979Order in the concordance group and Heegaard Floer homologyWe use the Heegaard-Floer homology correction terms defined by Ozsváth-Szabó to formulate a new obstruction for a knot to be of finite order in the smooth concordance group. This obstruction bears a formal resemblance to that of Casson and Gordon but is sensitive to the difference between the smooth versus topological category. As an application we obtain new lower bounds for the concordance order of small crossing knots.
Let ν be any integer-valued additive knot invariant that bounds the smooth 4-genus of a knot K , |ν(K)| ≤ g 4 (K), and determines the 4-ball genus of positive torus knots, ν(T p,q ) = (p − 1)(q − 1)/2. Either of the knot concordance invariants of Ozsváth-Szabó or Rasmussen, suitably normalized, have these properties. Let D ± (K, t) denote the positive or negative t-twisted double of K . We prove that if ν(D + (K, t)) = ±1, then ν(D − (K, t)) = 0. It is also shown that ν(D + (K, t)) = 1 for all t ≤ TB(K) and ν(D + (K, t)) = 0 for all t ≥ −TB(−K), where TB(K) denotes the Thurston-Bennequin number.A realization result is also presented: for any 2g × 2g Seifert matrix A and integer a, |a| ≤ g, there is a knot with Seifert form A and ν(K) = a. 57M27; 57M25
Murasugi discovered two criteria that must be satisfied by the Alexander polynomial of a periodic knot. We generalize these to the case of twisted Alexander polynomials. Examples demonstrate the application of these new criteria, including to knots with trivial Alexander polynomial, such as the two polynomial 1 knots with 11 crossings.Hartley found a restrictive condition satisfied by the Alexander polynomial of any freely periodic knot. We generalize this result to the twisted Alexander polynomial and illustrate the applicability of this extension in cases in which Hartley's criterion does not apply. 57M25, 57M27
Two knots in three-space are S-equivalent if they are indistinguishable by Seifert matrices. We show that S-equivalence is generated by the doubled-delta move on knot diagrams. It follows as a corollary that a knot has trivial Alexander polynomial if and only if it can be undone by doubled-delta moves.We consider tame, oriented knots in oriented S 3 , with equivalence being ambient isotopy. A Seifert surface for such a knot is an oriented surface whose boundary is the given knot, and whose orientation induces the given orientation on the knot. An oriented surface S in S 3 has a linking form * , * on the homology H 1 (S), where x, y is defined to be the linking number of the cycle x with the cycle y slightly pushed off S in a direction determined by the orientation of S. Given a knot K, choose a Seifert surface S for K and a basis for H 1 (S). Then the linking form is represented by an integer matrix M , which is called a Seifert matrix for K. Two knots are called S-equivalent if they have a common Seifert matrix (which is the same as saying that they have a common Seifert form). We sketch a proof at the end of the paper that this is an equivalence relation. The reader who wishes may ignore this proof and take S-equivalence to be the smallest equivalence relation that includes any pair of knots which have a common Seifert matrix. Two knots K and K ′ are then S-equivalent if and only if there exists a sequence K = K 1 , K 2 , . . . K m = K ′ , such that for all 1 ≤ i < m, K i and K i+1 have a common Seifert matrix. It makes no difference in the proof of Theorem A which definition we use, since in either case what we need to show is that two knots share a common Seifert matrix if and only if they are equivalent by certain diagram moves which we will call doubled-delta moves.The usual way to define S-equivalence is to define it first for matrices, and then to define it for knots by saying that knots with S-equivalent matrices are S-equivalent. See Gordon [3] and Kawauchi [5] for the standard definition, for further references, and for more detail on the following statements. S-equivalence of matrices was first introduced by Trotter in [13] under the name h-equivalence. Murasugi [10] and Rice [11] applied it to matrices obtained from knot diagrams. None of the abelian invariants, such as the Alexander polynomials, homology of cyclic and branched covers, or signatures, can distinguish between S-equivalent knots. It was shown by Levine [7] that in higher dimensions simple knots are characterized by S-equivalence. Two knots are S-equivalent if and only if their (integral) Blanchfield pairings are isometric. This follows from work of Levine [7] and Kearton [6], and was also proved by Trotter [14] from a purely algebraic point of view.A knot may be given by a regular projection in the usual way, with equivalence of diagrams given by the Reidemeister moves. For more details on knots, diagrams, and Seifert surfaces and matrices, see Rolfsen [12]. or Kawauchi [5]. We consider now the delta move and the doubled-de...
Suppose that K is a knot in S 3 with 2-fold branched cover M K . Our main result is the following. 0.1 Theorem. If |H 1 (M K )| = pm with p a prime congruent to 3 mod 4 and gcd(p, m) = 1, then K is of infinite order in the classical knot concordance group, C 1 .Our interest in this result is its application to the study of 4-torsion in the concordance group. There are 11 prime knots of 10 or fewer crossings, beginning with the knot 7 7 , that represent elements of order 4 in the algebraic concordance group. A simple calculation using this theorem yields: 0.2 Corollary. No prime knot with fewer than 11 crossings represents an element of order 4 in C 1 .Of greater interest than obstructing individual knots from being of order 4 is that the obstruction depends only on an abelian invariant of the knot. Hence corollaries like the next one concerning the Alexander polynomial of a knot, ∆ K (t), follow readily.0.3 Corollary. If ∆ K (t) = 5t 2 − 11t + 5 then K is of infinite order in C 1 .By way of contrast, according to Levine [L2], every knot with ∆ K (t) = 5t 2 − 11t + 5 is of order 4 in C 2k−1 for k > 1.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.