D. Rolfsen, Knots and links, Publish or Perish, Inc (1976).  Home/Search   Document Not in Database   Summary   Related Articles   Check

....p sphere can have any knot type, and similar but weaker results when the spheres are of specified knot type. Keywords : Statistical topology, higher dimensional links, random linking, entanglement complexity. 1. Introduction Knots and links have been studied for over one hundred years [1, 2] but only recently has there been an interest in statistical knot and link theory, where one asks for the probability that p spheres randomly embedded in Euclidean d space are topologically entangled (knotted or linked) 3] More precisely, one considers two embeddings which can be superimposed ....

D. Rolfsen, Knots and Links, Publish or Perish, Inc., Wilmington (1976).

....This line of investigation builds upon topological comparisons based upon knot theory [4, 5, 22] leading to adoption [1, 3, 16, 19] of the even stronger form of topological equivalence known as ambient isotopy. This stronger knot equivalence has been in the mathematical literature for some time [9, 11, 18], but it appears that recent applications of this theory in engineering modeling are novel. The example in Figure 2 shows how an unknotted curve can have a homeomorphic approximation which is badly knotted . There are two recently created criteria for such algorithmic approximations. For the ....

D. Rolfsen, Knots and Links, Publish or Perish, Berkeley, 1976.

....Euler characteristic. Furthermore, the restriction that is simply connected is not enough to deduce the topology of the boundaries # i even in this simple case. It is well known that all closed 3 manifolds are cobordant to S . In fact one can construct a cobordism with trivial fundamental group [20]. Thus, at least by the methods discussed here, the topology of the interior of a (4 1) dimensional ALADS spacetime is constrained but not completely characterized by the topology of the boundary at infinity. We mention in closing that the geodesic methods used to prove topological censorship ....

D. Rolfsen, Knots and Links, Publish or Perish, Berkeley, 1976.

....essential in M( for all but at most C(g) lines of slopes The following theorem shows that no such constant C(g) exist. Recall that with respect to the standard meridian longitude pair (m; l) of a knot K in S , a slope = pm ql on N(K) is represented by a rational number p=q or 1=0. See [Ro]. Thus M( M(p=q) Theorem 3.5. Let K be a non bred hyperbolic knot in S with genus g, and let M = S IntN(K) be the knot exterior. Then for any constant C there is an immersed essential surface F of genus 2g with the longitude of K as a coannular slope, such that F is compressible in ....

D. Rolfsen, Knots and Links, Publish or Perish, 1990.

....a Seifert matrix for the bered link K. Then the restriction map of the monodromy h, h : H 1 ( R) H 1 ( R) is represented by the matrix h = V V (1) in terms of the basis 1 ; n . The Alexander polynomial of a bered link is the characteristic polynomial of h . See [Rol76], Ch. 8, for more information on Seifert matrices and Alexander polynomials. From an ordered chord diagram L = f 1 ; n g, we de ne a bered link KL with ber L by starting with a disk in S and doing successive Murasugi sums [MM82] of Hopf bands as in Figure 2. That is, we embed ....

....associated to Dynkin diagrams. The simply laced minimal hyperbolic Coxeter system of smallest dimension is a triangle with a tail. The Coxeter link (see Figure 11) is uniquely determined in this case by the requirement of positivity, and equals the mirror of the 10 145 knot in Rolfsen s table [Rol76], which is (22; 3; 3 ) in Conway s notation [Con70] 2.5. Remarks on the Geometry of Coxeter systems and Coxeter links. We will assume throughout this section that Coxeter systems are irreducible, or equivalently the Coxeter graph is connected. A Coxeter system is nite if and only if its ....

D. Rolfsen, Knots and links, Publish or Perish, Inc, Berkeley, 1976.

....independent of knot orientations. Also, the linking number does not completely recognize linking. The Whitehead link in Figure 1, for example, has linking number zero, but is not separable. Surfaces. The linking number may be equivalently defined by other methods, including one based on surfaces [17]. A spanning surface for a knot k is an embedded surface with boundary k. An orientable spanning surface is a Seifert surface. Because it is orientable, we may label its two sides as positive and negative. We show examples of such surfaces for the Hopf link in Figure 3. Given a pair of oriented ....

....link and Seifert surfaces of its two unknots. Clearly, 1. This link is the 200th complex for data set H in Section 6. SEIFERT SURFACE LEMMA. k; k ) is the sum of the signed intersections between k and any Seifert surface for k. The proof is by a standard Seifert surface construction [17]. If the spanning surface is non orientable, we can still count how many times we pass through the surface, giving us the following weaker result. SPANNING SURFACE LEMMA. k; k ) mod 2) is the parity of the number of times k passes through any spanning surface for k. Graphs. We need to ....

ROLFSEN, D. Knots and Links. Publish or Perish, Inc., Houston, Texas, 1990.

....cubes all of whose 8 vertices are in R. The boundary of M consists of closed oriented two dimensional manifolds. Now each boundary component of M divides R into an outside and an inside: this is a classical theorem of algebraic topology (essentially the generalized Schoenflies Theorem, see [28]) Note that two boundary components of M are at d1 distance at least 1 2 . Let x 1 ; x 2 be two vertices in 1 R and x 2 points of M to which they are closest. We claim that x 2 are on the same boundary component of M . This is because (using connectedness of A) there is a path in ....

....components of Y correspond to different components of X: one can do this by pushing each local component of X slightly off of x. Note that this does not destroy the property of X of disconnecting cyl A n . Now X consists of closed two manifolds and by the generalized Schoenflies theorem [28], at least one component C of X cyl A disconnects cyl A n . be the set of edges corresponding to plaquettes in the neighborhood of C. Then is connected (in the sense that any two plaquettes are connected by a path of adjacent plaquettes) and separates 1 in cyl A. Corollary 5.2. ....

D. Rolfsen, Knots and links, Publish or Perish, Houston, 1990.

....p i 2 : p r . To see this, we must nd a natural surface of minimum genus bounded by K(r) and compute its Euler characteristic. By Theorem 12, Lemma 12.1 and Theorem 22 of [Sch53] a surface of minimum genus bounded by K(r) may be constructed by Seifert s algorithm, explained in Chapter 5 of [Ro76], from a representative of K(r) which has minimal braid index. We constructed such a representative in our proof of Part (1) To compute its Euler 22 characteristic, use the fact that is the number of Seifert circles minus the number of unsigned crossings (Exercises 2 and 10 on pages 119 and ....

....a representative of K(r) which has minimal braid index. We constructed such a representative in our proof of Part (1) To compute its Euler 22 characteristic, use the fact that is the number of Seifert circles minus the number of unsigned crossings (Exercises 2 and 10 on pages 119 and 121 of [Ro76]) By a theorem of Yamada [Ya] the number of Seifert circles is the same as the braid index, i.e. p 1 p 2 p r in our situation. The number of unsigned crossings is b r , where b i = p i 1)q i b i 1 p i and b 1 = p 1 1)q 1 and = p 1 p 2 p r b r . Adding up the contributions from ....

D. Rolfsen, Knots and Links, Publish or Perish

....2. Do the norms agree whenever L is bered 1 Notes and references. The Alexander polynomial of a knot was introduced in 1928 [Al] Fox treated the case of links and general groups via the free di erential calculus [Fox] For more on the Alexander polynomial of a knot, see [Mil] CF] Go] and [Rol]; for links, see [Hil] and [BZ] and for 3 manifolds, see [Tur] References for bered links include [Mur2] St] Har] and [Ga2] David Fried observed in the 1980s that the Thurston norm is related to the exponents of the Alexander polynomial in many examples. The Alexander ideal of a link is ....

....j i j = 1 for all i. They have also shown 2g(K) deg K for alternating knots [Cr] Mur1] In both cases an optimal surface is obtained using Seifert s algorithm, so this check can also be carried out using (b) Tables of knots and links. We now turn to examples drawn from the tables in [Rol]. These tables give diagrams for the prime knots up to 10 crossings and links up to 9 crossings, together with their Alexander polynomials. Notation such as 9 3 6 indicates the 6th link with 9 crossings and 3 components. It is known that 2g(K) deg K for all knots with 10 crossings or less (see ....

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D. Rolfsen. Knots and Links. Publish or Perish, Inc., 1976.

....Embedding graphs into S 3 extends, in a natural way, the classical knot theoretical problem of embedding one or more disjoint copies of the 1 sphere S 1 into S 3 where the resulting images are called knots and links, respectively. Classical terminology of knot theory can be found in [1] or [17], see [9] 11] 12] 16] for more recent introductions to the field. A topological graph is a 1 dimensional cell complex which is related to an abstract graph in the obvious way. In the following, always 4 regular graphs, which are allowed to have multiple edges or loops, are considered. ....

D. Rolfsen, Knots and Links, Publish or Perish, Berkeley, 2nd printing (1990).

....of knot orientations. Also, the linking number does not completely recognize linking. The Whitehead link in Figure 1(a) for example, has linking number zero, but is not separable. Surfaces. The linking number may be equivalently defined by other methods, including one based on surfaces [17]. A spanning surface for a knot k is an embedded surface with boundary k. An orientable spanning surface is a Seifert surface. Because it is orientable, we may label its two sides as positive and negative. We show examples of spanning surfaces for the Hopf link and Mobius strip in Figure 2. Given ....

....lemma asserts that this sum is independent of our the choice of h and s, and it is, in fact, the linking number. Seifert Surface Lemma. #(k, k # ) is the sum of the signed intersections between k # and any Seifert surface for k. The proof is by a standard Seifert surface construction [17]. If the spanning surface is non orientable, we can still count how many times we pass through the surface, giving us the following weaker result. Spanning Surface Lemma. #(k, k # ) mod 2) is the parity of the number of times k # passes through any spanning surface for k. Graphs. We need to ....

Rolfsen, D. Knots and Links. Publish or Perish, Inc., Houston, Texas, 1990.

....of G t G 0 . 4 Proof of Theorem 7 For the proof of Theorem 7, some knot theoretical definitions and results have to be given. The objects under consideration, as they are needed here, 8 are described more detailed in [12] For general knot theoretical terminology see, e.g. 1] 7] 8] [11]. In the following, a link is an embedding of a graph in R 3 that consists of one or more disjoint loops. A tangle is a part of a link diagram in form of a disk with four arcs emerging from it, see Fig. 2, where the tangle s position is indicated by e t a b c d t f t N(t) g t D(t) ....

D. Rolfsen, Knots and Links, Publish or Perish, Berkeley, 2nd printing (1990).

....of (1) to go through, the result is not always the Jones polynomial of a knot. This happens in particular in the cases, where 2k d 1. But the more thorough check using theorem 2.2 reveals the criterion to be more effective. A computer program is easily written and it showed for the Rolfsen [Ro, appendix] knots and their obverses that the criterion applies in 105 (that is, about one fifth) of the cases. Clearly, many cases are irrelevant, because other methods, e.g. the signature s, works. Nevertheless, in some cases, if the criterion applies for both mirror images, or considering one of them is ....

D. Rolfsen, Knots and links, Publish or Perish, 1976.

....homeomorphism # f on a boundary component of Y , it can not be the antipodal map. Thus the restriction of # f on #Y is the identity. Consider S 2 as the standard unit sphere in R 3 , and let x be the north pole. Then C = x [0, 1] is an arc in Y . By the Light Bulb Theorem [Ro, Page 257], we may change # f by isotopy rel #Y so that # f C = id. Let D 1 , D 2 be the upper and lower half disks of S 2 . Deform # f by an isotopy rel #Y so that it maps each D 1 t isometrically to itself, then further deform # f on D 2 [0, 1] rel #, so that it maps each D 2 t ....

D. Rolfsen, Knots and Links, Publish or Perish, 1976.

....[M] # 1 (K(p, q) n r) is cyclic if and only if n = rpq 1. Thus E(K 1 ) is an rpq 1 fold cyclic covering of E(K(p, q) But such a covering space is homeomorphic to E(K(p, q) This may follow from the fact that T (p, q) is a fibered knot with holonomy a periodic map of order pq, see e.g. [R]) # With Lemma 9 in hand, we need only to prove Theorem 1 for hyperbolic knots and satellite knots. Proof of Theorem 1 for hyperbolic knots: Suppose the complement E(K) of a hyperbolic knot K covers two knot complements E(K 1 ) and E(K 2 ) By Lemma 1, these coverings are cyclic. Let # 1 = # ....

D. Rolfsen, Knots and Links, Publish or Perish, 1976.

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D. Rolfsen, Knots and Links, Publish or Perish, 1976, p. 49; MR 58 #24236.

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D. Rolfsen. Knots and Links. Publish or Perish,Inc, Berkeley, 1976.

....defined as the algebraic intersection number of F and K , i.e. it is the sum of sign(x) over all x F #K.SinceH 2 (M,Z) this intersection number is well defined. When M is or S is also a knot, the above definition of linking number is equivalent to any of the eight definitions in [11]. Denote by i the knot K i with orientation reversed. We have the following basic property of ,K) which will be used in the paper repeatedly, in particular in Section 4 to investigate the role played by the Kojima Yamasaki # function in detecting the chirality of the link K 1 2 when lk(K ....

....is obtained from the Hopf link by Bing doublingn times (along any components) then it is # achiral if and only if #(#) 1)n 1 . 2 be the Hopf link. Since Whitehead link is of the form W(K 1 )#K 2 , by Corollary 3.2(2) it is absolutely chiral. K 2 be the link 8 in the table of [11], and let K 1 be the unknotted component. See Fig. 4(1) After surgery on K 1 the knot K 2 becomes K and K in S as shown in Fig. 4 (2) and (3) which are the knots 6 3 and 7 7 in the knot table of [11] By Lemma 2.3(1) L is absolutely chiral. 1) 2) 3) Fig. 4. 4. The # functions The ....

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D. Rolfsen, Knots and Links, Publish or Perish, Berkeley, CA, 1976.

....are words in x 1 , x 2g . Proof All these conclusions are quite easy to see from Figure 2.1 and the assumption that a g 1 have no intersectionswer the disk bounded by a g 1 . Figure 2. 1 3 Twisted Alexander Modules Let us first recall the definition of the Alexander module of a knot K (see [10]) Denote by M the knot complement S K.Let M M be the infinite cyclic covering. Consider the homology group H 1 ( M) H 1 ( M ; Z) Let t : 1 be the isomorphism induced by a generator of the deck transformations. Thenw e can think of M)asaZ[t, t 1 ] module. This is the ....

D. Rolfsen, Knots and links, Math. Lecture Series, Publish or Perish, 1976, 7.

....h 01 (f (T)g(U) g(T)f(U) h 11 g(T)g(U) or, in matrix notation, # H(T , U) # = br(T ) t H br(U) # The equivalence of the two formulae for # H(T , U) # may be demonstrated by means of Proposition 2.2. We turn now to geometric properties of the link H(T , U) Recall from [R1] that a link L in a solid torus V is said to be geometrically essential in V if each cross sectional disk of V meets L . The sublinks T N , U N of H(T , U) lie in solid tori V T , VU respectively, whose cores form a Hopf link. If T N , U N are geometrically essential in their respective ....

D. Rolfsen, Knots and Links, Publish or Perish, 1976.

....the direction of further investigations is indicated. 1 Virtual Knots and Links In classical knot theory, knots and links in 3 dimensional space are examined. As a main tool, projections of such links to an appropriate plane are considered, namely, the so called link diagrams (see, for example, [22], 1] 13] 16] 21] 19] The idea of virtual knot theory is to consider link diagrams where an additional crossing type is allowed. Thus, for an oriented diagram, there are three types of crossings: the classical positive or negative crossings and the virtual crossings (see Fig. 1) For a ....

D. Rolfsen, Knots and Links, Publish or Perish, Berkeley, 2nd printing (1990).

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D. Rolfsen, Knots and links, Publish or Perish, Inc (1976).

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D. Rolfsen, KNOTS, Publish or Perish, first edition (1976)

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D. Rolfsen, Knots and links, Publish or Perish, 1990.

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D. Rolfsen. Knots and Links. Publish or Perish, Inc., 1976.

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