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Abstract: In the previous paper, the author gave linear inequalities on the coefficients of the Alexander polynomials of alternating knots of genus two, which are best possible as linear inequalities on the coefficients of them. In this paper, we give infinitely many Alexander polynomials which satisfy the linear inequalities, but they are not realized by alternating knots. [Copyright &y& Elsevier]

It is known that the pass move is not an unknotting operation in classical knot theory. In this paper, we prove that the pass move is an unknotting operation in welded knot theory. [ABSTRACT FROM AUTHOR]

Let D be a link diagram and T a 4-tangle. By replacing each crossing of D by T , we get a new diagram D ⊗ T , called a link diagram with local symmetry or tensor product of D and T . In this paper, we will study polynomial invariants of the link diagram D ⊗ T with local symmetry in terms of D and T , and as an application, we will study the adequacy of D ⊗ T . [ABSTRACT FROM AUTHOR]

A delta-unknotting operation changes the second coefficient of the Conway polynomial of a knot k only one. And a delta-unknotting operation can be realized by two ordinary unknotting operations. In this paper we study the second coefficient of the Conway polynomial of the knot which is obtained from a knot by an ordinary unknotting operation related to a given delta unknotting operation. [ABSTRACT FROM AUTHOR]

The Γ-polynomial is an invariant of an oriented link in the 3-sphere, which is the common zeroth coefficient polynomial of both the HOMFLYPT and Kauffman polynomials. It is known that the HOMFLYPT and Kauffman polynomials, their 2-cable versions, and the satellite versions of the Alexander and Jones polynomials are invariant under mutation. On the other hand, there exists a mutant knot pair which is distinguished by the 3-cable version of the HOMFLYPT polynomial. In this paper, we show that the 3-cable version of the Γ-polynomial is invariant under mutation. [ABSTRACT FROM AUTHOR]

J.-B. Meilhan and the second author showed that any Milnor μ ¯ -invariant of length between 3 and 2 k + 1 can be represented as a combination of HOMFLYPT polynomial of knots obtained by certain band sum of the link components, if all μ ¯ -invariants of length ≤ k vanish. They also showed that their formula does not hold for length 2 k + 2 . In this paper, we improve their formula to give the μ ¯ -invariants of length 2 k + 2 by adding correction terms. The correction terms can be given by a combination of HOMFLYPT polynomial of knots determined by μ ¯ -invariants of length k + 1 . In particular, for any 4-component link the μ ¯ -invariants of length 4 are given by our formula, since all μ ¯ -invariants of length 1 vanish. [ABSTRACT FROM AUTHOR]

Abstract: We study the relations between the related crossings and the second and fourth coefficients of Conway polynomials of 2-adjacent knots, the relations between the crossings and the extreme coefficients of the related Homfly polynomials and explain how these coefficients determine the signs of the crossings. We give some interesting properties of the related Homfly polynomials. In fact, we show that 17 knots of the 19 exceptions given in A. Stoimenow's paper are not 2-adjacent. [Copyright &y& Elsevier]

Abstract: Bourgoin defined the notion of a twisted link which corresponds to a stable equivalence class of links in oriented thickenings. It is a generalization of a virtual link. Some invariants of virtual links are extended for twisted links including the knot group and the Jones polynomial. In this paper, we generalize a multivariable polynomial invariant of a virtual link to a twisted link. We also introduce a quandle of a twisted link. [Copyright &y& Elsevier]

Abstract: In this paper we define a lassoing on a link, a local addition of a trivial knot to a link. Let K be an s-component link with the Conway polynomial non-zero. Let L be a link which is obtained from K by r-iterated lassoings. The complete splitting number is greater than or equal to , and less than or equal to . In particular, we obtain from a knot by r-iterated component-lassoings an algebraically completely splittable link L with . Moreover, we construct a link L whose unlinking number is greater than . [Copyright &y& Elsevier]

Abstract: A twisted link is a generalization of a virtual link, which is related to a link diagram on a closed, possibly non-orientable surface. In this paper we generalize the Miyazawa polynomial invariant of a virtual link to an invariant of a twisted link in two formulae one of which is introduced by A. Ishii and the other by the author. [Copyright &y& Elsevier]