For any pair of knots of Gordian distance two, we construct an infinite family of knots which are ‘between' these two knots, that is, which differ from the given two knots by one crossing change. In particular, we prove that every knot of unknotting number two can be unknotted via infinitely many different knots of unknotting number on

BAADER, S.

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The Quarterly Journal of Mathematics Advance Access published on November 7, 2005Quart. J. Math. 57 (2006), 139–142; doi:10.1093/qmath/hai010NOTE ON CROSSING CHANGESby S. BAADER†(Mathematisches Institut der Universität Basel, Rheinsprung 21, CH-4051 Basel, Schweiz)[Received 2 February 2005]AbstractFor any pair of knots of Gordian distance two, we construct an infinite family of knots which are‘between’ these two knots, that is, which differ from the given two knots by one crossing change.In particular, we prove that every knot of unknotting number two can be unknotted via infinitelymany different knots of unknotting number one.1. Introduction and main resultIt is an elementary fact that every tame knot in S3 can be unknotted by a finite sequence of crossingchanges. Here, a crossing change is a strand passage operation along an embedded disc (see Fig. 1for an illustration).The Gordian unknotting number u(K) of a knot K is the minimal number of crossing changesneeded to transform K into the trivial knot [8]. More generally, the Gordian distance between twoknots is the minimal number of crossing changes needed to transform one knot into the other [1, 4].However, this process is by far not unique.THEOREM 1.1 For every pair of knots K and ˜K of Gordian distance two, there exist infinitely manynon-equivalent knots whose Gordian distance to K and ˜K is one.Theorem 1.1 has an interesting special case.COROLLARY 1.2 Every knot of unknotting number two can be unknotted via infinitely many differentknots of unknotting number one.Proof of Theorem 1.1. If two knots K and ˜K have Gordian distance two, then there exists a knotK0 which differs from K and ˜K by one crossing change. The knot K0 has a diagram with twodistinguished spots where crossing changes should take place to obtain K or ˜K , respectively. Thisset-up is shown in Fig. 2, where the two spots are labelled A and B; a crossing change at A or Btransforms K0 into K or ˜K , respectively.Moreover, we may assume that the two spots A and B are neighbouring, as shown in Fig. 3,because we can slide the clasp of one spot, say B, along the knot K0.Now we are ready to define a family of knots {Kn}n∈N diagrammatically. A diagram of the knot Knis shown in Fig. 4. It is understood that it coincides with the diagram of Fig. 3 outside the indicatedsection.†E-mail: baader@math-lab.unibas.ch139© The author 2005. Published by Oxford University Press. All rights reservedFor permissions, please email: journals.permissions@oxfordjournals.org140 S. BAADERFigure 1. A crossing change along a disc.Figure 2. A diagram of K0.The horizontal band is twisted −n times, in a manner that gives rise to 2n negative crossings. Thetwo vertical bands are twisted n times. We observe that all the knots Kn have the two required prop-erties: changing a crossing at A transforms Kn into K . Similarly, a crossing change at B transformsKn into ˜K .However, we have to prove that the family {Kn}n∈N contains infinitely many different knots. For thispurpose, we consider the Alexander polynomial written in Conway’s notation [2]. This polynomialin one variable z is normalized to one for the trivial knot and satisfies the following relation:P (z) − P (z) = zP (z).As we shall apply this skein relation several times, it is convenient to introduce a concise notationfor the knots arising from crossing changes and smoothings at different spots.Figure 3. A section of a diagram of K0.CROSSING CHANGES 141Figure 4. A family of knots {Kn}n∈N.Notation. For x, y, z ∈ Z ∪ {∞}, Kx y z denotes the knot with x full twists in the horizontal bandat the top left of the section as shown in Figs 3 and 4, y full twists in the left vertical band and z fulltwists in the right vertical band. The special case where x (y or z) is ∞ means that we have to smooththe diagram (in an oriented manner) at the corresponding place. As an example, K∞ 0 ∞ is shown inFig. 5.In particular, K−n n n denotes Kn (n ∈ N).LEMMA 1.3limn→∞1n2P(Kn; z) = −z2P(K∞ 0 ∞; z).Before we prove Lemma 1.3, let us complete the proof of Theorem 1.1. Suppose the family{Kn}n∈N contained only finitely many different knots, then the limit of Lemma 1.3 would be clearlyzero. However, this is not the case because P(K∞ 0 ∞; z) is not zero. Indeed, K∞ 0 ∞ is a connectedsum of a knot and two Hopf links. Therefore, its Conway polynomial P(K∞ 0 ∞; z) is a product of aConway polynomial of a knot and ±z2, which can certainly not be zero.Proof of Lemma 1.3. Applying the skein relation of the Conway polynomial n times at the crossingsof the left vertical band of K−n n n, we getP(K−n n n; z) = P(K−n 0 n; z) + nzP (K0 ∞ 0; z).Figure 5. K∞ 0 ∞.142 S. BAADERContinuing at the crossings of the right vertical band of P(K−n 0 n; z), we getP(K−n n n; z) = P(K−n 0 0; z) + nzP (K−n 0 ∞; z) + nzP (K0 ∞ 0; z).Finally, we apply the skein relation at the crossings of the horizontal bands of K−n 0 0 and K−n 0 ∞to obtainP(K−n n n; z) = P(K0 0 0; z) − nzP (K∞ 0 0; z)+ nz(P (K0 0 ∞; z) − nzP (K∞ 0 ∞; z))+ nzP (K0 ∞ 0; z).We conclude with some remarks and problems about unknotting operations.Given a knot K of unknotting number two, let U(K) be the set of all knots which lie between Kand the trivial knot, that is, knots which can be transformed into K and the trivial knot by one crossingchange. According to Theorem 1.1, the set U(K) has infinite (countable) cardinality. In particular,U(K) contains knots of arbitrarily high crossing number. Recently, Uchida found that the set U(51)associated with the torus knot 51 (in Rolfsen’s notation [7]) contains knots of arbitrary high bridgenumber. In contrast, the signature and the four-dimensional genus of all knots in U(51) equal one. Itmight be interesting to study the range of possible values of numerical invariants on the set U(K).We may also ask how much information on a knot K is encoded in the corresponding set U(K).So far, it seems very hard to compute the unknotting number of knots.A complete list of unknottingnumbers exists for knots up to seven crossings only. In the last decade, there has been considerableprogress on unknotting numbers of positive knots ([3] or [6]; see also [5] for recent results onunknotting numbers of prime knots). In contrast, very little is known about unknotting numbers ofconnected sums of positive and negative knots. For example, the unknotting number of the knot31#!51 is still unknown. Here ‘#’ stands for the connected sum operation and ‘!’ denotes the mirrorimage operation.AcknowledgementThe author would like to thank Yoshiaki Uchida who has had a great influence on this work.References1. M. Hirasawa and Y. Uchida, The Gordian complex of knots, J. Knot Theory Ramifications 11(2002), 363–368.2. L. H. Kauffman, The Conway polynomial, Topology 20 (1980), 101–108.3. P. B. Kronheimer and T. S. Mrowka, Gauge theory for embedded surfaces. I., Topology 32 (1993),773–826.4. H. Murakami, Some metrics on classical knots, Math. Ann. 270 (1985), 35–45.5. P. Ozsváth and Z. Szabó, Knots with Unknotting Number One and Heegard Floer Homology,arXiv: math.GT/0401426, January 2004.6. J. Rasmussen, Khovanov Homology and the Slice Genus, arXiv: math.GT/0402131, February2004.7. D. Rolfsen, Knots and Links, Mathematics Lecture Series, No. 7, Publish or Perish, Berkeley,1976.8. H. Wendt, Die gordische Auflösung von Knoten, Math. Z. 42 (1937), 680–696.

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