Narrow Results

In this paper we consider the generalized Smith conjecture of codimension greater than two, which says that no periodic tranformation of S^l can have the tame knotted S^h as fixed point set if l-h>2 and h>3. Using the Brieskorn spheres, this paper gives the explicit counterexamples to show that the conjecture is false in the DIFF category for the following cases: (i) l-h is even more than 2 and l is odd; (ii) 2l≤3(h+1) and h+1≡0 (mod 4).

Let be a 3-sphere embedded in the 5-sphere S⁵ (i = 1,2). Let and intersect transversely. Then the intersection is a disjoint collection of circles. Thus we obtain a pair of 1-links, C in , and a pair of 3-knots, in S⁵ (i = 1, 2). Conversely let (L₁, L₂) be a pair of 1-links and (X₁, X₂) be a pair of 3-knots. It is natural to ask whether the pair of 1-links (L₁, L₂) is obtained as the intersection of the 3-knots X₁ and X₂ as above. We give a complete answer to this question. Our answer gives a new geometric meaning of the Arf invariant of 1-links.

Let f : S³ → S⁵ be a smooth transverse immersion such that the self-intersection C consists of double points. Suppose that C is a single circle in S⁵. Then f^-1(C) in S³ is a 1-knot or a 2-component 1-link. There is a similar realization problem. We give a complete answer to this question.

We investigate a family of (reducible) representations of the braid groups corresponding to a specific solution to the Yang–Baxter equation. The images of under these representations are finite groups, and we identify them precisely as extensions of extra-special 2-groups. The decompositions of the representations into their irreducible constituents are determined, which allows us to relate them to the well-known Jones representations of factoring over Temperley–Lieb algebras and the corresponding link invariants.

An H(2)-move is an unknotting operation of a knot, which is performed by adding a half-twisted band. We define the H(2)-Gordian distance of two knots to be the minimum number of H(2)-moves needed to transform one into the other. We give several methods to estimate the H(2)-Gordian distance of knots. Then we give a table of H(2)-Gordian distances of knots with up to 7 crossings.

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.

In this paper, we discuss the region unknotting number of different classes of 2-bridge knots. In particular, we provide region unknotting number for the classes of 2-bridge knots whose Conway notations are C(m, n), C(m, 2, m), C(m, 2, m ± 1) and C(2, m, 2, n). By generalizing, we also provide a sharp upper bound for all the remaining classes of 2-bridge knots.

We show that every alternating link of two components and $12$ crossings can be reduced by $4$-moves to the trivial link or the Hopf link. It answers the question asked in one of the last papers by Slavik Jablan.

We introduce the notion of bicolored diagrams which are closely related to the region crossing changes. Moreover, we refine Cheng’s results on the region crossing changes and propose a certain way to calculate the Arf invariant of a proper link using a bicolored diagram.

Given a knot in the 3-sphere, the non-orientable 4-genus or 4-dimensional crosscap number of a knot is the minimal first Betti number of non-orientable surfaces, smoothly and properly embedded in the 4-ball, with boundary the knot. In this paper, we calculate the non-orientable 4-genus of knots with crossing number 10.

The non-orientable 4-genus of a knot $K$ in $S^3$ is defined to be the minimum first Betti number of a non-orientable surface $F$ smoothly embedded in $B^4$ so that $K$ bounds $F$. We will survey the tools used to compute the non-orientable 4-genus, and use various techniques to calculate this invariant for non-alternating 11 crossing knots. We will also view obstructions to a knot bounding a Möbius band given by the double branched cover of $S^3$ branched over $K$.

We define the 2-signatures, 2-nullities and Arf invariants (when possible) for links which are null-homologous modulo two in a rational homology three-sphere. We define these invariants using the Goeritz form on non-oriented spanning surfaces. We develop their cobordism properties from this point of view. We give a good way to index these invariants. We also define d-signatures and d-nullities for links which are null-homologous modulo d in a rational homology sphere from the point of view of branched covers. We index d-signatures and d-nullities and develop their cobordism properties. Finally we define Arf invariants (when possible) in a general closed 3-manifold using spin structures.

The Sato-Levine invariant for 2-component links with linking number zero is generalized to links with even linking numbers using not necessarily orientable surfaces. It is well defined modulo 4 and completely classifies “unoriented pass equivalence classes” of 2-component proper links. It is shown that it can be expressed in terms of the coefficients of the Conway polynomial and some applications to the Arf invariant are given.

Let L be a link in S³, and suppose all the linking numbers in L are even. We observe that the Arf invariant of L is determined in several different ways by the sublinks containing no more than three components of L.

Let F, G be a pair of quadratic forms defined over an arbitrary field k. We give a characterization for when every nontrivial zero of F = G = 0 defined over the algebraic closure of k is nonsingular. When chark ≠ 2, this result is well known. When chark = 2, the problem divides into two cases. If n is odd, we use the half-determinant, and if n is even, we use the Arf invariant for this characterization. The characterization depends only on the coefficients of the quadratic forms and operations taking place in the field k.

In this paper, we propose a new approach towards the classification of spherical fusion categories by their Frobenius–Schur exponents. We classify spherical fusion categories of Frobenius–Schur exponent 2 up to monoidal equivalence. We also classify modular categories of Frobenius–Schur exponent 2 up to braided monoidal equivalence. It turns out that the Gauss sum is a complete invariant for modular categories of Frobenius–Schur exponent 2. This result can be viewed as a categorical analog of Arf's theorem on the classification of non-degenerate quadratic forms over fields of characteristic 2.