Narrow Results

For knots in S³ criteria for free periodicity are obtained in terms of the bracket and the Jones polynomials. The criteria introduced in this paper generalize the result obtained by Murasugi for the Jones polynomials of periodic knots.

In the symmetric group S_n, we introduced the notion of crossing and linking numbers to each permutation. Then a unique factorization of a permutation is given due to its crossing number of its factors and how the factors are linked. Consequently we introduced a matrix associated to each permutation, which we used it as a tool to prove that positive braids with different matrices are not conjugate braids. Up to n≤5 it is proved that two positive permutation braids are conjugate if and only if they have the same matrix, and a complete calculation of these matrices with the associated link type is given.

We relate the tree part of the Århus integral to Milnor's μ-invariants of string-links in homology balls thus generalizing results of Habegger and Masbaum.

We use a simple geometric argument and small cancellation properties of link groups to prove that alternating links are non-trivial. Unlike most other proofs of this result, this proof uses only classic results in topology and combinatorial group theory.

We study Coxeter racks over ℤ_n and the knot and link invariants they define. We exploit the module structure of these racks to enhance the rack counting invariants and give examples showing that these enhanced invariants are stronger than the unenhanced rack counting invariants.

We show that all nontrivial embeddings of planar graphs on the torus contain a nontrivial knot or a nonsplit link. This is equivalent to showing that no minimally knotted planar spatial graphs on the torus exist that contain neither a nontrivial knot nor a nonsplit link all of whose components are unknots.

Any two knots admit orientation preserving homeomorphic Seifert surfaces, as can be seen by stabilizing. There is a generalization of a Seifert surface to the setting of links called a $C$-complex. In this paper, we ask when two links will admit orientation preserving homeomorphic $C$-complexes. In the case of 2-component links, we find that the pairwise linking number provides a complete obstruction. In the case of links with 3 or more components and zero pairwise linking number, Milnor’s triple linking number provides a complete obstruction.

We investigate properties of spatial graphs on the standard torus. It is known that nontrivial embeddings of planar graphs in the torus contain a nontrivial knot or a nonsplit link due to [2, 3]. Building on this and using the chirality of torus knots and links [9, 10], we prove that the nontrivial embeddings of simple 3-connected planar graphs in the standard torus are chiral. For the case that the spatial graph contains a nontrivial knot, the statement was shown by Castle et al. [5]. We give an alternative proof using minors instead of the Euler characteristic. To prove the case in which the graph embedding contains a nonsplit link, we show the chirality of Hopf ladders with at least three rungs, thus generalizing a theorem of Simon [12].

In [Dichromatic link invariants, Trans. Amer. Math. Soc.321(1) (1990) 197–229], Hoste and Kidwell investigated the skein theory of oriented dichromatic links in $S^3$. They introduced a multi-variable polynomial invariant $W$. We use special substitutions for some of the parameters of the invariant $W$ to show how to deduce invariants of finite type from $W$ using partial derivatives. Then we consider the 2-component 1-trivial dichromatic links. We study the Vassiliev invariants of the 2-component in the complement of the 1-component, which is equivalent to studying Vassiliev invariants for knots in $S^1\times D^2.$ We give combinatorial formulas for the type-zero and type-one invariants and we connect these invariants to existing invariants such as Aicardi's invariant. This provides us with a topological meaning of the first partial derivative, which is also shown to be universal as a type-one invariant.

This article concerns Fox colorings. We prove that if a link admits non-trivial $(2k+1)$-colorings, with prime $2k+1\ge 11$, it also admits non-trivial $(2k+1)$-colorings not involving colors $2k,2k-1$, nor $k$.

We obtain information on torsion in Khovanov cohomology by performing calculations directly over $\mathbb{Z}/p^k\mathbb{Z}$ for $p$ prime and $k\ge 2$. In particular, we get that the torus knots $T(9,10)$ and $T(9,11)$ contain torsion of order $9$ and $27$ in their Khovanov cohomology.

In this paper, we adjust the Yang–Baxter operators constructed by Jones for the HOMFLYPT polynomial. Then we compute the second homology for this family of Yang–Baxter operators. It has the potential to yield 2-cocycle invariants for links.

The possible $2$-variable link polynomial $P_L(v,z)$ for an oriented link $L$, which has width $2$ in the variable $v,$ is studied, where width of $P_L(v,z)$ is the minimal number of strands allowed by the index bound. It is shown that if the 2-variable link invariant $P_L(v,z)$ for an oriented link $L$ has width $k$ in the variable $v,$ then it is the same as the polynomial of a closed $k$-braid, $k=1,2.$ Also a complete list of $3$- braids of width $2,$ which close to knots, are given. Consequently it is shown that $P_L(v,z)$ determines $\widehat{\beta }$ for the full $3$-braid $\beta .$ Finally, the 2-variable polynomial for the non-full 3-braid is calculated.

Gram determinants earned traction among knot theorists after Witten’s presumption about the existence of a 3-manifold invariant connected to the Jones polynomial. Triggered by the creation of such an invariant by Reshetikhin and Turaev, several mathematicians have explored this line of research ever since. Gram determinants came into play by Lickorish’s skein theoretic approach to the invariant. The construction of different bilinear forms is possible through changes in the ambient surface of the Kauffman bracket skein module. Hence, different types of Gram determinants have arisen in knot theory throughout the years; some of these determinants are discussed here. In this paper, we introduce a new version of such a determinant from the Möbius band and prove some important results about its structure. In particular, we explore its connection to the annulus case and factors of its closed formula.

This is a survey of certain geometric aspects of inviscid and incompressible fluid flows, which are described by the solutions to the Euler equations. We will review Arnold’s theorem on the topological structure of stationary fluids in compact manifolds, and Moffatt’s theorem on the topological interpretation of helicity in terms of knot invariants. The recent realization theorem by Enciso and Peralta-Salas of vortex lines of arbitrarily complicated topology for stationary solutions to the Euler equations will also be introduced. The aim of this paper is not to provide detailed proofs of all the stated results but to introduce the main ideas and methods behind certain selected topics of the subject known as Topological Fluid Mechanics. This is the set of lecture notes, the author gave at the XXIV International Fall Workshop on Geometry and Physics held in Zaragoza (Spain) during September 2015.