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We introduce a family of polynomial invariants for flat virtual knots which extend the polynomial introduced by Kauffman and Richter, and we give several properties and examples.

We introduce some polynomial invariants for flat virtual links which are similar to the Jones–Kauffman polynomial, the Miyazawa polynomial and the arrow polynomial for virtual link diagrams, and we give several properties and examples.

We investigate embeddings from the set of long flat virtual knot diagrams to the set of long virtual knot diagrams so that we can construct invariants for long flat virtual knots. Also, we give properties and examples of several invariants for long flat virtual knots via these embeddings and invariants for long virtual knots.

We give a quotient of the ring so that the Miyazawa polynomial is a non-trivial invariant of welded links. Furthermore we show that this is also an invariant under the other forbidden move , and so it is a fused isotopy invariant. Also, we give some quotient ring so that the index polynomial can be an invariant for welded links.

We introduce invariants of flat virtual links which are induced from Vassiliev invariants of degree one for virtual links. Also we give several properties of these invariants for flat virtual links and examples. In particular, if the value of some invariants of flat virtual knots F are non-zero, then F is non-invertible so that every virtual knot overlying F is non-invertible.

We introduce a family of polynomial invariants for knotoids. Via the corresponding Gauss diagrams for knotoids, we investigate the method to calculate these polynomial invariants. Also, we give some properties and examples.

In this paper, we present the Goeritz matrix for checkerboard colorable virtual links or, equivalently, checkerboard colorable links in thickened surfaces Sg × [0, 1], which is an extension of the Goeritz matrix for classical knots and links in ℝ3. Using this, we show that the signature, nullity and determinant of classical oriented knots and links extend to those of checkerboard colorable oriented virtual links.

We introduce a polynomial invariant of virtual links that is non-trivial for many virtuals, but is trivial on classical links. Also this polynomial is sometimes useful to find the virtual crossing number of virtual knots. We give various properties of this polynomial and examples.

We introduce a polynomial invariant of long virtual knots that is non-trivial for many virtual knots, but is trivial on classical knots. Also this polynomial is sometimes useful for finding the virtual crossing number of long virtual knots. We give various properties of this polynomial and examples.

Approximate fibrations form a useful class of maps. By definition fibrators provide instant detection of maps in this class, and PL fibrators do the same in the PL category. We show that every closed s-hopfian t-aspherical manifold N with some algebraic conditions and X(N) 0 is a codimension-(2t + 2) PL fibrator.

We describe some conditions under which the product of two groups with certain property is a group with the same property, and we describe some conditions under which the product of hopfian manifolds is another hopfian manifold. As applications, we find some PL fibrators among the product of fibrators.

Approximate flbrations form a useful class of maps. By deflnition flbrators provide instant detection of maps in this class, and PL flbrators do the same in the PL category. We show that ev- ery closed s-hopflan t-aspherical manifold N with sparsely Abelian, hopflan fundamental group and ´(N) 6 0 is a codimension-(t + 1) PL flbrator.

Approximate fibrations form a useful class of maps. By definition fibrators provide instant detection of maps in this class, and PL fibrators do the same in the PL category. This paper formalizes a natural concept of partial asphericity and establishes fibrator properties of certain partially aspherical closed manifolds. One consequence is that any connected sum of aspherical PL manifolds with residually finite fundamental groups is a codimension-(2 n −2) PL fibrator.

The main result provides mild conditions under which a closed, orientable, PL 4-manifold $N = N_1\,\#\,N_2$ with $\pi_1(N_e)$ residually finite ( $e=1,2$ ) is a codimension-5 PL fibrator. The paper also presents a rich variety of conditions on a closed 4-manifold $N^4$ under which every PL map between manifolds, where the domain is orientable and all point inverses are copies of $N^4$ , must be an approximate fibration.

Let N be a Closed n-manifold with residually finite, torsion free (N) and finite H,(N). Suppose that k/(N)=0 for 1 (N) satisfies a certain condition.

LetNbe a closeds-Hopfiann-manifold with residually finite, torsion free π1(N) and finiteH1(N). Suppose that either πk(N) is finitely generated for allk≥ 2, or πk(N) ≅ 0 for 1

We introduce the odd index polynomial and the odd arrow polynomial for virtual links which are different from the original index polynomial and arrow polynomial.

We introduce a sequence of polynomial invariants for Gauss diagrams which are one-to-one correspondence with virtual knot diagrams. Also, we give some properties of these polynomials and examples.

We introduce the even index polynomial of virtual link diagrams and some even state sum polynomials which are the even Jones–Kauffman, even Miyazawa and even arrow polynomials. By simple computation, we can distinguish virtual links with these even polynomial invariants. Also, we give examples so that these invariants are different from the original ones.

Fibrators help detect approximate fibrations. A closed, connected n-manifold N is called a codimension-2 fibrator if each map p: M -s B defined on an (n + 2)-manifold M such that all fibre p-1 (b), b E B, are shape equivalent to N is an approximate fibration. The most natural objects N to study are s-Hopfian manifolds. In this note we give some necessary and sufficient conditions for s-Hopfian manifolds to be codimension-2 fibrators. Approximate fibrations, introduced and extensively studied by Coram and Duvall, form an important class of mappings, almost as efficacious as Hurewicz fibrations and cell-like maps. The advantage of this notion is that on one hand there exists an exact homotopy sequence but on the other hand there are more such approximate fibrations available. (See [3, 4] for the definition and usefulness of approximate fibrations.) Fibrators help detect approximate fibrations. A closed, connected n-manifold N is called a codimension-2 fibrator if each map p: M -> B defined on an (n +2)-manifold M such that all fibres p (b), b E B, are shape equivalent to N is an approximate fibration. The most natural objects N to study are s-Hopfian manifolds. All closed s-Hopfian manifolds with either trivial fundamental group or Hopfian fundamental group and nonzero Euler characteristic or hyperhopfian fundamental group are known to be codimension-2 fibrators [6, 7, 15, 16, 18]. Surprisingly few nonfibrators are known so far. The best well-known nonfibrators are closed manifolds that cyclically cover themselves nontrivially, such as S1 x S1 and IRP'#RPf [6, Theorem 4.2 ]. Recently Daverman gave some other kind of nonfibrator, which is L(p, q) x S3 [8, Example 2.1]. None of them gives necessary and sufficient conditions for s-Hopfian manifolds to be codimension-2 fibrators in general cases. In this note we give some necessary and sufficient conditions in some special cases. Received by the editors October 19, 1999. 2000 Mathematics Subject Classification. Primary 57N15, 55M25; Secondary 57M10, 54B15.

From a point of view of crystallizations as extended Heegaard diagrams, we introduce LCG (≡ linear cut and glue) moves with which we investigate relations between geometric T-transformations and crystallization moves. As applications of LCG moves, we verify various equivalences of Lins' crystallizations rin by means of LCG moves. In particular, we show that the two inequivalent Heegaard splittings of the Brieskorn homology sphere ∑(2,3,7) are equivalent under single stabilization of each Heegaard splitting by explicitly showing LCG moves associated with the stable equivalence.

In this paper, we show that ifNmis a closed manifold with hyperhopfian fundamental group,πi(N)=0for1

We introduce the odd Jones–Kauffman polynomial and odd Miyazawa polynomials of virtual link diagrams by using the parity of virtual link diagrams given in [Y. H. Im and K. I. Park, A parity and a multi-variable polynomial invariant for virtual links, J. Knot Theory Ramifications22(13) (2013), Article ID: 1350073, 18pp.], which are different from the original Jones–Kauffman and Miyazawa polynomials. Also, we give a family of parities and odd polynomials for virtual knots so that many virtual knots can be distinguished.

H. Dye defined the parity mapping for a virtual knot diagram, which is a map from the set of real crossings of the diagram to ℤ. The notion generalizes the parity which is studied extensively by V. Manturov. The mapping induces the ith writhe (i ∈ ℤ\{0}) which is an invariant of the representing virtual knot. She applied the parity mapping to introduce a grade to the Henrich S-invariant for a virtual knot, and showed that the invariants are Vassiliev invariants of degree one. Following it, we define the parity mappings for a virtual link diagram, and define the similar invariants as above for a virtual link by using the parity mappings. We show that some of the invariants are Vassiliev invariants of degree one. We also checked necessary conditions for invertibility and amphicheirality via the invariants.

We introduce a polynomial invariant with two variables for an oriented virtual knot, which refines the odd writhe polynomial with one variable due to Cheng by using a modified version of the warping degree. Our invariant is a Vassiliev invariant of degree one, reduces to one variable for a checkerboard colorable virtual knot, vanishes for classical knots, and detects non-invertibility and non-amphicheirality for some cases. We raise some examples to show effectiveness of our invariant. Moreover we define a similar invariant for a flat virtual knot.

We construct some polynomial invariants for virtual links by the recursive method, which are different from the index polynomial invariant defined in [Y. H. Im, K. Lee and S. Y. Lee, Index polynomial invariant of virtual links, J. Knot Theory Ramifications19(5) (2010) 709–725]. We show that these polynomials can distinguish whether virtual knots can be invertible or not although the index polynomial cannot distinguish the invertibility of virtual knots.

We introduce a four-variable index polynomial invariant of long virtual knots that is non-trivial for many long virtual knots, but is trivial for classical knots so that it is an extension of a one-variable index polynomial invariant introduced in [A polynomial invariant of long virtual knots, European J. Combin.30 (2009) 1289–1296]. We give various properties of this polynomial and examples. Also, we use this polynomial invariant for long virtual knots to distinguish virtual knots, and we obtain a polynomial invariant for long flat virtual knots which is very useful to determine whether long flat virtual knots are invertible or not.

In this paper, we investigate the behaviors of the index polynomial and the arrow polynomial for periodic virtual links. As applications, we give necessary conditions for a given virtual link to have period n in terms of those polynomials, which can be used to estimate the possible periods for a given virtual link with non-trivial index polynomial or non-trivial arrow polynomial.

We describe several conditions under which the product of hopfian manifolds is another hopfian manifold. As applications, the product F × A of a closed hopfian n-manifold F and a closed orientable aspherical m-manifold A is hopfian when either π 1 ( F ) is solvable and χ ( A )≠0 or π 1 ( F ) is finite. Also, the product of any rational homology n-sphere Σ n for which π 1 ( Σ n ) is finite and a closed orientable n-manifold N with π i ( N )=0 for 1 i n −1 is hopfian. Using such facts we investigate conditions under which products of codimension-2 (orientable) fibrators are again codimension-2 (orientable) fibrators.

In this paper, we show that any finite product N n of closed orientable surfaces of genus at least 2 satisfies the property that all proper, surjective mappings p from an orientable ( n +2)-manifold M to a 2-manifold B for which each p −1 ( b ) is homotopy equivalent to N n necessarily are approximate fibrations.

We introduce a polynomial invariant of flat virtual knots which is sometimes useful for determining whether given flat virtual knots are invertible or not, and for finding the virtual crossing number of flat virtual knots. Also we give several properties of the polynomial invariant for flat virtual knots and examples. In particular, we show that the conjecture given by Hrencecin and Kauffman (2003) [8] is true and so an infinite class of flat virtual knots {Un}(n=1,2,…) are mutually distinct.