Surface Knots Research Articles

The bridge number of surface links and kei colorings

Meier and Zupan introduced bridge trisections of surface links in S 4 $S^4$ as a 4-dimensional analogue to bridge decompositions of classical links, which gives a numerical invariant of surface links called the bridge number. We prove that there exist infinitely many surface knots with bridge number n $n$ for any integer n ⩾ 4 $n \geqslant 4$ . To prove it, we use colorings of surface links by keis and give lower bounds for the bridge number of surface links.

Dynamical ideals of non-commutative rings

A dynamical analog of the prime ideals for simple non-commutative rings is introduced. We prove a factorization theorem for the dynamical ideals. The result is used to classify the surface knots and links in the smooth four-dimensional manifolds.

Accurate and automated detection of surface knots on sawn timbers using YOLO-V5 model

Knot detection is a challenging problem for the wood industry. Traditional methodologies depend heavily on the features selected manually and therefore were not always accurate due to the variety of knot appearances. This paper proposes an automated framework for addressing the aforementioned problem by using the state-of-the-art YOLO-v5 (the fifth version of You Only Look Once) detector. The features of surface knots were learned and extracted adaptively, and then the knot defects were identified accurately even though the knots vary in terms of color and texture. The proposed method was compared with YOLO-v3 SPP and Faster R-CNN on two datasets. Experimental results demonstrated that YOLO-v5 model achieved the best performance for detecting surface knot defects. F-Score on Dataset 1 was 91.7% and that of Dataset 2 was up to 97.7%. Moreover, YOLO-v5 has clear advantages in terms of training speed and the size of the weight file. These advantages made YOLO-v5 more suitable for the detection of surface knots on sawn timbers and potential for timber grading.

Wilson Surfaces for Surface Knots: A Field Theoretic Route to Higher Knots

AbstractHolonomy invariants in strict higher gauge theory have been studied in depth aiming to applications to higher Chern–Simons theory. For a flat 2–connection, the holonomy of surface knots of arbitrary genus has been defined and its covariance properties under 1–gauge transformation and change of base data have been determined. Using quandle theory, a definition of trace over a crossed module such to yield surface knot invariants upon application to 2–holonomies has been given.

Knotted surfaces in $4$-manifolds by knot surgery and stabilization

Given a simply-connected closed 4-manifold $X$ and a smoothly embedded oriented surface $\Sigma$, various constructions based on Fintushel-Stern knot surgery have produced new surfaces in $X$ that are pairwise homeomorphic to $\Sigma$, but not diffeomorphic. We prove that for all known examples of surface knots constructed from knot surgery operations that preserve the fundamental group of the complement of surface knots, they become pairwise diffeomorphic after stabilizing by connected summing with one $S^2\tilde{\times}S^2$. When $X$ is spin, we show in addition that any surfaces obtained by a knot surgery whose complements have cyclic fundamental group become pairwise diffeomorphic after one stabilization by $S^2\tilde{\times}S^2$.

Frequencies and Severity of Injection-Site Lesions in Muscles from Rounds of Cow Carcasses

ObjectivesThe National Market Cow and Bull Beef Quality Audit (NMCBBQA) has been conducted 4 times: 1994, 1999, 2007, and 2016. Following the original audit attention was drawn to injection-site lesions and the need to further assess the frequency and severity. The 2016 audit reported a major decline in presence of surface knots or injection-site lesions compared with previous audits. The presence of lesions was discussed in detail at the strategy workshop held as the third phase of the 2016 NMCBBQA. Further processors expressed extreme concern of loss and the need to once again evaluate lesions as they had been evaluated in 1998, 1999, and 2000. Therefore, the objective of this study was to evaluate the frequency and presence of injection-site lesions in the outside round muscles of cows.Materials and MethodsAudits were conducted in 2017 on 1300 rounds from dairy and beef cows. Audits were conducted in 7 locations throughout the U.S. Outside round muscles were butterfly cut into 1.25-cm slices and if present lesions were counted, measured and categorized. The outside round consisted of 4 quadrants: Q1 identified as the most caudal end, closest to the shank, while Q4 identified as the most cranial end, included the biceps femoris muscle. Q2 and Q3 were evenly split between Q1 and Q4. Q1, Q2, and Q3 included both muscles. The depth of each lesion was measured from the outside surface (fat trimmed) to the innermost (center) of the lesion. The diameter was measured using the lengths of the lesion throughout the muscles. The lesions were classified using the 5-point system: classification 1: clear, lesion contains primarily clear connective tissue; classification 2: woody, lesion contains organized connective tissue and fat; classification 3: nodular, lesion contains nodules, a central foci, and granulomatous inflammation; classification 4: metallic, lesion contains mineralized remnants of muscle cells, typically bloody color; or classification 5: cystic, lesion contains fluid.ResultsOf the beef and dairy rounds evaluated 7 and 15%, respectively, had injection-site lesion(s) present. These frequencies were significantly decreased in 2017 from the 2000 audit where beef rounds were 13% points higher and dairy rounds were 20% points higher. The decline was even more significant from the 1998 audit where beef rounds were 24% points higher and dairy rounds were 45% points higher. The most common location of injection-site lesions was quadrant 2 and 3, which contained both the biceps femoris and semitendinosus muscles. Injection-site lesions were more frequent (P < 0.05) in the biceps femoris for both beef and dairy rounds. Clear and metallic lesions were the most common. Clear lesions accounted for 57% of injection-sites in both beef and dairy rounds while metallic lesions made up 23% of the total in beef and 25% in dairy.ConclusionOverall, there was a dramatic decline in the frequency (P < 0.05) of injection-site lesions since the 1998, 1999, and 2000 audits. Educational programs, such as Beef Quality Assurance, have increased improvements in management practices for both the beef and dairy industries

On higher holonomy invariants in higher gauge theory I

This is the first of a series of two technical papers devoted to the analysis of holonomy invariants in strict higher gauge theory with end applications in higher Chern–Simons theory. For a flat 2-connection, we define the 2-holonomy of surface knots of arbitrary genus and determine its covariance properties under 1-gauge transformation and change of base data.

FUNDAMENTAL BIQUANDLES OF RIBBON 2-KNOTS AND RIBBON TORUS-KNOTS WITH ISOMORPHIC FUNDAMENTAL QUANDLES

The fundamental quandles and biquandles are invariants of classical knots and surface knots. It is unknown whether there exist classical or surface knots which have isomorphic fundamental quandles and distinct fundamental biquandles. We show that ribbon 2-knots or ribbon torus-knots with isomorphic fundamental quandles have isomorphic fundamental biquandles. For this purpose, we give a method for obtaining a presentation of the fundamental biquandle of a ribbon 2-knot/torus-knot from its fundamental quandle.

UNKNOTTING NUMBERS AND TRIPLE POINT CANCELLING NUMBERS OF TORUS-COVERING KNOTS

It is known that any surface knot can be transformed to an unknotted surface knot or a surface knot which has a diagram with no triple points by a finite number of 1-handle additions. The minimum number of such 1-handles is called the unknotting number or the triple point cancelling number, respectively. In this paper, we give upper bounds and lower bounds of unknotting numbers and triple point cancelling numbers of torus-covering knots, which are surface knots in the form of coverings over the standard torus T. Upper bounds are given by using m-charts on T presenting torus-covering knots, and lower bounds are given by using quandle colorings and quandle cocycle invariants.

THE ALGEBRA OF RACK AND QUANDLE COHOMOLOGY

This paper presents the first complete calculation of the cohomology of any nontrivial quandle, establishing that this cohomology exhibits a very rich and interesting algebraic structure. Rack and quandle cohomology have been applied in recent years to attack a number of problems in the theory of knots and their generalizations like virtual knots and higher-dimensional knots. An example of this is estimating the minimal number of triple points of surface knots [E. Hatakenaka, An estimate of the triple point numbers of surface knots by quandle cocycle invariants, Topology Appl139(1–3) (2004) 129–144.]. The theoretical importance of rack cohomology is exemplified by a theorem [R. Fenn, C. Rourke and B. Sanderson, James bundles and applications, Proc. London Math. Soc. (3)89(1) (2004) 217–240] identifying the homotopy groups of a rack space with a group of bordism classes of high-dimensional knots. There are also relations with other fields, like the study of solutions of the Yang–Baxter equations.

The lower bound of the 𝑤-indices of surface links via quandle cocycle invariants

The w w -index of a surface link F F is the minimal number of the triple points of surface braids representing F F . In this paper, for a given 3 3 -cocycle, we consider the minimal number of the w w -indices of surface links whose quandle cocycle invariants associated with f f are non-trivial, and denote it ω ( f ) \omega (f) . In particular, we show that ω ( θ 3 ) = 6 \omega (\theta _3)=6 and ω ( θ p ) ≥ 7 \omega (\theta _p)\geq 7 , where θ n \theta _n is Mochizuki’s 3 3 -cocycle of the dihedral quandle of order n n and p p is an odd prime integer ≠ 3 \not =3 . As a consequence, for a given non-negative integer g g , there are surface knots with genus g g with the w w -index 6 6 .

Widths of surface knots

We study surface knots in 4‐space by using generic planar projections. These projections have fold points and cusps as their singularities and the image of the singular point set divides the plane into several regions. The width (or the total width) of a surface knot is a numerical invariant related to the number of points in the inverse image of a point in each of the regions. We determine the widths of certain surface knots and characterize those surface knots with small total widths. Relation to the surface braid index is also studied. 57Q45; 57M25

On 1-handle surgery and finite type invariants of surface knots

Vassiliev's 1-knot invariants are defined in a combinatorial way as finite type invariants. The notion of a finite type invariant associated with given local operations is defined. In this paper we prove that a local operation called a 1-handle surgery is an unknotting operation for generically immersed surfaces in 4-space and investigate the finite type invariants of such surfaces in 4-space associated with this operation.