3-manifold into any compact 3-manifold., Comment: 24 pages, 9 figures

Let K be a knot in the 3-sphere S 3 , and a disk in S 3 meeting K transversely in the interior. For non-triviality we assume that | \ K| 2 over all isotopies of K in S 3 @ . Let K ,n( S 3 ) be a knot obtained from K by n twistings along the disk . If the original knot is unknotted in S 3 , we call K ,n a twisted knot. We describe for which pair (K,) and an integer n, the twisted knot K ,n is a torus knot, a satellite knot or a hyperbolic knot.

Let V be the standard solid torus in S3. Let Kp, 2 be the (p, 2)-torus knot in V such that Kp, 2 meets a meridian disk D of V in two points with the winding number zero and the 2-string tangle TKp, 2 obtained by cutting along D is a rational tangle. We compute the Casson invariant of the cyclic covering space of S3 branched over a satellite knot whose companion is any 2-bridge knot D(b1,…,b2m) and pattern is (V, Kp, 2).

J. Hempel's definition of the distance of a Heegaard surface generalizes to a complexity for a knot which is in bridge position with respect to a Heegaard surface. Our main result is that the distance of a knot in bridge position is bounded above by twice the genus, plus the number of boundary components, of an essential surface in the knot complement. As a consequence knots constructed via sufficiently high powers of pseudo-Anosov maps have minimal bridge presentations which are thin., Comment: 14 pages, 4 figures

We write $K_1 \geq K_2$ for two prime knots $K_1,K_2$ if there exists a surjective group homomorphism from $G(K_1)$ onto $G(K_2)$ where $G(K_1), G(K_2)$ are the knot groups of $K_1,K_2$, respectively. In this paper, we determine this partial order for the knots in Rolfsen's knot table.

Let K be a knot in the 3-sphere S 3 and D ad isk inS 3 meeting K transversely more than once in the interior. For nontriviality we assume that |D ∩ K| 2 over all isotopies of K in S 3 − ∂D .L et KD,n (⊂ S 3 ) be a knot obtained from K by n twisting along the disk D. We prove that if K is a trivial knot and KD,n is a graph knot, then |n| 1o rK and D form a special pair which we call an “exceptional pair”. As a corollary, if (K, D) is not an exceptional pair, then by twisting unknot K more than once (in the positive or the negative direction) along the disk D, we always obtain a knot with positive Gromov volume. We will also show that there are infinitely many graph knots each of which is obtained from a trivial knot by twisting, but its companion knot cannot be obtained in such a manner.  2004 Elsevier B.V. All rights reserved. MSC: 57M25

We will construct a new fundamental polyhedron for the figure-eight knot group by using the generators introduced in Sato [I. Kra, B. Maskit (Eds.), The First Ahlfors–Bers Colloquium, in: Contemp. Math., 2000, p. 271] related to Jorgensen groups.

The paper computes the noncommutative A-ideal of the figure-eight knot, a noncommutative generalization of the A-polynomial. It is shown that if a knot has the same A-ideal as the figure-eight knot, then all colored Kauffman brackets are the same as those of the figure eight knot.

Let D be a clasp disk in S3 with n singularities and let U be a regular neighborhood of D. Say D bounds the knot Γ. We show that the 3-complex X=(U\Γ)∪ c * Bd(U) 3-deforms to [Formula: see text]. In particular, π1(X)=ℤ for all 3-complexes X constructed in this manner. We observe that each X in this class corresponds to an unknotting scheme for the participating knot Γ.

In this paper, we count the number of SL2(Fq)-representations of torus knot group up to conjugacy. For the finite field Fq case, the counting methods are interested in its own right. Explicit formulae of the effective counting are given in this paper.

We derive, from the A -polynomial of a knot, a single variable polynomial for the knot, called C -polynomial, and explore topological and geometrical information about the knot encoded in the C -polynomial.

We study the way a strongly irreducible Heegaard surface $\Sigma$ intersects a knot exterior $X$ embedded in a 3-manifold, and show that if $\Sigma \cap \partial X$ consists of simple closed curves which are essential in both $\Sigma$ and $\partial X$, then the intersection $X \cap \Sigma$ consists of meridional annuli only. As an application we show that when considering two Heegaard surfaces that intersect essentially and spinally (cf. Rubinstein and Shcarlemann) any embedded torus in the union of the two bounds a solid torus., Comment: 12 pages

A knot K is called n-adjacent to the unknot, if K admits a projection containing n generalized crossings such that changing any m (no larger than n) of them yields a projection of the unknot. We show that a non-trivial satellite knot K is n-adjacent to the unknot, for some n>0, if and only if it is n-adjacent to the unknot in any companion solid torus. In particular, every model knot of K is n-adjacent to the unknot. Along the way of proving these results, we also show that 2-bridge knots of the form K_{p/q}, where p/q=[2q_1,2q_2] for some integers q_1,q_2, are precisely those knots that have genus one and are 2-adjacent to the unknot., 13 pages, 3 figures. to appear in Topology and Its Applications

For a hyperbolic knot K in S3, a toroidal surgery on K is integral or half-integral. It is well known that no 2-bridge knot admits non-integral toroidal surgery. In this paper, we show that the (-2, 3, 7)-pretzel knot and its mirror image are the only 3-bridge knots that admit non-integral toroidal surgery.

A knot K in a closed connected orientable 3-manifold M is called a double-torus knot, if it is in a genus two Heegaard splitting surface H of M. We give a necessary and sufficient condition for a double-torus knot to be the trivial knot in words of meridian disks of genus two handlebodies obtained by splitting M along H.