Low Dimensional Knot Ahmad Lutfi Dec 2024 (8)

1 Dedication Without the following, this paper would have never seen the light, hence it is my greatest pleasure to mention the following helpers, not tutors, in ascending Alphabetic order: 1. Google: main research buddy, mainly with ’books’ and ’scholar’. 2.Numdam: for the help with LateX’s syntax understanding and comprehension by the share of tex papers, besides offering academic work in English and French. 3.https://www.mathgenealogy.org/id.phpid=310110Simon Jubert: for recommending ”Overleaf.com”, a thanks might be not be enough, as without him, all effort, progress, and work would have gone in vein, with the wind, ”Une grande merci à vous”. 4. Slmath: for offering resources, both physical and online, in addition to offering a set of references and links, and their welcoming hospitality. This page would be intentionally left blank if we would not wish to inform about that. Contents 1 Dedication 2 Introduction 3 Instructions for preparing your manuscript 4 Adding figures and tables 4.1 Adding figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Monodromy 5.1 Adding tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Euler Characteristic 7 Low Dimensional Topology 7.0.1 Sub-manifold (definition) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.0.2 Presentations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Homotopy Equivalences in Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Knots 8.1 knot types . . . . . . . . . . . . . 8.1.1 Briad . . . . . . . . . . . . 8.1.2 Contact Geometry . . . . . 8.2 Seifert Surface . . . . . . . . . . . 8.2.1 Definition . . . . . . . . . . 8.2.2 Motivation . . . . . . . . . 8.2.3 Seifert Matrix . . . . . . . . 8.2.4 Goeritz matrix . . . . . . . 8.2.5 Goeritz-Seifert Relationship 8.2.6 padic integers . . . . . . . 8.3 knot meaning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 knot polynomials 9.1 Alexander polynomial . . . . 9.2 Kaufman Bracket . . . . . . . 9.3 Jones polynomial . . . . . . . 9.3.1 Alternating diagrams . 9.4 Oscar Simony’s Torus Knots 9.5 Seifert Polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Contact Geometry 10.1 Structure Forms . . . . . . . . . . . . . 10.2 Odd, Even Dimensions . . . . . . . . . . 10.2.1 Example (Field of Planes) . . . . 10.3 Contact Structures as Symplectic Cones 10.4 Odd Dimension Advantage . . . . . . . 10.5 graphs and Legendre varieties . . . . . . 10.6 Markov Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Atiyah Theory 11.1 Atiyah conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.1 Strong Atiyah Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Appendix 12.1 Morphism Review . . . . . 12.2 Isomorphism . . . . . . . . 12.2.1 Generators . . . . . 12.2.2 Cap Product . . . 12.2.3 Poincaré Duality . . 12.2.4 Use of Isomorphism 12.3 Gauss-Bonnet . . . . . . . . 12.3.1 Normal Bundle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Levi-Cevita 13.1 Sub-manifolds . . . . . . . . . . . . . . 13.2 Haken transform . . . . . . . . . . . . 13.3 Fundamental Group . . . . . . . . . . 13.3.1 Fundamental Group on knots 13.3.2 On a Topological Space . . . . 13.3.3 Relaxed Atiyah Cohomology . 13.3.4 On a group . . . . . . . . . . . 13.3.5 On a Scheme Group . . . . . . 13.3.6 Cone . . . . . . . . . . . . . . 13.4 Co-fiber . . . . . . . . . . . . . . . . . 13.4.1 Cone . . . . . . . . . . . . . . 13.4.2 On a Vector Space . . . . . . . 13.4.3 On a type . . . . . . . . . . . . 13.4.4 On a Manifold . . . . . . . . . 13.4.5 On a Category . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Conclusion 15 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction to Low-Dimension Topology and Knots Ahmad Lutfi January 1, 2025 Abstract In this paper, we examine knots from a beginner’s perspective, along with related terminology which may aid the researcher if understood from a different angle, in addition to the Monodromy, Euler characteristic function, Fundamental groups which are touched upon, focusing on knots their types and homologies, and polynomials. besides that a short appendix, finally we conclude with future recommendations. Introduction 2 Introduction ancientGreece Low-dimensional topology is a dense subject, with topics yet to be explored, however, its prominent part is about the knots. Excluding the Babylonian culture, the first knot known to man can possibly be the Gordian knot. The knot’s name is still mysterious and subject to research. Having suffered hard times, and relying only on an Oracle’s prophecy, Lydians started searching for their savior king, finding him in Midas as a slave, d and at Midas, they crowned their king, who led a fierce battle in Gordain , slaying and cutting open the Gordian’s knot, ending the Phylogian’s reign knot, becoming a hero (and subsequently a hero’s journey to rule the world, most of. Note this legend resembles Alexander The Great, whom this legend is allegedly written, with also some mystery. In addition, the Gordian Knot emphasizes the prominent position of Kybele and Zeus, between religion and legend.passing then throughout the renaissance and Leonardo da Vinci, passing to the modern world. A more recently,[?] have studied The fourth and sixth Da Vinci-Druer knot, There is a hexagonal lattice core that is invariant (Under the reflection of the points set p). In result, they were able to generate two new infinities, where each Link can have an infinite, number of knots, leading to WW2 and the beginning of maturity of Knots. 3 Instructions for preparing your manuscript There are specific instructions for preparing your article for the DH Benelux Journal. 1 . 4 Adding figures and tables This section has two sub-sections. 4.1 Adding figures If you include figures or tables, use reference keys (e.g. Figure 1), to create an unambiguous reference from the text to each figure and table, e.g. to Table 6. 1 See http://journal.dhbenelux.org/submission/preparing-the-final-version-of-your-manuscript/ Figure 1: This is where the caption of the figure goes. Remember, figure captions go below the figure. Table 1: Table caption. Captions for tables go above the table, unlike for figures. 5 Monodromy a Monodromy1 A Monadromy Representations can be used to build a commutative diagram out of 2 different homologies [?] [?] builds upon that by stating the Monodromy group can be either symmetric Sn or Asymmetric (An) . Another application is Monodromy covers of the Riemann sphere by the generic Riemann surface. Monodoromy, Cohomology, and -1 [?] extends their study of Cohomology milnor have extensively studied Cohomologies stating that the Monodromy operator can act on some cohomology group of the Milnor fiber at some point of the hypersurface f − 1(0)Cn . T he monodromy can be used an unipotent pseudoreflection , and that at the local monodromy is ( -1 ) times a unipotent pseudoreflection . In the next section , we will give examples in characteristic two, where we do not know any way to avoid [?] The Monodromy theorem could imply a family of smooth , complex , projective manifolds over the punctured disk , after a finite base change the monodromy can be made unipotent . [?] Another use of monodromy is to build a commutative of diagrams [?] 5.1 a Adding tables 6 Euler Characteristic a ccording to [?] Euler characteristic is a classical, well-understood topological invariant that has appeared in numerous applications, including in the context of random fields. E.g. Let X be a CW complex of dimension k, then an Euler characteristic is given by the formula: sumki=1 (−1)i .ni sumki=1 (−1)i .ni (2) sumki=1 (−1)i .ni when a source term θ is introduced in the equation: 1 ∂g(t) = − (g(t)) + θ, ∂t β (3) Theorem 1. Let X be a compact KÃhler manifold endowed with a smooth closed (1, 1)-form θ satisfying (??). Then lim g (β) (t) = P (g0 + tθ) (:= g(t)) β→∞ in the weak topology of currents, where P is a (non-linear) projection operator onto the space of positive currents. Moreover, the metrics g (β) (t) are uniformly bounded on any fixed time interval [0, T ]. degenerate on large portions of X, the support is defined as: X(t) := (dVg(t) ) (4) g(t) = g0 + tθ on X(t), in the almost everywhere sense. As a consequence, typically the volume form dVg(t) has a sharp discontinuity over the boundary of X(t) (the function g(t)n /ω n is typically discontinuous at some point on the boundary of the contact set X(t)) showing that the limiting (degenerate) L∞ -metrics g(t) are not continuous and hence 0 -convergence in the previous theorem cannot hold, in general. h it could also provides a stable summary of a space/dataset over an entire range of parameter choices/scales.[?] totalEnergy Plus in recent paper it has been proven that the Euler characteristic χ(G) of G is equal to the total energy: (5) χ(G) = Σx,y g(x, y) = E(G) furthermore F urthermore, χ(G)equaltothenumberof positiveminusthenumberof negativeeigenvaluesof L χ(G) = numP ositiveEigenvalues(L) − numN egativeEigenvalues(L) but how can we make use of it? This depends primarily on dimensionality. In low dimensional space, Euler Characteristic is made use of in Persistent Homology (PH), which is a powerful tool in topological data analysis, can track and analyze the shape of the data.[?] description:What (it does): also in the , Low Dimensional Space the use of Persistent Homology, which provides a stable summary of a space/dataset over an entire range of parameter choices/scales. 2.Higher-dimensional Space Use topological invariant parameter spaces: clearly enables more discriminative summaries. Motivation The goal of is to this paper illustrates that Euler surfaces can provide insight into the data over multidimensional parameter spaces. This is the header row 1 3 2 4 this is a cell in the first row this is a cell in the second row G reeks offer(2): 1. Complete list of objects 2. Model for these math objects Issues Either(2):a complete list does not exist at all Why(2): as either(2): — 1 set objects but countable Or 2.Model meaning is not clear, -(Importance of a clear model meaning) as it should reflect (many) shared properties by the objects it represents —- 7 Low Dimensional Topology Can be divided into two main classes: —- 1. Surfaces 2. Manifolds (3-manifolds, or 2-manifolds) List 2 1. Surfaces 1.1.Circle − homeomorphisms[givenbysurf ace − dif f eomorphisms] 1.2.Extremalsurf ace − dif f eomorphisms1.3.T heconjugacyproblem1.4.Homotopyequivalences 2.Manifold 2.1. Presentations 2.2. Hierarchies 2.3. Classification 2.4. Homotopy Equivalences —Homotopy Equivalences the goal os to split up the homotopy equivalences into simpler pieces in smaller math objects, to study each object separately. surface, in M? is called “essential” if the inclusion S ¡ M defines an injection I7,(S, @if inclusion S ∈ + in Π1 (S, δS) → − Π1 (M, δM ) of the relative fundamental groups. Note the fundamental group of any surface with non-empty boundary is free. 7.0.1 Sub-manifold (definition) surfaces Can consist of 3 main parts: 1. Presentations 2. Triangulations 3. Branched coverings Peesentations 7.0.2 Presentations For presentations,In particular, we can have either one of (3) different specifications either: 1. Heegaard 2. Polyeder or 3. Surgery The existence of characteristic submanifolds provides us with a splitting result similar to those discussed for homotopy equivalences between surfaces. 7.1 Homotopy Equivalences in Manifolds done after the classification of Haken 3-manifolds, we begin studying the Homotopy Equivalences between the manifolds, which are similar to those between surfaces. If the characteristic submanifolds exist, they provide us with the splitting result. homotopy equivalences between surfaces In order to describe this splitting result, let M and M’ be Haken 3-manifolds (closed or not!) and denote by V and V’ the characteristic submanifolds in M and M’, Why study Pseudo-Anosov as diffeomorphisms are part of homotopy equivalences. (Thus it is practical to study them). Plus, the split up a homotopy equivalences is: f ′ : M → M ′ which is morphed into homotopy equivalences between l-bundles and Seifert fiber spaces. Since V and V’ consist of those, then they also are diffeomorphisms between simple 3-manifolds. The homotopy equivalences are between Seifert fiber spaces. As far as homotopy equivalences or in particular, between, I-bundlesand Seifert fibrespaces. where I-bundles and Seifert fibre spaces and (@V—¢GM) consist of essential annuli and tori. Further, the equivalence can be deformed so that the fiber structure is invariant. Note that between surfaces (closed or not) we can reduce the study of Homotopy Equivalences to that one of Pseudo-Anosov diffeomorphisms. (Thus, studying Pseudo-Anosov diffeomorphisms is a good start). 8 Knots 8.1 knot types there are (3) main types 1. braids 2. Cobordism 3. Tangles 8.1.1 Briad 8.1.2 Contact Geometry 8.2 8.2.1 Seifert Surface Definition A two-variable polynomial PK On Oriented knot or link K Types diagram D for an oriented knot, We distinguish between (2) crossings based on links orientation: 1. Positive crossing c + (D): left on top of right link and 2. Negative Crossing c − (D): Right is on top of left link. crossing number, c(D) c(D) = c+ + c− Algebraic crossing number, c(D) c(D) = c+ − c− (6) 8.2.2 Motivation The goal is to find an isotopy, for a given Seifert Surface, which is regarded as a disc with bands, so that the bands could cross over, where: 1. Plus (+): denotes one side of the surface, and 2. Minus(-) refers to the other. The modification shown in Figure 2 produces an orientable surface obtainable from the indicated knot projection by shading. 8.2.3 Seifert Matrix Since any non-singular Seifert matrix for K is 2hx2h, where 2h = deg(A), it follows from Witt’s cancellation theorem, over any local ring in which 2 is invertible, which forms A + AT Where non-singular Seifert matrices A are all congruent. 8.2.4 Goeritz matrix This matrix coincides with A + AT for some Seifert matrix A associated with F. Hence, for any Seifert matrix A of K, A + AT is in the equivalence class of Goeritz matrices of K. 8.2.5 Goeritz-Seifert Relationship For some Seifert matrix B, where the Goeritz matrix will then be B + B which is an S-equivalent to A. By joining the two bands of the I-handle, at each band-crossing. 8.2.6 padic integers for the padic integers, this holds as p is odd, and p ∈. Hence, if relation A + AT is even, then the genus of A + AT is invariant in . 8.3 9 9.1 knot meaning knot polynomials Alexander polynomial The Alexander polynomial can be derived from the group of a knot (or link). In 1984, the Alexander and the Jones polynomials were generalized to the skein (module) ( or skein calculus). Furthermore, the Alexander polynomial can be derived from the group of a knot (or link). 9.2 Kaufman Bracket Given an Invariant (D) satisfied < D >∈ Z[, u, A, B] of an unoriented link diagram D called the Kaufman bracket where it gives a variant of the Jones polynomial for oriented links. Namely, for A = t − L′ 4 Where D is an oriented diagram of L. 9.3 Jones polynomial Jones polynomial provides a simple tool for recognizing knots and their permutations. Furthermore, Kaufman has found another approach to the Jones polynomial. 9.3.1 Alternating diagrams according to przytycki1998classical Uses Jones type polynomials, as follows: 1.T1: Alternating diagram with no nugatory crossings, of an alternating link realizes the minimal number of crossings among all diagrams representing the link 2.T2: Two alternating diagrams, with no nugatory crossings, of the same oriented link have the same Tait (or writhe) number, i.e. the signed sum of all crossings of the diagram with the convention 3.T3: Two alternating diagrams, with no nugatory crossings, of the same link are related by a sequence of flypes. Figure 2: Caption: Knot Meaning F rom the figure above, one has to distinguish 2 separate (2) functions of a graph: 1.Compression/ decompression 2.interpretation 1.Compression/ decompression for the latter, where the graph is decompressable into 3 distinct parts, or states whichever they are. seemingly one could also compress back those 3 symbols back into one unified graph. we speculate that it is convenient if one can send one compressed symbol with stakeholders yo communicate with, this model perhaps assumes that the stakeholder can correctly dissect the large symbols into its primary (3) unique symbols. 2 . interpretation: there are only 3 distinct states, with (2) orientations either t=-1 negative. or t=0 neutral. or t= 1 positive. for the orientations, they can either be cut along the x axis or the y-axis. this variety is good as one can use as many as 9.4 Oscar Simony’s Torus Knots T he fundamental problem in knot theory distinguishes nonequivalent knots. one gets the L(3,l) from S3 space using the Betti numbers; more precisely one shows that the first homology group is nontrivial and it’s clearly understood that it is a 3-torsion group. He did not state, however that it can be used to distinguish the trefoil knot from the unknot, furthermore, He doesn’t state, however,it was later discovered that this method is used to distinguish the unknot, from the trefoil knot. D0 ’Non-crossing’ of (2) curves It consists of two non-touching curved-arrows, that are symmetric along the y-axis. although they converge along the middle (i.e. say at origin (0,0) then diverge afterwards, as one goes right, while the other continues left 2. D− (2) straight lines crossings, where left arrow is on to of right on. 3. @D+ (2) straight lines, crossing: where the right arrow is on top of the left one. In fact if we rewrite v as a parametrized vector, with respect to bi 9.5 Seifert Polynomial a ccording to [?] As for a polynomial expression P K, WITH 2 variables: n, z respectively,then for a twovariable polynomial, PK(v, z), of the oriented link K, with an interval k ∈ [e, E] Where: (7) P K(v, z) = k=E X k=e As v ranges on the interval [e, E] Thus, We can write the polynomial as: P K = Zpk0 = (1/v)pk+ − vpk− (8) 10 Contact Geometry 1 .Let a be a differential form of degree 1 2. A differential form of class C ° ° on with degree 1,a smooth variety M . There is a special case Where Outside the singular locus of a , the equation of P aa = 0 defines a ck-map of: (typeAmap ) ∗ F tangent to M such that (typeAmap ) ∗ F is tangent to M Hence, we can say that: ck − map is tangent to M . 10.1 Structure Forms hus there are: 1. Two forms a and a′ let’s define equivalent Pfaff equations* with: Hyperplane fields, which are isomorphic on, s1 . There exists a diffeomorphism cp of M and a function f , Where it has non-zeros, everywhere on M, hence it’s a non-vanishing condition, such that: T α · F alpha (9) S uppose that the form a does not vanish on M . This is valid, when the Frobenius condition is equal to: (10) Frobenius Condition = aAda = 0 subsubsectionnon-vanishing assumption of a Suppose that the form a does not vanish on M . This is valid, when the Frobenius condition is equal to: (11) Frobenius condition = aAda = 0 This condition can be verified at every point of M , the equation a = 0 is completely integrable, and the field F generates a foliation of co-dimension 1. 10.2 Odd, Even Dimensions IF the dimension of M Q is m: (12) dim(M Q) = m And if the number 2n + 1 is odd, Then there exists an equation, given by: P ( ai ) (13) When dimension Number dim = 2n, then number is even, and the function is generic. The number is dim = 2n + 1, then the Number is odd , and the function exists, For any given variety of co-dimension 3. 10.2.1 Example (Field of Planes) the field of planes defined by xdy = dz on the variety M 3 , by the diffeomorphism, It Is simply a quotient of R3 , satisfying this relation: (14) 10.3 (x, y, z) → (−x, y + 1, −z) Contact Structures as Symplectic Cones A nother angle of contact structures on M is to look at the Symplectic Cones in the cotangent fiber of M , because F is a contact structure on M if and only if the cone in T ∗ M is of the algebraic 1-forms. Thus, one should study contact structures along with Symplectic structures, As these (2) structure types, occur together in analysis and geometry problems. 10.4 Odd Dimension Advantage I t has been demonstrated that for any closed, orientable manifold with an odd dimension i.e. 2n + 1 possessing a transversally-oriented, almost complex tangent hyperplane field, is the boundary for an almost complex compact manifold of dimension 2n + 2. (Please note this value has significance in Number Theory ). F urthermore, odd dimensional manifolds are interesting because any odd-dimensional manifold M (with dimensions dim(M ) = 2n + 1) whose structural group reduces to U (n) can always be equipped with a contact structure - a one advantage of an odd dimension. 10.5 graphs and Legendre varieties ′ T hese graphs are the prototypes of Legendre varieties. The geometry of this structure F0 constitutes of an approach to the integration of nonlinear first-order partial differential equations. Legendre Varieties: These graphs represent the prototypes of Legendre Varieties where the geometry of this structure ′ F0 constitutes one approach to integrating partial differential equations, which are both first-order, and non-linear. ′ Where the structure F0 is: 1. a field 2. a standard structure satisfying: ′ F0 ∈ R2n+1 (15) Please note these structures can be drawn in (3) Dimensions w.r.t. R3 . 2 2 Let’s consider S2 n + 1 at every point x ∈ Sn+1 , The tangent space at Sn+1 contains a unique, complex hypermap equal to: (16) hypermap(tangent space) = (Cn + 1) This equation is the hypermap for the contact: (17) hypermap(contactGeometry) = F0 ∈ x From that hypermap, we can form a sequences si (18) si = (p0 , , P n, Qn) The list of equations are polypolar of (Cn + 1, F 0) And of level n. 10.6 Markov Surface I f all the saddle points on the edge of the pocket, and is centered on P , Then the saddle points are positive, and the strand the rotates constantly, in the counterclockwise direction. Now, after a complete turn of d onto r0 , it should return to its initial position, assuming the geometry is circular. e.g. lets say such that: S ′′ > A′′ (19) ′′ Or as an Acceleration: Acceleration(S)” > Acceleration(A) Or simply as a distance: (20) distance(S) > distance (A) N ote that if S ′′ is strictly greater than A′′ there shall exist a pocket of V , having only one negative saddle point on its edge. S econdly, when a Markov surface V has a pocket TI , whose boundary contains only one negative saddle point, then it is easy to find a support isotopy in M , which deforms V into a Markov Surface V with a pocket TI . t his can be described dynamically by Markov Surface’s sections on the planes. with angle rotation d, where the kast variable it is considered as a time variable. 11 Atiyah Theory [?] mentons that, Atiyah can construct the L2 -Betti numbers of a compact Riemannian manifold; They are defined in terms of the spectrum of the Laplace operator (on the universal covering of M). By Atiyah’s L2 index theorem, which can be used e.g. to compute the Euler Characteristic function. 11.1 Atiyah conjecture a ccording to [?]: Atiyah group can construct the L2 -Betti numbers of a compact Riemannian manifold; They are defined in terms of the spectrum of the Laplace operator (on the universal covering of M). By Atiyah’s L2 -index theorem, they can be used e.g. to compute the Euler characteristic function. N ote that the Atiyah group is likely applied to Artin’s braid groups which are Abelian, torsion-free, nilpotent groups, mostly compatible with trivial ones, But could also apply for non-trivial groups under some conditions, such that, Only if their quotients are Abelian, and torsion-free groups. It means that every finitely-generated subgroup of G is contained in an A∗-group, and GXfy This means that G has a normal A subgroup H such that the quotient is a y-group: G H (21) ( Definition 1.1, [?]): For a discrete group G, is also a set multiple of Least Common Multiple(LCM): (22) lcm(G) := F |F ≤ GandF < inf 11.1.1 T Strong Atiyah Conjecture he strong Atiyah conjecture over KG group says: lcm(G)dimG (kerA) ∈ Q∀A ∈ Mn (QG) (23) f or all A ∈ Z Where: K ⊂ Z Moreover: (24) lcm(G) · dimG (kerA) ∈ Mn (KG) Where K ⊂ C Is always a sub-ring of the complex numbers which is closed ( under complex conjugation). 12 Appendix 12.1 Morphism Review 12.2 Isomorphism a ccording to [?] A transform on a collection that is This collection is partially ordered collection by inclusion of some open sets: (25) U ≡ Mn It is good to add that the Isomorphism can be given by evaluating a generator which forms its fundamental class: (26) A n isomorphism is given by evaluating a generator which is its fundamental class. E.g. (27) W F undamentalclass(isomorphism) = Hn (B1 , ∂B1 ) D = [Mn ]∩ : Hk (Mn : R) → Hnk (Mn ; R) here D is an isomorphism for all k. 12.2.1 Generators N ote the group of “generators” ( for the covering space ) is well-defined on Compact Subspaces, each of which is contained in a chart. By induction, we could conclude that the result’s validity in this step, true for (for k= 1,...,n), for every compact subsets A that are contained in a chart: (28) 12.2.2 A⊂U Cap Product I t is a product operation for any X a topological space, and R a commutative ring. The Cap product ∩ property satisfies the following rather Odd Naturality property: This is what makes working with such spaces useful. (29) f(α) ∩ φ = f(α ∩ f (φ)) Furthermore, the cap product is used In Cohomology, where it is represented as an adjoint (of transforms) to the cup products. 12.2.3 Poincaré Duality Poincaré Duality is an isomorphism for all k (Theorem 16). This theorem implies If F is a field and Mn is a closed F-oriented manifold with fundamental class [Mn ], then: (30) F undamentalclass = [Mn ] ⊂ Hn For (Mn ; F ), then the pairing reduces into F : (31) I Hk (Mn ; F )Hnk (Mn ; F ) → F n addition, it’s useful to generalize the theorem from a compact, into non-compact manifolds setting. 12.2.4 12.3 Use of Isomorphism Gauss-Bonnet Concerning the Euler characteristics function can be decomposed into (2) parts as follows: (32) M =< (T (M )), [M ] > ′ Please note that we can replace ω svalueasf ollows : (33) < [FA ], [M ] >= Z ω=ı M Z κdA M Thus the above lemma states that: (34) 2πıM =< [FA ], [M ] > Where: < [FA ], [M ] > 2πı It is also interesting to add this is equivalent to the classical Gauss-Bonnet theorem, stating that: (35) (36) M= Z κdA = 2πM = 2π(2 − 2g) M Where g is the genus of the Riemann Surface M. I n [?]: By representation theorem,let’s amalgamate G over K This amalgamation is isomorphic to E and G|C is trivial. but since each cell: C, Cl[E(G) − A(G)(C)] Provides an H-cobordism between the boundaries: A(Ḡ)(C) and Ē(C) And by following Theorem 4.11’s proof, the classifying maps of Gand E are homotopic; then it follows that: (37) 12.3.1 g≈f Normal Bundle L et M be an m-dimensional homology manifold, withPL-embedded property in a q-dimensional homology manifold Q. There exists a normal Dq−m bundle to MinQ, whose isomorphism class depends only on the PL-concordance class of the embedding. The set of isomorphism classes of bundles over a ceil complex K is in (I-1)-correspondence with the set of isomorphism classes. Since A is onto, Then: (38) A(E) ≈ A(F ) Hence E ≈ F and dis(1 − 1) dimension. Then define AHn (X) for the set of isomorphism classes of S n−1 -bundles (or Dn -bundles) H(n) be the (Kan) A-semigroup of which a typical m-simplex is an isomorphism between Am xsn−1 and itself: (39) BH(n) = B · P rol0 (40) γ = P rol0 T is of dimension (l − 1), consider a bundle E over X but there is a quotient E/Y : E/Y = f ∗ γ is defined for some A-map f : f : Y + BH(n) Then there exists g which follows g : X → BH(n) hence by inducing BH(n), one can successfully extend f extending f, having a linear map in terms of X variable: f, g : X + BH(n) such that the non-linear map js as follows : (41) f ∗ yrg ∗ y A bundle over 2, where Z ≈ (X)xI E/Y = f ∗ y, for some SA-map f: Y + BH(n) the isomorphism Linear relationship For maps: f, g : X + BH(n) such that: (42) f ∗γ =g∗γ (Corollary 4.8). Note: that isomorphism map is an orientation-preserving isomorphism between : ∆m xS n−1 and itself. Note that the bundle has (2) variables, Where Z is a map, s.t. us a ∆-map Z = absXxI: Where the following convergence is satisfied: (43) f ≈g But since BH(n)isKan, then the equality is satisfied: (44) f ≈g 13 13.1 Levi-Cevita Sub-manifolds I s an abstract manifold, where a manifold is living inside a manifold. There are at least (3) variants of sub-manifolds: 1. Embedded sub-mainfold: E.g. Any manifold of dimension d can be embedded in the Euclidean space E ( 2d + 1). 2. Immersed manifold: One allows the manifold to cross itself. 3.Hyper-surface: Is a sub-manifold and is one dimension less than the Ambient manifold. Sub-manifolds are useful, as each has a natural tangent space to our current space under study. There is yet an open question to discuss: Can a hyper-surface contain a few sub-manifolds , Is a surface a manifold, or at can it be at least But what makes a surface different from a manifold?) but can a surface be a manifold? In other words, are surface properties the same as those found in a manifold? I.e. Is a surface equivalent, quasi-equivalent, or non-equivalent to a manifold? Similarity-wise, are sub-manifolds self-similar to a hypersurface? 13.2 Haken transform For the 3-manifold case, it’s a transformation, as it carries a hyperbolic structure that is easy to construct, ( whereas this map is different from the I-bundle over the Klein bottle). However, the Haken transformation is quite different for surfaces while interesting, it can get quite complicated, as the hyperbolic structures for this transformation are difficult to construct. 13.3 13.3.1 Fundamental Group Fundamental Group on knots F group on knots The fundamental group was essentially introduced by Poincare in his 1895 paper. Cohomology The continuous Cohomoloygy of the pro-p complete, where h is am isomorphism. For infinitely many quotients, And H is torison-free, then factorization property is fulfilled. Note that in cohomology, we might have to replace F with a non-trivial subgroup. 13.3.2 On a Topological Space L et be a Hausdorff space then there is a dimension ( 1-1) correspondence between subgroups and its covering spaces. Then one can construct non-trivial covering spaces, using Hurewicz’s topologized fundamental group? But How? For a group X, we can use step-homotopies, Instead of continuous deformations. The path α1 is from σ-homotopic to the path an , If there exists a finite sequence of paths: al, ..., an Where: Where: n − 1, ai(t), ai + 1(t) Strong Atiyah Cohomology ] It is defined by Artin’s Pure braid group, all conditions are satisfied. Whereas the classes of groups, which are finitely-generated, and free. Defined on surface groups, primitive for Realtor groups, knot groups. And primitive link groups. Where these building blocks, construct H as an iterated semi-direct product. Thirdly, the quotient always acts trivially on the abelianization of the kernel ker. 13.3.3 Relaxed Atiyah Cohomology O n the U-free quotient, which it can be further relaxed. In Cohomology (we might have to replace F with a non-trivial group). In order to find a factorization through an elementary amenable group, while having as little torsion as possible. To this case, Theorem can be extended, an assumption, on the torsion-free quotient, which can be relaxed A re contained in a common set σ − homotopic where as all sequences of paths share the same starting and ending points. (hence sequences are repeatable). Atiyah’s Background Theory I f there exists an L ∈ N then the inverse image also exists G/H that is a quotient group, With a projection: G (lemma 2.4). π:G→ H We can rewrite G/H in terms of different primes, considering the positive values only: (45) ( G ) = pn1 1 pnk k H H ence, the inverse image Hp Can be called the Sylov Subgroup of the Group G. We initially call k ∗ G to be the crossed product of a Skew Field k, anf group G is abelien, but If F is a set of finite subgroups, then the crossed product is called an Ore Domain having a right quotient ring which is a a Skew Field k. LCM T here exists a Least Common Multiple (lcm) for the crossed product KG or KHp , Where G is a group, and Hp can be built from it,as the latter forms disjoint subsets of the former group. Where Hp ≡ G. Atiyah’s Suggestion L et us construct Hp from G I.e. Hp ≡ G, thus the lcm can be f orallAinMn (KG) as: (46) L · lcm(G)dimG (kerA) ∈ Z Additionally, lcm can be expressed in Terms of the Crossed Product KG: (47) L · lcm(G)dimG (kerA) ∈ Z That is for ∀A ∈ Mn (KG). Strong Atiyah M eanwhile, evidence suggests the Hp implies the Strong Atiyah Conjecture for G. (Since Hp are disjoint subgroups of G). Hence, there exists a Least Common Multiple (lcm) on Hp or on the crossed product KHp , in general, satisfying the condition: f orallAinMn (KHp ), with a general form: (48) L · lcm(Hp )dimHp (kerA) ∈ Z Thus, the LCM is under the condition of: f orallAinMn (KHp ). Equivalency Classes in Topological Space T wo paths are equivalent if they are: σ-homotopic for every σ Along with the usual definition of product and the inverse of paths. In addition, the set of equivalence classes of all paths is closed at a point of y0 ∈ Y , forming a group. 13.3.4 On a group group1 According to the textbfVan Kampen theorem: Any group can be realized as the fundamental group of some CW-complex, and pushouts of groups”. Such groups include free groups as fundamental groups of wedges of circles. [?] group2 group topology is used to study spaces with a complicated local structure. In the past one used historical shape-theoretic approach, where spaces are approximated by a combinatorial object is a polyhedra; on the other hand, pro-groups have taken the place of groups, However, one can directly approach such spaces by transferring topological data into a homotopy-invariant structure. Group-last Where spaces are approximated by a combinatorial object i.e. a polyhedra, on the other hand, and pro-groups have taken the place of groups, In addition, group topology is used to study spaces with complicated local structure. In the past, in fact, one uses historical shape-theoretic approach, where spaces are approximated by polyhedra and p0, Such that N takes the place of groups, However, One can directly approach such spaces, By transferring topological data into a homotopy-invariant structure. Line Bundle 13.3.5 On a Scheme Group The author defines a scheme of line bundles (of degree zero), obtains an embedding of group schemes, and defines X line bundles of order p. Finally, the author finds that the canonical homomorphism is not an isomorphic map. Please note that for the Abelian case, If X is any projective variety, then the P ic0 (X) base changes correctly, as it is a connected component of the Picard scheme. Note: For the Abelian case, if X is any projective variety, then the P ic0 (X) base changes correctly, as it is a connected component of the Picard scheme. SubsectionCo-fiber Let the co-fibration be: ι : A → X Where the Of ι is the Co-fiber. The Co-fiber or the quotient space is defined as: (49) X/A = X Where the equivalence relation for two points a, b ∈ A is given by: (50) ι(a) ι(b) Whereas A satisfied the following inclusion map: (51) 13.3.6 ι:A⊂X Cone Since any map: f : X → Y is homotopic to a co-fibration with a co-fiber, Hence it maps a cone: Cone ≡ Y ∪f c(X)(52) Please note that the co-fibration map is equivalent to the mapping Cone: Cone ≡ X ∪i c(A)(53) The only difference observed between the two formulas of a cone is the change of variables: Namely, from f to i, and from X to A, while the mapping cone can also be referred to as the homotopy co-fiber of f . It is also good to mention that the quotient space: X/A 13.4 Co-fiber Let the co-fibration be: ι : A → X Where the Of ι is the Co-fiber. The Co-fiber or the quotient space is defined as: (54) X/A = X/ Where the equivalence relation for two points a, b ∈ A is given by: (55) ι(a) ι(b) Whereas A satisfied the following inclusion map: (56) 13.4.1 ι:A⊂X Cone Since any map f : X → Y is homotopic to a co-fibration with a co-fiber, Hence it maps a cone: Cone ≡ Y ∪f c(X)(57) Please note that the co-fibration map is equivalent to the mapping Cone: (58) Cone ≡ X ∪i c(A) The only difference observed between the two formulas of a Cone is The change of variables Namely, from f to i, and from X to A, Whereas the Mapping Cone can also be referred to as the Homotopy Co-fiber of f . It is also good to mention that the quotient space X/A is also known as the Co-fiber (On the co-fibration map): (59) Co-fibration ≡ ι : A → X Furthermore, by the inclusion of Y Into the mapping Cone, not the it satisfies the following co-fibration relationship: (60) Co-fibration ≡ Y ⊂ Y ∪f c(X) also known as the Co-fiber (On the co-fibration map): (61) Co-fibration ≡ ι : A → X Furthermore, by the inclusion of Y Into the mapping Cone, not the it satisfies the following co-fibration relationship: (62) Co-fibration ≡ Y ⊂ Y ∪f c(X) 13.4.2 On a Vector Space 13.4.3 On a type 13.4.4 On a Manifold 13.4.5 On a Category [?] 14 Conclusion W e have reviewed popular knots, and some relevant theory behind them in building enough intuition and basic understanding of the subject matter. I t turns out that one of the great uses of knots is performing a surgery , with a goal to somehow remove them; This would require further work, as a separate research topic on its own. I n the end , another possible line of research might also be done, by discovering characters and their representations that belong to representation theory primarily. U nraveling unknown territories might also be interesting to explore in the hope of finding connections to relevant geometric constructs in the near future. I n both cases, it is interesting to know more about knots. Furthermore, the fundamental group has a decent amount of available literature, that might be a separate topic which is worthy to explore and write about in the near future. F inally, finding out about the meaning of knots is fine, however,on one hand it might be good to explain further theory, explaining propositions corollaries, and theorems in the pure field. On the other hand, it would also be a worthwhile pursuit, worthy to further develop,unraveling that which is hidden, shedding some light upon uncovering hidden connections and relationships with other fields in the applied field. hence, whether applied or theoretical, knots are interesting to study, learn about, and hence provide more value, while strengthening one’s understanding in return. 15 References [ fickers2012towards]articlebodyhistoryFickers, A., 2012: Towards A New Digital Historicism? Doing History In The Age Of Abundance. View: Journal of European Television History and Culture, 1, 19–26, https://doi.org/10.18146/2213-0969. [2][herrera2017generating]articlebodyhistory Herrera, Blas, and Albert Samper. ”Generating Infinite Links as Periodic Tilings of the da Vinci–Dürer Knots.” Mathematical Intelligencer 39, no. 3 (2017): 18-23. ( 2)[2][maxwell2013qualitative]bookbodyresearchMaxwell, J. A., 2013: Qualitative Research Design: An Interactive Approach: An Interactive Approach. (3)[3]misc[owens2014start]articlebodyresearchOwens T., 2014: Where to Start on Research Questions in the Digital Humanities. TrevorOwens.Org(2014) (4)[4][gibbons1992kinks]articlebodyeulercosmos Gibbons, G. W., and S. W. Hawking, 1992:Kinks and topology change, APS Physical Letters Review, (5)[5][beltramo2021euler]articlebodyeulerknotimageBeltramo, Gabriele, Rayna Andreeva, Ylenia Giarratano, Miguel O. Bernabeu, Rik Sarkar, and Primoz Skraba. ”Euler characteristic surfaces.” arXiv preprint arXiv:2102.08260 (2021) [antosh2024characterization]articlebodyeuler ()[5][numdamEntrelacementsxE9quations]miscbodyknotscorbordism linksBennequin, Daniel. ”Linkings and Pfaff’s equations.” Russian Mathematical Surveys 44, no. 3 (1989): 1. [6][martin1971homology]articleappendixmorphism Martin, John R. ”Fixed point sets of homeomorphisms of metric products.” Proceedings of the American Mathematical Society 103, no. 4 (1988): 1293-1298. [6]Martin (1971) title=Bundles, Homotopy, and Manifolds ()[7][cohen2023bundles]articleappendixmorphismCohen, Ralph L. Bundles, Homotopy, and Manifolds. (2023). ()[8][linnell2007atiyah]articleappendixAtiyahLinnell, Peter, and Thomas Schick. ”The Atiyah conjecture and Artinian rings.” arXiv preprint arXiv:0711.3328 (2007)