Journal of Knot Theory and Its Ramifications

We give elementary proofs of certain relations in the mapping class group of a closed surface of genus 2, MCG(F_{2, 0}). We generalize portions of these to relations in quandles with certain types of elements, associated to a relatively broad class of groups (including mapping class groups), and derive further similar quandle relations. We show these quandle relations correspond to 2-cycles in the homology of racks of tuples of quandle elements, and thence to families of commutation relations back in the groups. The recurrence of some of these phenomena within higher level structures is also explored, as are multiple types of modifications yielding different relations. The constructions are quite malleable in this respect.

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.

Let K be a knot in S³ that admits two disjoint, separating, essential surfaces with boundary, where one of them has an accidental annulus. Then, K admits infinitely many closed essential surfaces unless the annulus is "centered". This implies for example that any non-satellite knot that admits a 4-puntured essential sphere and any alternating knot that admits a closed essential surface of genus 2 admit infinitely many closed essential surfaces.

We study the relation between the maximum z-degree of Kauffman polynomial and the determinant for a quasi-alternating link. This gives a stronger criterion for deciding whether a non-alternating link is quasi-alternating or not than the known criterion in terms of Q-polynomials. Also, we determine all quasi-alternating links with determinant 5.

Let L = L₁ ∪ L₂ ∪ ⋯ ∪ L_n be a link in ℝ³ such that L_i is a trivial link for each i, 1 ≤ i ≤ n. Let P₁, P₂,…,P_n be mutually distinct flat planes in ℝ³ such that no two of them are parallel. Then there is a link in ℝ³ such that L is ambient isotopic to L^′ and for each i, 1 ≤ i ≤ n.

In this paper, a method is given to calculate the Jones polynomial of the 6-plat presentations of knots by using a representation of the braid group 𝔹₆ into a group of 5 × 5 matrices. We also can calculate the Jones polynomial of the 2n-plat presentations of knots by generalizing the method for the 6-plat presentations of knots. Also, it helps us to detect 3-bridge knots in 3-plat presentations.

Flat plumbing basket surfaces of links were introduced to study the geometry of the complement of the links. These flat plumbing basket surfaces can be presented by a sequential presentation known as flat plumbing basket code first found by Furihata, Hirasawa and Kobayashi. The minimum number of flat plumbings to obtain flat plumbing basket surfaces of a link is defined to be the flat plumbing basket number of the given link. In this paper, we use these sequential presentations to find the complete classification theorem of prime knots whose flat plumbing basket number is 6 or less. As applications, this result improves the work of Hirose and Nakashima which finds the flat plumbing basket number of prime knots up to nine crossings.