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Unknotting Number and Knot Diagram 1 Introduction

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Unknotting Number and Knot Diagram 1 Introduction REVISTA MATEMÁTICA de la Universidad Complutense de Madrid Volumen 9, número 2: 1996 http://dx.doi.org/10.5209/rev_REMA.1996.v9.n2.17585 Unknotting number and knot diagram Yasutaka NAKANISHI Abstract TImis note is a continuation of [Nl], Wbere we have discusged tbe unknotting number of knots With rspect tía knot diagrams. Wc wilI show that for every minimum-crossing knot-diagram among ah unknotting-number-one two-bridge knot there exist crossings whose exchange yields tIme trivial knot, ib tbe tbird Tait conjecture is true. 1 Introduction The unknotting rtumber uQc) is defined tobe the minimum number of ex- changes of over- and under- crossing required to convertk into the trivial knot over all knot díagrams representing 1v. The unknotting number is intuitive and attractive for knot theorists and there is no algorithm to compute it for a given knot until noW. Qur motivation is to fixid 811 ap- proach. For a gíven knot, there are finíte number of minimum crossíng knot diagrams. If tIme condition “all knot diagrams” is restricted to “al! minimum crossing knot diagrams”, then We cannot obtain tIme unknot- ting nmnbers as in [Nl] and [B}: The minirnum crossing diagram of the knot lOs as lxi Fig. 1(a) (or 514 in the Conway notation [C]) is unique up to homeomorphísm of projected 2-sphere, where excImanges of tWO crossings never yield the trivial knot, however the unknotting number of 108 is tWo. (There is another diagram of the knot lOs Where exehanges 1991 Maihematice Subject Classlfication. 57M25. Servido Publicaciones Univ. Complutense. Madrid, 1996. 360 Yasutaka Nakanishi of two erossings yield the trivial knot). The author is afraid that this sítuation is not exceptional. He has raigal a question: Let K be a mrnimum crossing knot diagram of a knot 1v. Can we conved K into K’ With u(k’) .c u(1v) by exehange of a croising? II the auswer is affirmative for alí knots, we can give an algorithm. For tIme diagram as in Fig. 1 (a), we can conved the knot lOs into 62 wíth u(62) = 1 by exchange of the croissing mariced by the asterisk. (a) (b) (c) Fig. 1 In the aboye, it is necessary that K is a minimum cróssing diagram: Fig. 1 (b) and (c) illustrate tWo diagram of the famous ConWay 11 crossing lcnot. The central diagram (b) has the mínimum number of crossings, and exchange of the crossing marked by tIme asterisk yields the trivial knot. TIme right one (c) is not a mnumum crossing diagram, and there is no crossing Whose exchange yields the trivial knot. In fact, it is seen iii [N2] that, for any unknotting-number-one knot, there exist a diagram where is no crossing whose exchange yields the trivial knot. In tImis note, We give an affirmative answer for tIme class of unknotting- number-one tWo-bridge knots. Kanenobu aud Murakami [KM] char- acterized the unlcnotting-nuxnber-one tWo-bridge knots as knots repre- sented iii tIme Conway notation by C(a, ~1•~2< , a,,, ±2<a,,~, a~, —al). Unknotting number ¡md ... 361 In section 2, ~e deform the diagram aboye into the alternating one While keeping the crossing Whose exchange yields tlie trivial lcnot. Since a reduced alternating diagram mnst have the minimum number of cross- ings, by Kauffman, Murasugi [Ml and ThistlethWaite [Th], We have the followíng. TImeorem A. For a certain minimum crossing diagram of art unknotting- number-one tino-bridge krtot, titere extit crossing tu/tose exehange yields ihe trivial knot. By the third Tait conjecture, all knot diagrams with the mínimum crossings are transformed by a finite sequence of flypes. Here, a croissing whose exchange yields the trivial knot survives after fiyping. Hence, We have the required result: Theorem B. For evertj minimum crossing diagram of any givert unknotting-number-one tino-bridge knot, there extit crossirtgs in/tíase ex- chartge yields tite trivial krtot, up tía tite third Tait conjecture (i.e. if ¡¿ti conjecture turus Lo be true). 2. Deformation and Proof Proposition 1([KMJ). Tite unknottirtg-number-one tino-bridge ¡moLa can be represented in tite Oortway notation by C(a, a 1, ~2, - ,a,, +2, By the proposition aboye, an unknotting-number-one tWo-bridge knot can be illustrated as in Fig. 2. Here, a, a2, G4,~• mean nunibers of right-half-twist and a1, a~, - - mean numbers of left-half-tWist. The Fig. 2 shows the case « = 3, aj = 2, a2 3,.... And a single exchange at each crossing marked by an asterisk yields tIme trivial lcnot. ... ••. ./ZY$42i~jj¿~jjjjj11 NI» a a1 (12 a,. 2 —a,. —a2 —(Ii Fig. 2 In this section, We deform the diagram aboye into an alternating díagram. 362 Yasu talca Nakanishi In tite first step, We deform a diagram so that every pair of integers a 1, a1+i(1 =i =rt —1) has the same signas in Fig. 3. We rotate by ir the central part of three strings from the second crossing corresponding ta ~ through the second last crossing corresponding to —a¿~’ and tIme number of crossíngs is decreased by two with preser-ving a tWo-bridge form aud the crossings mariced by the asterisks. -½~IL~ a, ~ —a~+~ —a~ Fig. 3 Since the nunher of crossings is finite, a diagram can be deformed so that every pair of integers aj, a¿+i(1 =1 n — 1) has the same sign by a finite sequence of operations in tite first step. In tlie second step, we deform a diagram hito an alternating diagram with preservíng the crossings marked by the asterisks. There are at most twa possibilities of non-alternating after the flrst step; tite pair of a and Unknotting number and ... 363 a¡, and eíther one of the pair of a,, and +2 or tIme pair of +2 aud —a,.. t~- —4 a a1 Fig. 4 Por the former case, We push out the subarc from tIme last crossing cor- responding to a through tIme flrst crossing corresponding to a1 as in Fig. 4. Por tIme latter case, we pusIm out tIme subarc from tIme last crossing corresponding to a,. through the first crossing corresponding to +2 (or tIme subarc from tIme last crossing corresponding to +2 thraugh the first 364 Yasutalca Nakanishi crossing corresponding to -a,.) as in Fig. 5. —4 a~ —2 u— 2 —a, —>1 Fig. 5 TIme diagram after the second step is an alternating diagram and it is already reduced. By the folloWing result by Kauffman, Murasugi [M] and ThistlethWaite [Th], it is a minimum crossing diagram and it has at least <me crossing Whose exchange yield the trivial knot. Proposition 2 (The second lMit conjecture [M], [Th]). Tino (con- nected and proper) alternating diagrame of art alternaLíng knot ¡¿ave tite samA number of crossings. Unknotting number and ... 365 A flype is defined to be an operation on knot diagram as lxi Fig. 6. Here We remariced that the resulta of exchanges of the crosaings mariced by the asterisks have the sanie knot type. So we can say a crossing whose exchange yields the trivial knot survives after flyping. IIn-m Fig. 6 FolloWing the third TaU conjecture [T], all reduced alternating dia- grams of the sanie bat type can be deformed into eacIm other by a finite sequence of flypes. Hence, there exist crossíngs Whose exchange yields the trivial knot on every mínimum crossing diagram of an unknotting- number-one tWo-bridge knot. Acknawledgement. The author Would like to expresa his gratítude to Professor 3. M. Montesinos for his suggestions and showing English correction. References ¡II] 8. A. Bleiler, A note mi unknotting number, Math. Proc. Cambrídge Phils. Soc. 96 (1984), 469-471. [C] 3. H. ConWay, An enumeratior¿ of knoL and lira/es, ami titeir algebraic properties, Computational problem in Abstract Algebra (Oxford 1967), Pergamon Press, 1970, Pp. 329-358. [KM] T. Kanenobu and H. Murakami, Tino-bridge knots initit unknotting number one, Proc. Amer. Math. Soc 98 (1986), 499-502. 366 Yasutaka Nakanishi [MJ K. Murasugí, iones potynomials arad clasaical conjectures ira IctioL theory, Topology 26 (1987), 187-194. [Nl] Y. Nakanishi, UrtknotLirzg numbers and krtot diagrams with tite imum crossirags, Math. Seminar Notes Kobe Univ. 11 (1983), 257- 258. 1N21 Y. Naicanishí, Union arad Langle, to appear in Proc. Amer. Math. Soc. IT] P. G. Tait, Ort knots, Scientific paper 1, Cambridge Univ. Press, London, 1989, pp. 273-347. [Th] M. B. Thistlethwaíte, A spanraing tree expansion of Lhe iones poly- rwmial, Topology 26 (1987), 297-309. Department of Mathematics Recibido: 16 de Marzo de 1995 Kobe University Nada-ku, Kobe, 657 Japan E-mail addres: nakanisiamath.s.kobe-u.ac.jp.
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