Narrow Results

We define a local move on a ribbon 2-knot diagram, called an HC-move. We show that it is an unknotting operation for a ribbon 2-knot, and that the application of a single HC-move to a ribbon 2-knot changes the second derivative at t=1 of its normalized Alexander polynomial by either ±2 or 0. This result is applied to the calculation of the HC-unknotting numbers of ribbon 2-knots. We also consider a relation with a 1-handle unknotting operation.

We give a basis for the space of the Vassiliev knot invariants of order 6, which implies a sixth order Vassiliev knot invariant is determined by its HOMFLY and Kauffman polynomials. Furthermore, we show that there is a seventh order invariant which cannot be obtained from these polynomials of a knot.

In this note, we will study Δ link homotopy (or self Δ-equivalence), which is an equivalence relation of ordered and oriented link types. Previously, a necessary condition is given in the terms of Conway polynomials for two link types to be Δ link homotopic. A pair of numerical invariants δ₁ and δ₂ classifies all (ordered and oriented) prime 2-component link types with seven crossings or less up to Δ link homotopy. We will show here that for any pair of integers n₁ and n₂ there exists a 2-component link κ such that δ₁(κ) = n₁ and δ₂(κ) = n₂ provided that at least one of n₁ and n₂ is even.

The purpose of this note is to give a formula for the Casson knot invariant (the second coefficient of the Conway polynomial) of a 2-bridge knot, calculating without using a continued fraction expansion.

It is an open question whether there are Vassiliev invariants that can distinguish an oriented knot from its inverse, i.e., the knot with the opposite orientation. In this article, an example is given for a first order Vassiliev invariant that takes different values on a virtual knot and its inverse. The Vassiliev invariant is derived from the Conway polynomial for virtual knots. Furthermore, it is shown that the zeroth order Vassiliev invariant coming from the Conway polynomial cannot distinguish a virtual link from its inverse and that it vanishes for virtual knots.

The transforms of two oriented parallel strands to a k-half twist of two strands are called t_k-move and -move respectively depending on the orientations of the two strands. In this paper we give criterions to detect whether a knot K can be transformed to a knot K' by t_2k-moves and -moves respectively and if so, we give some results on how many moves are needed in these transformations respectively, by using some Vassiliev invariants. Moreover we give a relation between the Δ-move and the t_2k-move by considering the coefficient of z² in the Conway polynomial of a knot, which is a Vassiliev invariant of degree 2.

We consider how C_n-move has an influence on the Vassiliev invariant of order m. In other words, we investigate the variance of the value of concrete Vassiliev invariant of order m by a C_n-move. We choose the mth coefficient of the Conway polynomial, a_m, as a concrete Vassiliev invariant of order m, and study the variance of that by a C_n-move for n≤4. And we get that when we give a natural number k, there is a pair of knots satisfying that they are transformed into each other by a C_n-move and the difference of a_m of them is equal to k in almost case under the condition that n<m and n≤4.

Nakanishi and Shibuya gave a relation between link homotopy and quasi self delta-equivalence. And they also gave a necessary condition for two links to be self delta-equivalent by using the multivariable Alexander polynomial. Link homotopy and quasi self delta-equivalence are also called self C₁-equivalence and quasi self C₂-equivalence respectively. In this paper, we generalize their results. In Sec. 1, we give a relation between self C_k-equivalence and quasi self C_k+1-equivalence. In Secs. 2 and 3, we give necessary conditions for two links to be self C_k-equivalent by using the multivariable Conway potential function and the Conway polynomial respectively.

Formulas previously presented for the Casson–Walker invariant are generalized to Lescop's extension. These formulas in terms of linking numbers and surgery coefficients compute the change in Lescop's invariant under crossing changes in a framed link presenting a 3-manifold. This leads us to revisit an old formula for a coefficient of the Conway polynomial. Finally we compute Lescop's invariant of several lens spaces.

Geometric aspects of the filtration on classical links by k-quasi-isotopy are discussed, including the effect of Whitehead doubling, relations with Smythe's n-splitting and Kobayashi's k-contractibility. One observation is: ω-quasi-isotopy is equivalent to PL isotopy for links in a homotopy 3-sphere (respectively, contractible open 3-manifold) M if and only if M is homeomorphic to S³ (respectively, ℝ³). As a byproduct of the proof of the "if" part, we obtain that every compact subset of an acyclic open set in a compact orientable 3-manifold M is contained in a PL homology 3-ball in M.

We show that k-quasi-isotopy implies (k + 1)-cobordism of Cochran and Orr. If z^m-1 (c₀+c₁z²+⋯+c_nz²ⁿ) denotes the Conway polynomial of an m-component link, it follows that the residue class of c_k modulo gcd(c₀,…,c_k-1) is invariant under k-quasi-isotopy. Another corollary is that each Cochran's derived invariant β^k is also invariant under k-quasi-isotopy, and therefore assumes the same value on all PL links, sufficiently C⁰-close to a given topological link. This overcomes an algebraic obstacle encountered by Kojima and Yamasaki, who "became aware of impossibility to define" for wild links what for PL links is equivalent to the formal power series ∑βⁿzⁿ by a change of variable.

Let V be the standard solid torus in S³. Let K_{p, 2} be the (p, 2)-torus knot in V such that K_{p, 2} meets a meridian disk D of V in two points with the winding number zero and the 2-string tangle TK_{p, 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,…,b_2m) and pattern is (V, K_{p, 2}).

It is well-known that the coefficient of z^m of the Conway polynomial is a Vassiiev invariant of order m. In this paper, we show that for any given pair of a natural number n and a knot K, there exist infinitely many knots whose Vassiliev invariants of order less than or equal to n and Conway polynomials coincide with those of K.

By the works of Gusarov [2] and Habiro [3], it is known that a local move called the C_n move is strongly related to Vassiliev invariants of order less than n. The coefficient of the zⁿ term in the Conway polynomial is known to be a Vassiliev invariant of order n. In this note, we will consider to what degree the relationship is strong with respect to Conway polynomial. Let K be a knot, and K^C_n the set of knots obtained from a knot K by a single C_n move. Let be the set of the Conway polynomials for a set of knots . Our main result is the following: There exists a pair of knots K₁, K₂ such that ∇K₁ = ∇K₂ and . In other words, the C_n Gordian complex is not homogeneous with respect to Conway polynomial.

Levine showed that the Conway polynomial of a link is a product of two factors: one is the Conway polynomial of a knot which is obtained from the link by banding together the components; and the other is determined by the -invariants of a string link with the link as its closure. We give another description of the latter factor: the determinant of a matrix whose entries are linking pairings in the infinite cyclic covering space of the knot complement, which take values in the quotient field of ℤ[t, t^-1]. In addition, we give a relation between the Taylor expansion of a linking pairing around t = 1 and derivation on links which is invented by Cochran. In fact, the coefficients of the powers of t - 1 will be the linking numbers of certain derived links in S³. Therefore, the first non-vanishing coefficient of the Conway polynomial is determined by the linking numbers in S³. This generalizes a result of Hoste.

It is shown that two knots can be transformed into each other by C_n-moves if and only if they have the same Vassiliev invariants of order less than n. Consequently, a C_n-move cannot change the Vassiliev invariants of order less than n and may change those of order more than or equal to n. In this paper, we consider the coefficient of the Conway polynomial as a Vassiliev invariant and show that a C_n-move changes the nth coefficient of the Conway polynomial by ±2, or 0. And for the 2mth coefficient (2m > n), it can change by p or p + 1 for any given integer p.

We describe the Polyak–Viro arrow diagram formulas for the coefficients of the Conway polynomial. As a consequence, we obtain the Conway polynomial as a state sum over some subsets of the crossings of the knot diagram. It turns out to be a simplification of a special case of Jaeger's state model for the HOMFLY polynomial.

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

A sharp move is a local move introduced by Murakami. In this note, we will introduce local moves generated by a single sharp move. We give an infinite family of composite knots with sharp unknotting number one. We study Conway polynomials of knots with sharp unknotting number one.

An oriented 2-component link is called band-trivializable, if it can be unknotted by a single band surgery. We consider whether a given 2-component link is band-trivializable or not. Then we can completely determine the band-trivializability for the prime links with up to 9 crossings. We use the signature, the Jones and Q polynomials, and the Arf invariant. Since a band-trivializable link has 4-ball genus zero, we also give a table for the 4-ball genus of the prime links with up to 9 crossings. Furthermore, we give an additional answer to the problem of whether a (2n + 1)-crossing 2-bridge knot is related to a (2, 2n) torus link or not by a band surgery for n = 3, 4, which comes from the study of a DNA site-specific recombination.

We give a counterexample to the generalized Kawauchi conjecture on the Conway polynomial of achiral knots which asserts that the Conway polynomial C(z) of an achiral knot satisfies the splitting property C(z) = F(z)F(-z) for a polynomial F(z) with integer coefficients. We show that the Bonahon–Siebenmann decomposition of an achiral and alternating knot is reflected in the Conway polynomial. More explicitly, the generalized Kawauchi conjecture is true for quasi-arborescent knots and counterexamples in the class of alternating knots must be quasi-polyhedral.