Kunio Murasugi

• University of Toronto

We study the zeros of Alexander polynomials of three classes of arborescent links. In the first class, the zeros are real (and negative) or modulus one. In the second class, the zeros are real (and positive). In the third class, the zeros are real or modulus one. For this purpose, we modify their Alexander polynomials into other real polynomials, w...

We study distribution of the zeros of the Alexander polynomials of knots and links in S 3 . After a brief introduction of various stabilities of multivariate polynomials, we present recent results on stable Alexander polynomials.

In this paper, we study distribution of the zeros of the Alexander polynomials of knots and links in S^3. We call a knot or link "real stable" (resp. "circular stable") if all the zeros of its Alexander polynomial are real (resp. unit complex). We give a general construction of real stable and circular stable knots and links. We also study pairs of...

We prove that for any zero {\alpha} of the Alexander polynomial of a two-bridge knot, -3 < Re({\alpha}) < 6. Furthermore, for a large class of two-bridge knots we prove -1<Re({\alpha}).

Let M be a non-abelian semi-direct product of a cyclic group ℤ/n and an elementary abelian p-group A = ⊕k(ℤ/p) of order pk, p being a prime and gcd(n,p) = 1. Suppose that the knot group G(K) of a knot K in the 3-sphere is represented on M. Then we conjectured (and later proved) that the twisted Alexander polynomial Δγ,K(t) associated to γ : G(K) →...

Let p be an odd prime and D_p a dihedral group of order 2p. Let \rho: G(K) --> D_p --> GL(p,Z) be a non-abelian representation of the knot group G(K) of a knot K in 3-sphere. Let \Delta_{\rho,K} (t) be the twisted Alexander polynomial of K associated to \rho. Let H(p) is the set of 2-bridge knots K, such that G(K) is mapped onto a non-trivial free...

Suppose the knot group G(K) of a knot K has a non-abelian representation \rho on A_4 \subset GL(4,Z). We conjecture that the twisted Alexander polynomial of K associated to \rho is of the form: \Delta_K(t)/(1-t) \phi(t^3), where \Delta_K (t) is the Alexander polynomial of K and \phi(t^3) is an integer polynomial in t^3. We prove the conjecture for...

A torti-rational knot, denoted by K(2α, β|r), is a knot obtained from the 2-bridge link B(2α, β) by applying Dehn twists an arbitrary number of times, r, along one component of B(2α, β). We determine the genus of K(2α, β|r) and solve a question of when K(2α, β|r) is fibered. In most cases, the Alexander polynomials determine the genus and fiberedne...

Let H(p) be the set of 2-bridge knots K(r), 0<r<1, such that there is a meridian-preserving epimorphism from G(K(r)), the knot group, to G(K(1/p)) with p odd. Then there is an algebraic integer s0 such that for any K(r) in H(p), G(K(r)) has a parabolic representation ρ into SL(2, ℤ[s0]) ⊂SL(2, ℂ). Let $\widetilde{\Delta}_{\rho, K(r)}(t)$ be the twi...

A knot (or link) invariant, by its very definition, as discussed in the previous chapter, does not change its value if we apply one of the elementary knot moves. As we have already seen, it is often useful to project the knot onto the plane, and then study the knot via its regular diagram. If we wish to pursue this line of thought, we must now ask...

Knot theory is a concept in algebraic topology that has found applications to a variety of mathematical problems as well as to problems in computer science, biological and medical research, and mathematical physics. This book is directed to a broad audience of researchers, beginning graduate students, and senior undergraduate students in these fiel...

Necessary and sufficient conditions are given for a satellite knot to be fibered. Any knot $\tilde k$ embeds in an unknotted solid torus $\tilde V$ with arbitrary winding number in such a way that no satellite knot with pattern $(\tilde V, \tilde k)$ is fibered. In particular, there exist nonfibered satellite knots with fibered pattern and companio...

A double torus knot $K$ is a knot embedded in a Heegaard surface $H$ of genus 2, and $K$ is non-separating if $H \setminus K$ is connected. In this paper, we determine the genus of a non-separating double torus knot that is a band-connected sum of two torus knots. We build a bridge between an algebraic condition and a geometric requirement (Theorem...

Without Abstract

For Montesinos knots, we explicitly construct Seifert surfaces of minimal genus and solve the question of when they are fibred knots. For those of tunnel number one, we show that they are mostly fibred if their Alexander polynomials (of proper degrees) are monic.

We show how the Alexander/Conway link polynomial occurs in the context of planar even valence graphs, refining the notion of the number of their spanning trees. Then we apply knot theory to deduce several statements about this graph polynomial, in particular estimates for its coefficients and relations between congruences of the number of vertices...

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

1. Introduction & Foundations. 2. The Braid Group. 3. World Problem. 4. Special types of braids. 5. Quotient groups of the braid group. 6. Isotopy of braids. 7. Homotopy braid theory. 8. From knots to braids. 9. Markov's theorem. 10. Knot invariants. 11. Braid groups on surfaces. 12. Algebraic equations. Appendix I: Group theory. Appendix II: Topol...

In Theorem 2.2 of Chapter 2 we showed that Bn has a particularly readable/compact presentation. But, since the subgroup B2 of the braid group Bn is of infinite order, Bn for n ≥ 2 is not a finite group.

As a consequence of Markov’s theorem (Theorem 1.1, Chapter 9), we may use braids, and the theory of braids, as a basis for finding knot (or link) invariants. One of the most elegant examples of this method is the so-called Alexander polynomial, [Al2], named after the American mathematician J.W. Alexander, and denoted by ΔK (t), where K denotes a kn...

Before we begin to develop the theory of braids with a distinctive diagrammatic, geometric slant, we would like to spend a few moments looking at some examples of the context of braids.

The concept of homotopy of braids can be traced back to the seminal work of Artin [Ar2]. Before we peruse its exact nature, we would like in this section to introduce informally the notion of homotopy of braids.

In what may be called an empirical approach to braid theory, in Chapter 1, we intuited that two braids are equivalent if we can deform one into the other by pushing, pulling and any other means that will not causing the braid to break, with the condition that all this happens within a cube.

In trying to establish a theory of braids, the most primitive question we may ask is, How many different (non-equivalent) braids are there?

In Section 4 of the previous chapter, starting with a diagram D of an oriented knot K, we described a method that allowed us to find a separating simple closed curve L on the plane ℝ2. This, in turn, led to a braided link (D, L), which we then used to extract a braid ß. Coming full circle, the closure of ß, denoted by ß, is equivalent to K.

From the work of Section 3 in the previous chapter, a n-braid β is equivalent to another n-braid β ' if (and only if) the corresponding elements in the n-braid group Bn are equal. By abuse of notation, we shall denote these algebraic elements also by βandβ ',respectively. Therefore, a primary requirement is to find a practical method that will allo...

In the previous chapter, we discussed the braid group for the surfaces §2 and ℙ2. The methods used were rather ad hoc, looking at each case separately. To try and establish the braid group for the more general case of manifolds in dimensions greater than or equal to 2, we need a more methodical approach. Such an approach exists and has been develop...

The art of braiding or platting has enriched human civilization over many centuries, probably even as far back as the dawn of humanity. Usually, braids and plaits are thought of as being formed by entwining together several strands of hair or string. In this chapter, contrary to the above, we will investigate the platting of leather from just a sin...

Suppose we begin with the trivial 4-braid in a cube, Figure 1.1(a). Fixing every face except the bottom face of the cube, let us rotate the bottom face of the cube twice around a vertical axis that connects the centre of the top face with the centre of the bottom face. On completion of this double twist, the trivial braid (in the cube) now has the...

A knot, succinctly, is a simple closed polygonal curve in ℝ3, however, for the purposes of this book, we will usually think of a knot as a simple closed smooth curve, see Figure 1.1.

We examine how graphs and techniques from graph theory can be transferred into knot theory. In particular, we look at how a certain graph polynomial may be used to calculate the braid index, minimum crossing number of specific knots and links. We also show how the graph polynomial is related to recent knot invariants, most significantly, the skein...

Our aim is to give a proof of the Melvin-Morton Conjecture—the “truncated” Jones polynomial is equal to the reciprocal of the Alexander polynomial—in terms of techniques from knot and graph theory. In this paper, we show that the Melvin-Morton Conjecture holds for 3-braids. The techniques developed in this paper form a basis for a proof of the Melv...

Towards the end of the 1980s in the midst of the Jones revolution, V.A. Vassiliev introduced a new concept that has had profound significance in the immediate aftermath of the Jones revolution in knot theory [V]. The importance of these so-called Vassiliev invariants lies in that they may be used to study Jones-type invariants more systematically.

Using the vertex model interpretation of the coloured (generalised) Jones polynomial of a link L, we show that if the colour of the ith component is Ni+mir, then modulo tr−1 this coloured Jones polynomial is congruent, up to a product of calculable factors, to the coloured Jones polynomial with the colour of the ith component Ni, where Ni and r are...

The recent progress made toward solving the determination problems of the minimal crossing number and braid index of a knot is discussed. Some relationships among these invariants and the bridge number are also discussed.

We show that, at least for an alternating fibered link or 2-bridge link L, there is an exact formula which expresses the braid index b(L) of L as a function of the 2-variable generalization PL(l, m) of the Jones polynomial.

If PL(v,z) = Σbi(v)zi is the skein polynomial of a link L, and D = D1 * D2 is the diagram which is a planar star (Murasugi) product of D1 and D2 then bϕ(D)(v) = bϕ(D1)·bϕ(D2)(v) where ϕ(D) = n(D)– (s(D) – 1) and n(D) denotes the number of crossings of D, and s(D) is the number of Seifert circles of D.

To each weighted graph τ, two invariants, a polynomial Pτ(x, y, z) and the signature σ(τ), are defined. The various partial degress of Pτ(x, y, z) and σ(τ) are expressed in terms of maximal spanning graphs of τ. Furthermore, one unexpected property of Tutte’s dichromate is proved. These results are applied to knots or links in S3.

Let L be a link in S3 which has a prime period and L* be its factor link. Several relationships between the Jones polynomials of L and L* are proved. As an application, it is shown that some knot cannot have a certain period.

Let L be an alternating link and be its reduced (or proper) alternating diagram. Let w() denote the writhe of [3], i.e. the number of positive crossings minus the number of negative crossings. Let VL(t) be the Jones polynomial of L [2]. Let dmax VL(t) and dmin VL(t) denote the maximal and minimal degrees of VL(t), respectively. Furthermore, let σ(L...

A new concordant invariant of a link in 3-sphere is introduced through the general theory of the abelian covering linkage invariants.

Let $J_K(t) = a_rt^r + \cdots + a_st^s, r > s$, be the Jones polynomial of a knot K in S3. For an alternating knot, it is proved that r - s is bounded by the number of double points in any alternating projection of K. This upper bound is attained by many alternating knots, including 2-bridge knots, and therefore, for these knots, r - s gives the mi...

Let Jk(t) = artr +….+asts, r > s, be the Jones polynomial of a knot K in S3. For an alternating knot, it is proved that r — s is bounded by the number of double points in any alternating projection of K. This upper bound is attained by many alternating knots, including 2-bridge knots, and therefore, for these knots, r-s gives the minimum number of...

A conjecture of Fox about the coefficients of the Alexander polynomial of an alternating knot is proved for alternating algebraic (or arborescent) knots, which include two-bridge knots.

In this volume, which is dedicated to H. Seifert, are papers based on talks given at the Isle of Thorns conference on low dimensional topology held in 1982.

Let p be a prime and let L be a 2-component link in S3. We define a numerical invariant, called p-height of L, using a tower of successive p-fold branched cyclic coverings of L, and show, in particular, 2-height is algorithmically determined for any 2-component link. Some relationships between p-height and known link type invariants are also establ...

Let L = X U Y be an oriented 2-component link in S3. In this paper we will define two different types of polynomials which are ambient isotopic invariants of L. One is associated with a cyclic cover branched along one of their components, an the other is associated with a metabelian cover of L. This invariants are defined for any link unless the li...

Let Δl(x, y) be a Alexander polynomial of a link l of two components X and Y in S3. Denote by Arf (Z) the Arf invariant of Z, a knot or a proper link [9](Received June 27 1983)

If $M$ is an abelian branched covering of $S^3$ along a link $L$, the order of $H_1(M)$ can be expressed in terms of (i) the Alexander polynomials of $L$ and of its sublinks, and (ii) a "redundancy" function characteristic of the monodromy-group. In 1954, the first author thus generalized a result of Fox (for $L$ a knot, in which case the monodromy...

If M M is an abelian branched covering of S 3 {S^3} along a link L L , the order of H 1 ( M ) {H_1}(M) can be expressed in terms of (i) the Alexander polynomials of L L and of its sublinks, and (ii) a "redundancy" function characteristic of the monodromy-group. In 1954, the first author thus generalized a result of Fox (for L L a knot, in which cas...

A (tame) knot k n in S ³ is said to have period n if there exists a homeomorphism ϕ : S ³ → S ³ , necessarily orientationpreserving, such that (i) the fixed point set of ϕ is a circle disjoint from k n (ii) ϕ ( k n ) = k n ; (iii) ϕ has order n . Several necessary conditions for a knot to have period n have already been established in the literatu...

A knot K in a 3-sphere S3 is said to have period n [9] [13] )or to be a periodoc knot of order n) if there is a rotation of S3 with period n and axis A, where ... http://www.tulips.tsukuba.ac.jp/mylimedio/dl/page.do?issueid=450523&tocid=100040741&page=331-347

There have been few published results concerning the relationship between the homology groups of branched and unbranched covering spaces of knots, despite the fact that these invariants are such powerful invariants for distinguishing knot types and have long been recognised as such [ 8 ]. It is well known that a simple relationship exists between t...

Let K be a knot in a manifold M. Corresponding to a representation of Π 1 (M — K) into a transitive group of permutations there is a branched covering space of M. K is covered by which may be a link of several components. The set of linking numbers between the various components of has long been recognised as a useful knot invariant. Bankwitz and S...

It is proved that a homomorph of the group of trefoil knot cannot be the group of a 2-knot in 4-sphere.

It is proved that a homomorph of the group of trefoil knot cannot be the group of a 2-knot in 4-sphere.

In this paper, we will prove, as a consequence of the main theorem, THEOREM A. (See Corollary 2.6). The group of an alternating knot, for which the leading coefficient of the knot polynomial is a prime power, is residually finite and solvable.

Some conditions for the knot group to be an R-group, i.e., the group in which the extraction of roots is unique, will be discussed in this paper. In particular, the group of a product knot is an R-group iff the knot group of each component is an R-group. For a fibred knot, a sufficient condition for its group to be an R-group will be given.

J. S. Birman has conjectured that, when a knot is represented by a closed braid on a minimal number n of strands, the conjugacy class of the braid exhausts the set of braids in B closing to define the knot. Counterexamples are given to disprove the conjecture, even when it is weakened to refer only to oriented knots.

The aim of this paper is to show that the commutator subgroup of the alternating knot group is the (proper or improper) free product of free groups with isomorphic subgroups amalgamated.

The aim of this paper is to show that the commutator subgroup of the alternating knot group is the (proper or improper) free product of free groups with isomorphic subgroups amalgamated.

Without Abstract

We introduce two graph invariants and show the relation between them. As a consequence of this relation, we show that one of these graph invariants is related to the Tutte polynomial.

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