trefoil knot fundamental group

Indeed, they do and we shall present this interpretation with the notation of [1]. 'It's not the space itself - it's what the knot is doing in 3-dimensional space that matters, and this theorem captures that precisely' Theorem 2 (Gordon-Luecke). But, take for example the trefoil knot animated at the top of the page. And, S 3 minus the trefoil knot is secretly the same as SL(2,R)/SL(2,Z)! The fundamental group ˇ 1(S n) is the nth Sieradski group with presentation hy 1;:::;y njy 1 = y 2y n;y 2 = y 3y 1;y 3 = y 4y 2;:::;y n= y 1y n 1i: Remark 1.5. We will be especially interested by the trefoil knot that underlies work of the rst author [2] as well as the gure-of-eight knot, the Whitehead link and the Borromean The knot group (or fundamental group of an oriented link in general) can be computed in the Wirtinger presentation by a relatively simple algorithm. The square knot SK and the granny knot GK are a well known example of a pair of distinct knots with isomorphic fundamental groups. The chapter ends with a proof of the existence of nontrivial knots. CiteSeerX - Document Details (Isaac Councill, Lee Giles, Pradeep Teregowda): Abstract. Therefore, the corresponding Alexander ideal is a principal ideal, and any generator is called an Alexander polynomial of the knot, denoted . The fundamental groups of these knots are therefore amalgamated free products of two copies of the trefoil group. CiteSeerX - Document Details (Isaac Councill, Lee Giles, Pradeep Teregowda): Abstract. So I was playing around with things and started wondering if you could express all the right handed trefoil knot projections and all the left handed trefoil knot projections as two different groups (say R and L) under the same operation of reidmeister moves. The non-commutative generalization of the A-polynomial of a knot of Cooper, Culler, Gillet, Long and Shalen [4] was introduced in [6]. • At the moment, I am most interested in the case where is the trefoil knot, because in this case the knot group is isomorphic to the braid group on 3 strands. This is known as the unknot. Therefore . Amazingly, there are many in-equivalent knots. Knot Groups and the Wirtinger Presentation De nition 2.1. The knot group of a knot awith base point b2S3 Im(a) is the fundamental group of the knot complement of a, with bas the base point. For a proof of this theorem, see Rolfsen (1990). Now I think I understand all the argument and in the end using van Kampen we get that that it's x, y: x 2 = y 3 . Let us state the following important result. Example: c a b Figure 3. The use of the fundamental group allows the definition of algebraic quantities with-out reference to diagrams for the knot. Find equations for the other. Consider, for example, the mirror image of the trefoil knot. Quantum Information in the Protein Codes, 3-manifolds and the Kummer Surface. the knot group. The square knot SK and the granny knot GK are a well-known example of a pair of distinct knots with isomorphic fundamental groups. a three-manifold with finite fundamental group is a topic of long-standing interest, particularly the case of cyclic surgeries. So far it is not known if the Poincaré homology 3-sphere is the only homology 3-sphere with nontrivial finite fundamental group. We classify these knots by using SL(2,)-representations of the fundamental groups of the 2-fold branched covering spaces. We show that Gn(SK) and Gn(GK) are non-isomorphic for all n≥2. The relation in the fundamental group induces the interaction term. A knot diagram of the trefoil knot, the simplest non-trivial knot In the mathematical field of topology , knot theory is the study of mathematical knots . x(t) = sint+2sin2t y(t) = cost−2cos2t z(t) = −sin3t The knotgroupof a knot is the fundamental group of the complement on the knot. Simon's Conjecture (see [16, Problem 1.12(D)]) asserts the following: Conjecture 1.1. π1(S3 \K) surjects onto only finitely many distinct knot groups. By Klee Irwin and Michel Planat. Example The group Br ( 4 ) Br(4) (simplifying notation as before ) has generators u , v , w u,v,w and relations: The problem remains open, although substantial . Proof: The Euler Poincare characteristic of X is . Given a basis {fk} for the first homology group H 1 . 6(1) or (2), where the incoming undercrossing arc . The three-dimensional space surrounding a knot K-the knot complement S3 nK- is an example of a three-manifold [1, 26]. 2. computer for t ∈ [0,2π]. Distance one lens space fillings and band surgery on the trefoil knot 2441 Figure 1: The links L 1 and L 2 differ in a three-ball in which a rational tangle . In each of these cases we obtain the trefoil knot. A special case of this problem is one of the fundamental questions of Knot Theory: Given a knot, is it the unknot? The trefoil knot is not equivalent diffeomorphic to S1 × RRR2), but the fundamental group of the complement of the knotted curve turns out to be the fundamental group for the complement of the trefoil knot. For starters, the 3-strand braid group is also the fundamental group of S 3 minus the trefoil knot! Proof. When S3 p/q(K) has cyclic fundamental group, i.e. Theorem: If X is a topological group which is compact triangulable for which the identity is not isolated then its Euler Poincare characteristic is 0. We show that G n (SK) and G n (GK) are non-isomorphic for all n ≥ 2. December 2005; Revista Matematica . Note that the (p,q) torus knot in one full torus is just the (q,p) torus knot in the other one. The trivial knot is defined. a fundamental group for a topological space. The knot group of the unknot is Z. • This paper is devoted to provide evidence to support . Is it possible to transform this knot so that it looks like the unknot? Figure 2 Figure 2 shows a picture of the trefoil knot projected into the plane. This generalization consists of a finitely generated left ideal of polynomials in the quantum plane, the non- commutative A-ideal, and was defined based on Kauffman bracket skein modules, by deforming the ideal generated by the A-polynomial with respect to a . The fundamental groups of two isotopic spaces are isomorphic, so the fundamental groups of equivalent knots are isomorphic. Here are two knots, the unknot and the trefoil: Example 2.1. You can read about it (in English) in Moritz Epple's paper in Historia Mathematica 22 (1995), 371-401. We will be especially interested by the trefoil knot that underlies work of the rst author [15] as well as the gure-of-eight knot, the Whitehead link and the . For our purpose, the fundamental group ˇ 1 of M3 does the job. This framework also brings into play the powerful techniques of algebraic topology, for instance, homology theory. In fact, if we can prove that the fundamental groups π1(R3\K 1) and π1(R3\K2) are not isomorphic, then we know that the knots K1 and K2 are not equivalent. 'It's not the space itself - it's what the knot is doing in 3-dimensional space that matters, and this theorem captures that precisely' Theorem 2 (Gordon-Luecke). n A2 osf the fibre Therefore, finally we get a complete set of operations to realize any quantum circuit: a 1-qubit operation by the knot group of the trefoil knot and a 2-qubit operation by the complement of the link (Hopf link for instance). If R3 nK 1 ˇR3 nK 2 then K 1 ˘=K 2 Remark 4. This is also known as the "trefoil knot group", i.e., the fundamental group of the complement of a trefoil knot. So, the fundamental group of this space is the same as the fundamental group of S 3 minus a trefoil knot. The trefoil knot is the simplest example of nontrivial knot, so it seems remarkable that, not long after the discovery of the fundamental group of a topological space, Max Dehn (1914) succeeded in proving that the trefoil knot and its mirror image had isomorphic groups, but their knot types were distinct. So AN INTRODUCTION TO KNOT THEORY AND THE KNOT GROUP 5 complement itself could be considered a knot invariant, albeit a very useless one on its own. For the group criteria, closure holds by definition. The Fundamental Group of a Link Notes for Math 148 Jim Hoste December 12, 2012 1 The Wirtinger Presentation Associated to every knot or link in R3 (or in S3) we can associate a group, called the fundamental group of the complement of the link, or more simply, the fundamental group of the link. The fundamental group of a knot can be also defined in a combinatorial way as follows: consider a diagram of the knot and a crossing in diagram, as in Fig. These "generalized knot groups" were introduced independently by Wada and Kelly, and contain the fundamental group as a subgroup. the knot. Associativiy clearly holds. However, two distinct knots can have isomorphic fundamental groups. K)surjects onto the fundamental group of the trefoil knot complement. The fundamental groups of each piece are isomorphic to Z, while the intersection deformation retracts to the torus minus the trefoil, which is an annulus, so also has fundamental group Z. Consider, for example, the mirror image of the trefoil knot. Trefoil knot Fundamental Group The knot group is the fundamental group of the knot complement (R3-K) The fundamental group is the set of the product of homotopy classes (loops that have the same base point and there is a path that maps one loop to the other) For the group criteria, closure holds by definition. Group Geometrical Axioms for Magic States of Quantum Computing. A representation used to represent the fundamental group of the trefoil knot, applied to the two-variable HOMFLY polynomials, yields simple polynomials which distinguish knots of low crossing numbers. For any coprime integers p and q, the tori (p,q) and (q,p) are isotopic. For starters, the 3-strand braid group is also the fundamental group of S 3 minus the trefoil knot! A filling Dehn surface in a 3-manifold M is a generically immersed surface in M that induces a cellular decomposition of M. Given a tame link L in M there is a filling Dehn sphere of M that "trivializes" (diametrically splits) it. Knot groups are, by definition the fundamental group of the complement of a knot K embedded in . first at a single component link we will use the trefoil knot shown in Fig-ure 1.3. Contents Abstract iii . spaces have isomorphic fundamental groups, the knot group is an invariant of the isotopy class of a knot. The trefoil knot is the simplest form of knot, however it is a fundamental part of learning the knot theory. sentations of the fundamental group of a link into G are described by the multivariable Alexander polynomial of the link. Example 2.2. The last quandle can be identified with the Dehn quandle of the torus and the cord quandle on a 2-sphere with four punctures. The knotted S2 in S4 is obtained by spinning a trefoil knot in the manner According to Wikipedia "The trefoil can be obtained by joining together the two loose ends of a common overhand knot, resulting in a knotted loop." The name itself is derived from the clover - trefoil plant. Thus any loop in space can be moved into the union of the 0, 1, and 2-handles. The granny knot is the connect sum of two left- or two right-handed trefoils, while the square knot is the connect sum of a left- and a right-handed trefoil. One might speculate that, in general, the braid groups occur naturally as fundamental groups. While inspired by knots which appear in daily life, such as those in shoelaces and rope, a mathematical knot differs in that the ends are joined so it cannot be undone, the simplest knot . By (ii) the fundamental group of the Poincaré homology 3-sphere is the binary icosahedral group /120 • This group has order 120 and trivial abelianization. trefoil knot manifold (or of hyperbolic 3-manifolds) Alexander's theorem states that every knot or link can be represented as a closed braid [14]. By Klee Irwin, Michel Planat, and Marcelo Amaral. Examples The unknot has knot group isomorphic to Z. the (3, 5) ((2, 5), (2, 3)) torus knot. (2, 3)) torus knot. For our purpose, the fundamental group ˇ 1 of M3 does the job. The group presentations are well-known once the manifolds are recognized as branched cyclic covers of S3, branched over the gure-eight knot and the trefoil knot. trivial finite fundamental group [9, §62]. Proof. The three-dimensional space surrounding a knot K-the knot complement S3 nK- is an example of a three-manifold [1, 24]. simply call the fundamental group of S3 \K, the group of K or the knot (link) group. This paper has two-fold goal: it provides gentle introduction to Knot Theory starting from 3-coloring, the concept introduced by R. Fox to allow undergraduate students to see that the trefoil knot is non-trivial, and ending with statistical mechanics. This allows to construct filling Dehn surfaces in the coverings of M branched over L. It is shown that one of the simplest filling Dehn spheres of S3 (Banchoff . Now, for a simple loop, that's an easy question. This group is nonabelian, since it surjects onto the symmetric group S 3. By punctured M3, or more briefly pune M3, we mean a space homeomorphic to M3 minus a point; and by bounded punctured M3 we mean a space homeomorphic to M3 minus a (tame) open 3-cell. Quantum Computation and Measurements from an Exotic Space-Time R 4. Answer (1 of 3): There are even examples which are closed manifolds: Poincare dodecahedral space is a closed 3-manifold with fundamental group the binary icosahedral group \widetilde{A}_5. Indeed, the fundamental group of the (p;q . The trefoil knot is the simplest example of nontrivial knot, so it seems remarkable that, not long after the discovery of the fundamental group of a topological space, Max Dehn (1914) succeeded in proving that the trefoil knot and its mirror image had isomorphic groups, but their knot types were distinct. The fundamental group sometimes allows us to answer the question 'R3 nK 1 ˇ R3 nK 2?' and therefore turn this into an . Hence for the trefoil knot, we have the presentation: π1 (T ) = π1 (S 3 \ N (T )) = (a, b, c | a = b−1 cb, b = c−1 ac, c = a−1 ba). Trefoil with surface and 0-framed longitude (red) Trefoil with meridian (red) A Hopf link with several elements from the fundamental group of its complement in 3 0-FRAMED LONGITUDE. This allows to construct filling Dehn surfaces in the coverings of M branched over L. It is shown that one of the simplest filling Dehn spheres of S3 (Banchoff . For example, the knot group of the trefoil knot is known to be the braid group B 3 , {\displaystyle B_{3},} which gives another example of a non-abelian fundamental group. Chapter III presents two sophisticated methods of calculating This is the simplest nontrivial example and should serve as a template for others. It is invariant under ambient isotopy. the right-hand trefoil knot (or, reversing orientation, from -1-surgery on the left-hand trefoil knot). Fundamental group of the complemente of the Trefoil Knot 2 Now I have just read about how to compute π 1 ( R 3 − K) where K is the trefoil knot. Figure 2 shows that trefoil knot, which is not equivalent to the unknot. Chapter II defines knot group and calculates the knot group for the trefoil, figure eight and square knots. Knot groups are the groups that appear as fundamental groups of where is a knot. If R3 nK 1 ˇR3 nK 2 then K 1 ˘=K 2 Remark 4. The three-dimensional space surrounding a knot K-the knot complement S3 \ K- is an example of a three-manifold [1, 26]. We will be especially interested by the trefoil knot that underlies work of the rst author [15] as well as the gure-of-eight knot, the Whitehead link and the . Answer (1 of 6): I was a little bit unsatisfied with other answers and so I propose here mine. 2. There exist infinitely many ribbon knots in Si with fundamental group the trefoil knot group, but with non-isomorphic n2 {as ZTI\-modules). Specifically, the fundamental group is given: CiteSeerX - Document Details (Isaac Councill, Lee Giles, Pradeep Teregowda): We follow the program set out in the author's first year report to extract the algebraic information eponymous with this document associated to a diagram of a trefoil knot. Accordingly, we begin by fixing notation for the trefoil group T. 1 and R3\K2 have isomorphic fundamental groups. We will be especially interested by the trefoil knot that underlies work of the first author [2] as well as the figure-of-eight knot, the Whitehead link and the Borromean For example, we have computed the fundamental group of the trefoil knot and the fundamental group of the cinquefoil knot. We can label the strands and crossings as indicated in the figure. Then subdivide your space into a slight thickening of the torus (minus the thickened knot) and a slight thickening of the complement. The knot group is the braid group of three strands used for anyons too. Unfortunately, it can be very difficult to verify whether or not the resulting groups of two different knots are isomorphic, so the knot group is limited in its practical use. group Z, so is the abelianization of the knot group. ment of the knot in three space, R3 −K, and form its fundamental group. Thus the knot group is a knot invariant. Any path in space is homotopic to a path that misses the 3-handle. Given a knot K we may construct a group Gn(K) from the fundamental group of K by adjoining an nth root of the meridian that commutes with the corresponding longitude. K)surjects onto the fundamental group of the trefoil knot complement. Nonetheless, it is a powerful invariant. The interaction terms are known from the Ising model. Write a formula for an extension to a tubular neighborhood. On the other hand, not every knot group is bi-orderable; for example, it is not di cult to see that the knot group of a non-trivial torus knot is not bi-orderable. fundamental group π1 of M3 does the job. Then the row corresponding to crossing 1 . As before, if an ambient isotopy existed, then the complements of the two embeddings of the line would be diffeomorphic and hence would have isomorphic fundamental groups. and filed under math. It can be used to find Milnor's invariants The knot group of a knot K is the fundamental group of the com- Suciu constructed infinitely many ribbon 2-knots in S4 whose knot groups are isomorphic to the trefoil knot group. It is de ned up to multiplication by units of , i.e., elements of the form ti for some integer i. For example, while using the knot group, the trefoil knot cannot be deformed into the -gure-eight knot (Figure 2:1) by showing the corresponding knot groups are not isomorphic. describing thereby in an intrinsic way part of the asymptotic Brownian behavior in the fundamental group of the . These "generalised knot groups " were introduced independently by Wada and Kelly, and contain the fundamental group as a subgroup. These "generalised knot groups " were introduced independently by Wada and Kelly, and contain the fundamental group as a subgroup. fundamental group of the complement of the trefoil knot, and that is what Heegaard got by extending Riemann's surfaces to 3-manifolds and projecting sterographically onto SA3 the self-intereections of a 4-manifold immersed in real 6-space. In which case either there is an algebraic curve component of genus 0 which is the component of the character variety of the trefoil knot group containing an irreducible representation or there is a component of dimension at least.

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