Low Dimensional Knot Ahmad Lutfi Dec 2024-

keywords = 57Kxx, 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.Simon Jubert: for recommending ”Overleaf.com”, a thanks might 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. Contents 1 Dedication 2 Introduction 3 Monodromy 3.1 Monodoromy, Cohomology, and -1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Euler Characteristic 5 Low Dimensional Topology 5.0.1 Sub-manifold (definition) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.0.2 Presentations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Knots 6.1 knot types . . . . . . . . 6.1.1 Briad . . . . . . . 6.1.2 Cobordism . . . . 6.1.3 Contact Geometry 6.2 knot meaning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Polynomials on Knots 7.1 Alexander Polynomial . . . . 7.2 Kaufman Bracket . . . . . . . 7.3 Jones polynomial . . . . . . . 7.3.1 Alternating diagrams . 7.4 Oscar Simony’s Torus Knots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Contact Geometry 8.1 Structure Forms . . . . . . . . . . . . . 8.1.1 non-vanishing assumption of a . 8.2 Odd, Even Dimensions . . . . . . . . . . 8.2.1 Example (Field of Planes) . . . . 8.3 Contact Structures as Symplectic Cones 8.4 Odd Dimension Advantage . . . . . . . 8.5 graphs and Legendre varieties . . . . . . 8.6 Atiyah Theory . . . . . . . . . . . . . . 8.6.1 Atiyah conjecture . . . . . . . . . 8.6.2 Braid groups . . . . . . . . . . . 8.6.3 Least Common Multiple (LCM) 8.6.4 Strong Atiyah Conjecture . . . 8.6.5 sub-ring C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Appendix 9.1 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Levi-Cevita 11.1 Sub-manifolds . . . . . . . . . . . . . 11.2 Haken transform . . . . . . . . . . . 11.2.1 On a Topological Space . . . 11.2.2 Relaxed Atiyah Cohomology 11.2.3 On a group . . . . . . . . . . 11.2.4 On a Scheme Group . . . . . 11.3 Co-fiber . . . . . . . . . . . . . . . . 11.3.1 Cone . . . . . . . . . . . . . 11.4 Co-fiber . . . . . . . . . . . . . . . . 11.4.1 Cone . . . . . . . . . . . . . 11.4.2 On a Vector Space . . . . . . 11.4.3 On a type . . . . . . . . . . . 11.4.4 On a Manifold . . . . . . . . 11.4.5 On a Category . . . . . . . . 12 Conclusion 13 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction to Low-Dimension Topology and Knots Ahmad Lutfi January 7, 2025 Abstract In this paper, we examine knots from different angles, we provide a beginner’s perspective, along with providing related terminology, which may aid the researcher, 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. 2 Introduction ] Low-dimensional topology is a dense subject, with topics yet to be explored, however, its prominent part is about the knots. Leaving out 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, 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 might 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, leading to WW2 and where the beginning of maturity of Knots is being marked by history, as peace rules over ,there was never never such a great opportunity where old literature is being compiled to pick up where scientists had left of where break through s are a possibility. In [herrera2017generating], they have studied The fourth and sixth Da Vinci-Druer knot, which have been analyzed for various knot orientations ( with different number of crossings, whereas the circle is being the simplest, with a goal of discovering the invariant property. In fact, it turns out that 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 infinite sequences, where each Link can have an infinite, number of knots, L ow-dimensional topology is a dense subject, with topics yet to be explored, however, its prominent part is about the knots. Leaving out 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, 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 might 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 withLeonardo Da Vinci, then passing to the modern world, and especially in the post WW2 era, where the world has just got back on its feet, trying to live on with peace and harmony again; it was an era where old literature has hada a chance been able to get reviewed by scientists whom without their help, there wouldn’t have been any breakthroughs nor a progress of humanity. In [herrera2017generating], mentions Da Vinci and Druer’s knots (and their patterns with different orientations, and crossings (a line is over another line). [BS17](Blas and Semper. (2017) have studied The fourth and sixth Da Vinci-Druer knot, where the study begs a big question, with a goal to find out whether the Invariance property is valid or not (i.e. whether there are some repeating self-similar properties). In fact, it turns out that 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 infinite sequences, each of which have infinite number of knots (on a Lie Algebra). I t is interesting because this not only It might be beautiful to carefully analyze such artifact with attention, then to see such knots as one not only observe knots being analyzed, but also they are able to see them under a given context of a broader diagram (i.e. of which renders the knots pattern dense with the background as an additional layer of complexity; this is regardless of Da Vinci’s and Druer’s knots as an original work of art, of dancing as patterns, on some curvy on wooden plates, which are non-linear, , rendering the plate as a part of sacred geometry. and might require a separate study Artistic-wise, one only takes a look just to fell in awe with such creation, or better yet, one has to find some larger pattern as an appropriate back-ground , for a given knot, which hence, by finding the best fit into (that) the however, it’s more of a beautiful art, (that is curvy and feminine). In art, beauty is a fuzzy interpretation of some picture, hence there’s no best but only good possibilities with a latent back ground generator of creative feminine curvatures. In other words, if we consider a subject being a knot as a one piece of puzzle, of many open-ended possible fillings whose function is explicitly unknown.This marks the end of a personal, alas, an immature critique of such a great historical art. 3 Monodromy ] [Kappes2011-id] states that A Monodromy Representations can be used to build a commutative diagram out of (2) different homologies (that are acyclic). Meanwhile, in [GuralnickS hareshianS taf f ord2 012]: It is conjectured that by stating the monodromy group can be either symmetric ( Sn )or Asymmetric ( An ). However, another application of Monodromy is to cover the Riemann sphere by the generic Riemann surface. 3.1 Monodoromy, Cohomology, and -1 ] M eanwhile [BoriesV eys2 016] they have extend their study, to include Milnor’s work, who has extensively studied co-homologies, saying the Monodromy operator can act on some co-homology group of the Milnor fiber at some point of the hypersurface: (1) f − 1(0)LenCn ] The Monodromy can be used an unipotent pseudoreflection, and that at ∞ besides that the local Monodromy is valued ( -1 ) times a unipotent pseudo-reflection . An example in the characteristic (function of two variables, where we do not know any way to avoid. ([KAT Z2023]) F urther, 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. Meanwhile, another use of a Monodromy is one can build a commutative diagram of (2) connecting homologies[Hassett2013 − dv]. a 4 Euler Characteristic A ccording to [beltramo2021euler]: Since Euler characteristic is a classical it is a well-understood topological invariant that has appeared in numerous applications, including in the context of random fields. E.g. Let Euler Characteristic function χ be a CW -Complex of dimension k: (2) T P dim(CW − Complexχ) = k hen an Euler Characteristic is given by the formula: Sum = sumki=1 (−1)i .ni (3) Meanwhile, Sum is given by the following Euler Characteristic function: (4) Sum = k X ((−1)i · ni ) = χ(T )) i=1 Where T = total time instances of this function ∀t ∈ T , whereas: t > 0, t < ∞ i = 1k (−1)i · ni (5) By degenerating a manifold, it’s not only it’s possible but also rapid, as it b an hour be done in (5) steps: 1. θ: A source , when is introduced (as a perturbation in Ric the Ricci equation: (6) ∂g(t) 1 = − Ric(g(t)) + θ, ∂t β when a source term θ is introduced in the equation: 1 ∂g(t) = − (g(t)) + θ, ∂t β (7) Theorem 1. Let X be a compact Kahler manifold endowed with a smooth closed (1, 1)-form θ satisfying (??). Then (8) lim g (β) (t) = P (g0 + tθ) (:= g(t)) β→∞ note that the value converged to some value g(t) at t. 3 .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 ]. Note thus, er we can degenerate Degenerate, over a large portions of X, which is defined by the Support Supp of the differential of g(t). i.e. the support is defined as: (9) X(t) := Supp(g(t) ) 4 . Degenerate (the support Supp, in the corresponding manifold), by some support value Supp Degenerate on large portions of the characteristic function χ, i.e. from the support supp, we can retrieve the chi, by multiplying by dV , at time t, by a map g is defined in terms ofχ as: (10) (11) χ(t) := Supp(dVg(t) ) g(t) = g0 + tθ onχ(t) H owever, for the g(t) equation , the general equation for g(t) can be calculated as follows: (12) g(t) = g0 + tθ onχ(t), 6 . Finally, in almost everywhere sense. As a consequence, typically the volume form dVg(t) has a sharp discontinuity over the boundary of X(t) (i.e. 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 Cc0 -convergence in the previous theorem cannot hold, in general. ] it could also provides a stable summary of a space/dataset over an entire range of parameter choices/scales.[beltramo2021eul Note that theorem cannot hold, in general. i t could also provides a stable summary of a space/dataset over an entire range of parameter’s choices, or scales.[beltramo2010221] totalEnergy Plus in recent paper it has been proven that The Euler characteristic χ(G) of G is equal to the total energy: (13) χ(G) = Σx,y g(x, y) = E(G) furthermore F urthermore, χ(G)equaltothenumberof positiveminusthenumberof negativeeigenvaluesof L Euler characteristic (χ) as an energy function: totalEnergy In addition, in recent paper it has been proven that the Euler characteristic χ(G) of G is can be expressed as a the total energy: X (14) E(G) = χ(G) = g(x, y) = E x,y F urthermore, the χ(G) can be equal to the number of positive, minus the negative number of eigenvalues of L: (15) E(G) = χ(G) = num(P ositiveEigenvalues(L)) − num(N egativeEigenvalues(L)) B ut how can we make use of it? This depends primarily on dimensionality. Indeed, in low dimensional space, the Euler Characteristic (χ) makes use of Persistent Homology (PH), which is a powerful tool in topological Data Analysis, can track and analyze the shape of the data.[antosh2024characterization] description:What (it does): also in the Low Dimensional Space, we can make use of the Persistent Homology, Which provides a stable summary of a space/dataset (over an entire range of parameter choices/scales). 2.Higher-dimensional Space By the Use of topological-invariant parameter spaces: this clearly enables more discriminative summaries. textbfMotivation Euler surfaces’ motivation is such that they provide an insight into data, over multidimensional parameter spaces (in ). o ffer(2): 1. Complete list of objects 2. Model for these mathematical objects Issues: Either(2): a complete list does not exist at all But, Why is that? well, this is because either one of the (2) options, either : 1 Set objects are uncountable Or 2.Model meaning is not clear ] Hence the Importance of a clear model meaning, as it should reflect (many of the) shared properties by the objects that it represents. 5 Low Dimensional Topology I t can be divided into two main classes: Below we provide (2) lists: list1, and list2 : list1(classification) 1. Surfaces 2. Manifolds (3-manifolds, or 2-manifolds) List2(detailed): 1. Surfaces 1.1. Circle-homeomorphisms [given by surface-diffeomorphisms] 1.2. Extremal surface-diffeomorphisms 1.3. The conjugacy problem 1.4. Homotopy equivalences 2.Manifold 2.1. Presentations 2.2. Hierarchies 2.3. Classification 2.4. Homotopy Equivalences Homotopy Equivalences The goal is to split up the homotopy equivalences into simpler pieces in smaller math objects, to study each object separately. T he surface, S ∈ M +andiscalled\essential”if theinclusionS ¡ Mdef inesaninjectionmapI7,(S, @)if inclusionison; suf raceS ∈+ I n Π1 (S, δS) → Π1 (M, δM ) its a morphism of probability of change of variables , from surface into a manifold. the space does not only change but has a role ion changing the relative fundamental groups. Note The fundamental group of any surface with non-empty boundary is free. 5.0.1 Sub-manifold (definition) Surfaces can consist of 3 main parts: 1. Presentations 2. Triangulations 3. Branched coverings 5.0.2 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 sub-manifolds provide us with a splitting result, similar to those discussed for homotopy equivalences between surfaces. Q. How to let the submanifolds: v, v′, and the manifolds m m’ be equivalent to each other? manifolds(mfds and sub-manifolds . 6 6.1 Knots knot types there are (5) main types: 1. Links 2. Braids 3. Corbordism 4. Tangles 5. Cables 6.1.1 6.1.2 6.1.3 6.2 7 7.1 Briad Cobordism Contact Geometry knot meaning Polynomials on Knots Alexander Polynomial The Alexander polynomial can be derived from the group of a knot (or link) (regardless of structure being oriented, or non-oriented). Furthermore, the Alexander polynomial can be derived from the group of a knot (or link). 7.2 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. However for an oriented Link diagram L, then A is defined as follows:L Namely, for A = t − L′4 (16) Where D is an oriented diagram of L. 7.3 Jones polynomial Jones polynomial provides a simple tool for recognizing knots and their permutations. Furthermore, Kaufman has found another approach to the Jones polynomial. 7.3.1 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. 7.4 Oscar Simony’s Torus Knots The fundamental problem in knot theory distinguishes nonequivalent knots. Figure 1: Caption: Knot Meaning F rom the figure above, one has to distinguish (2) separate functions of a graph: 1.Compression/ decompression 2.interpretation 1 .Compression/ decompression for the latter, where the graph is de-compressable 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 to 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 is negative 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 (one wishes to). o ne 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 , theparametrized Post-Jones Polynomial In 1984, the Alexander and the Jones polynomials were generalized to the skein (module) ( or skein calculus). 8 Contact Geometry In [numdamEntrelacementsxE9quations]: 1 .Let a be a differential form of degree 1 2. A differential form of class C on (a manifold) with somedegree o1, a smooth variety M . a There is a special case Where Outside the singular locus of a , the equation of P a = 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 . 8.1 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 (17) S uppose that the form a does not vanish on M . This is valid, when the Frobenius condition is equal to: (18) 8.1.1 Frobenius Condition = aAda = 0 non-vanishing assumption of a Suppose that the form a does not vanish on M . This is valid, when the Frobenius condition is equal to: (19) 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. 8.2 Odd, Even Dimensions IF the dimension of M Q is m: (20) dim(M Q) = m And if the number 2n + 1 is odd, Then there exists an equation, given by: P ( ai ) (21) 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. 8.2.1 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: (22) 8.3 (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. 8.4 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. 8.5 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: (23) ′ 2 F0 ∈ Rn+1 lease note these structures can be drawn in (3) Dimensions w.r.t. R3 . 2 2 2 Let’s consider Sn+1 at every point x ∈ Sn+1 , The tangent space at Sn+1 contains a unique, complex hypermap equal to: P (24) hypermap(tangent space) = (Cn + 1) This equation is the hypermap for the contact: (25) hypermap(contactGeometry) = F0 ∈ x From that hypermap, we can form a sequences si (26) si = (p0 , · · · , P n, Qn) The list of equations are poly-polar of (Cn + 1, F 0) And of level n. 8.6 Atiyah Theory 8.6.1 Atiyah conjecture a ccording to [linnell2007atiyah]: Atiyah group can construct the L2 -Betti numbers of a compact Riemaniann manifold; They are defined in terms of the spectrum of the Laplace operator (on the universal covering of M). one use of Atiyah’s L2 -index theorem, which can be used to compute the χ Euler Characteristic function. 8.6.2 Braid groups N ote that the Atiyah group is likely applied to Artin’s braid groups which are Abelian, torsion-free, nilpotent groups, that are 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 (27) 8.6.3 ( Least Common Multiple (LCM) Definition 1.1, [linnell2007atiyah]): For a discrete group G, is also a set multiple of Least Common Multiple(LCM): (28) lcm(G) := ||F ||F ≤ Gand|F | < ∞| 8.6.4 T Strong Atiyah Conjecture he strong Atiyah conjecture over KG group says: (29) lcm(G)dimG (kerA) ∈ Q That’s ∀A ∈ Mn (QG) T his equation is valid under the condition: For all A ∈ Z Where: K ⊂ Z Moreover, we can make use of the lcm alongside the kernel ker as follows: (30) lcm(G) · dimG (kerA) ∈ Mn (KG) 8.6.5 sub-ring C Where K ⊂ C Is always a sub-ring of the complex numbers which is closed ( under complex conjugation). W hen K = C For a given X y, and if K = C: Atiyah’s groups are finite Abelian, then their class are finitely generated. Where X and y are classes of groups, then (31) G ∈ LX Group G I t means that every finitely-generated subgroup of G is contained in an A∗-group, and G ∈ fy This means that G has a normal A subgroup H such that the quotient is a y-group: G H (32) 9 9.1 9.1.1 Appendix Matrices Seifert Matrix Since any Seifert matrix that is non-singular 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. 9.1.2 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. 9.1.3 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. 9.1.4 p Adic 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 . 9.2 Manifolds 9.3 Homotopy Equivalences in Manifolds ] it is done after the classification of Haken 3-manifolds, we begin studying the Homotopy Equivalences between manifolds, which are similar to those between surfaces. If the Characteristic function of Submanifolds exist, then they can provide us with Splitting Result. the Homotopy Equivalences between surfaces, is a goal to find, in order to describe this splitting result. E.g.letMandM′ be Haken 3-manifolds (closed or not!) and denote by V and V ′ the characteristic submanifolds in the on the defined M and M ′ ′, but Why study Pseudo-Anosov: As diffeomorphisms are part of the homotopy equivalences. (Thus it is practical to study them, on the Pseudo-Anosov Space). 9.4 9.4.1 Surfaces Seifert Space Plus, the split up a homotopy equivalences is: first the map 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 the Seifert fiber spaces. As far as homotopy equivalences or in particular, between, I-bundlesand Seifert fibrespaces. where I-bundles and Seifert Fiber 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). Definition On an Oriented knot or a link (K): A two-variable polynomial (P K), (translated from French, numdam) In a Diagram (D): for an oriented knot, based solely on the links’ orientation, we can distinguish between the (2) main crossings: 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− (33) Algebraic crossing number, c̃(D), which is also equal to: c̃(D) = c+ − c− (34) So, for a given Seifert Surface, the goal is to find an isotopy, 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. Please note that The modification produces an Orientable Surface obtainable from the indicated knot projection by shading. 9.4.2 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′′ (35) Or as an Acceleration: 9.5 Isomorphism A ccording to [cohen2023bundles]: A transform on a collection that is This collection is partially ordered collection by inclusion of some open sets: (36) U ≡ Mn It is good to add that the Isomorphism can be given by evaluating its generator, which forms its fundamental class: (37) F undamentalclass(isomorphism) = Hn (B1 , ∂B1 ) A n isomorphism is given by evaluating a generator which is its fundamental class. E.g. (38) W D = [Mn ]∩ : Hk (Mn : R) → Hn−k (Mn ; R) here D is an isomorphism for all k. 9.5.1 Generators N ote the group of “generators” ( for the covering space ) should be well-defined, on Compact Subspaces, each of which is contained in a chart. Thus, by induction, we could conclude that the following result is valid, and is true ∀i ∈ (k = 1, · · · , n) i.e. for every compact subsets A that are contained in a chart (which in turn is a subset of the Universal subset U : (39) 9.5.2 A⊂U Cup 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. f∗ (α) ∩ φ = f∗ (α ∩ f ∗ (φ)) (40) F urthermore, the cap product is used In Co-homology, where it is represented as an adjoint (of transforms) into the cup products. 9.5.3 Cap Product ∩ ] 9.5.4 Poincaré Duality P oincaré 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: (41) F undamentalclass = [Mn ] ⊂ Hn For (Mn , F ), then the pairing reduces into F : (42) Hk (Mn , F ) × Hn−k (Mn , F ) → F I n addition, it might be useful to generalize the theorem from a compact, into non-compact manifolds setting. 10 Algebra ] Overall review: From a Set to group to Category, and finally Frobenius Group: 1. Group 2. Category 3. Scheme 4.Module 5. Frobenius (how is their theorem is compatible with t 10.1 10.1.1 Fundamental Group Fundamental Group on knots Fisagrouponknots, thenT hef undamentalgroupisessentiallyintroducedbyP oincareinhis(1895)paper[] 10.2 Gauss-Bonnet Concerning the Euler characteristics function can be decomposed into (2) parts as follows: (43) χ(T (M )[M ] Note that we can replace ω’s value as follows: (44) < [FA ], [M ] >= Z ω=ı M Z κdA M Thus, and in other words, the above lemma states that: (45) < [FA ], [M ] >= 2πıχM Where: (46) χM = < [FA ], [M ] > 2πı It is also interesting to add this is equivalent to the classical Gauss-Bonnet theorem, stating that: Z (47) κdA = 2πχM = 2π(2 − 2g) M Where g is the genus of the Riemann Surface Mdocument. I n [ martin1971homology]: By the 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)] It provides an H-cobordism between the boundaries: A(Ḡ)(C) and Ē(C) And by following (Theorem 4.11, in [])’s proof, the classifying maps of Gand E are homotopic; Then it follows that g approximates f : g∼ =f (48) 10.2.1 Normal Bundle L et M be an m-dimensional homology manifold, with PL-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: (49) 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: (50) BH(n) = B · P rol0 (51) γ = P rol0 Please note that T is of dimension (l − 1), can be 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 :→ BH(n) Hence by inducing BH(n), one can successfully extend f By extending f , while having a linear map in terms of χ variable: f, g : X + BH(n) such that the non-linear map is as follows : (52) f ∗ yrg ∗ y Q. Are f and g always independent? g is the morphism to BH while f has a value of both BH ply some γ So putting things in order g should happen first, then because of g occurred (with end value of BH) f might also occur A bundle over 2, where Z ∼ = (X)xI While the quotient is true E/Y = f ∗ y, As it is equaly to f, y multiplipilcation, where the map f : f : Y + BH(n) The isomorphism representing a Linear relationship. However, for maps: f, g : X + BH(n) such that: (53) f ∗γ =g∗γ (Corollary 4.8). N ote: an 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, and we have a ∆-map and Z: (54) Z = ∆xI However, if that is verified, then this implies the Z, χ homeomorphism: Z∼ =χ (55) But Since BH(n)isKan, then the family of classes f , g equality is satisfied: f∼ =g (56) Where, if f ⊂ Z, g ⊂ χ. We can say the same is true when their values are strictly positive: f∼ =g (57) Abelian Knot (Definition): An Abelian knot is a knot which are n- adjacent to the unknot.[askitas2002knot] Ambient Knot (definition ): is part of tame (wild) knots. See also: ultra knot, limit knot. 11 11.1 I Levi-Cevita Sub-manifolds s an abstract manifold, where a manifold is living inside another 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: EuclideanSpace ≡ 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. T here is yet an open question to discuss: Can a hyper-surface contain a few sub-manifolds. I s a surface a manifold, or at can it be at just the latter, at least ? But what makes a surface different from a manifold?) But can a surface be a manifold? I n 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? 11.2 Haken transform F or the 3-manifold case, 0 transformation, as it carries a hyperbolic structure that is easy to construct, ( whereas this map isl different from the I-bundle over the Klein bottle). H owever, the Haken transformation is quite different for surfaces while interesting, it can get quite complicated, as the hyperbolic structures for this transformation are difficult (but interesting ) to construct. Co-homology The Continuous Co-homology of the pro − p is complete, where h is an isomorphism. For infinitely many quotients, And H is torison-free, then factorization property is fulfilled. Note that in Co-homology, we might have to replace F with a non-trivial subgroup. 11.2.1 On a Topological Space Metrics Spaces Hausdorff space L et be a Hausdorff space then there is a dimension (i − 1) correspondence between sub-groups 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 pathα1 is from σ-homotopic to the path an , If there exists a finite sequence of paths: al , , an Where: (58) α1 σ ∼ = αn (59) α1 σ ∼ = αn Where the relevant sequence is: n − 1, ai (t), ai+1 (t) Strong Atiyah Cohomology I t 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. 11.2.2 O Relaxed Atiyah Cohomology n the U-free quotient, which it can be further relaxed. Cohomology, non-trivial I n 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 n interesting observation is that they are 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 there is also an inverse image 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: (60) |( 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, and group G is Abelian, 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 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: G ≡ Hp . Atiyah’s Suggestion L et us construct Hp from G I.e. Hp ≡ G, thus the lcm can be f orallAinMn (KG) as: (61) L · lcm(G), dimG (kerA) ∈ Z Additionally, lcm can be expressed in Terms of the Crossed Product KG: 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: (62) 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. 11.2.3 On a group A ccording 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.([dugundji1950topologized]) T he 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. ] 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 the 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. Q. What makes a structure homotopy invariant? ( hint: it is graphical) 11.2.4 On a Scheme Group Line Bundle 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. 11.3 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 (a: (63) X/A = X Note that A equals: A = I Where the equivalence relation for two points: a, b ∈ A which, when one applies ι on both a,b, hence, one can also define the following congruence: (64) ι(a) ι(b) Whereas A satisfied the following inclusion map: (65) 11.3.1 ι:A⊂X Cone Since any map f : (66) f :X→Y is homotopic to a co-fibration with a co-fiber, Hence it maps a cone: (67) Cone ≡ Y ∪f c(X) Please note that the co-fibration map is equivalent to the mapping Cone: (68) 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, 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 11.4 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: (69) X/A = X Where the equivalence relation for two points a, b ∈ A is given by: (70) ι(a) ι(b) Whereas A satisfied the following inclusion map: (71) ι:A⊂X — 11.4.1 Cone Since any map f : X → Y is homotopic to a co-fibration with a co-fiber, Hence it maps a cone: (72) Cone ≡ Y ∪f c(X) Please note that the co-fibration map is equivalent to the mapping Cone: (73) 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): (74) Co-fibration ≡ ι : A → X Furthermore, by the inclusion of Y Into the mapping Cone, not the it satisfies the following co-fibration relationship: (75) Co-fibration ≡ Y ⊂ Y ∪f c(X) also known as the Co-fiber (On the co-fibration map): (76) Co-fibration ≡ ι : A → X Furthermore, by the inclusion of Y Into the mapping Cone, not the it satisfies the following co-fibration relationship: (77) Co-fibration ≡ Y ⊂ Y ∪f c(X) 11.4.2 On a Vector Space 11.4.3 On a type 11.4.4 On a Manifold 11.4.5 On a Category for full classification see : [johannson1986classification] 12 Conclusion W e have reviewed popular knots, besides some valuable low-dimensional concepts, and some relevant theory behind them, 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. F urthermore, 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, by 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, while shedding some light upon it, 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 them, in return. 13 References ] [A24](1)[4][antosh2024characterization]articlebodyeuler [AK02](2)[askitas2002knot]Askitas, Nikos, and Efstratia Kalfagianni. ”On knot adjacency.” Topology and its Applications 126, no. 1-2 (2002): 63-81. [B89](3) [numdamEntrelacementsxE9quations]-Misc-Body-Knots-Corbordism-Links-Bennequin, Daniel. ”Linkings and Pfaff’s equations.” Russian Mathematical Surveys 44, no. 3 (1989): 1. [BV16](4) (8) [1][BoriesV eys2 016]articlemonodromy Bories, Bart, and Willem Veys. 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