A note on unavoidable sets for a spherical curve of reductivity four
aa r X i v : . [ m a t h . G T ] O c t A note on unavoidable sets for a sphericalcurve of reductivity four
Kenji Kashiwabara , Ayaka ShimizuOctober 12, 2018
Abstract
The reductivity of a spherical curve is the minimal number of alocal transformation called an inverse-half-twisted splice required toobtain a reducible spherical curve from the spherical curve. It is un-known if there exists a spherical curve whose reductivity is four. Inthis paper, an unavoidable set of configurations for a spherical curvewith reductivity four is given by focusing on 5-gons. It has also beenunknown if there exists a reduced spherical curve which has no 2-gonsand 3-gons of type A, B and C. This paper gives the answer to thequestion by constructing such a spherical curve. A spherical curve is a closed curve on S , where self-intersections, called crossings , are double points intersecting transversely. In this paper, sphericalcurves are considered up to ambient isotopy of S , and two spherical curveswhich are transformed into each other by a reflection are assumed to be thesame spherical curve. A spherical curve is trivial if it has no crossings. Aspherical curve P is reducible if one can draw a circle on S which intersects P transversely at just one crossing of P . Otherwise, it is said to be reduced .An inverse-half-twisted splice , denoted by HS − , at a crossing of a sphericalcurve P is a splice on P which yields another spherical curve (not a link1igure 1: An inverse-half-twisted splice operation at a crossing c . Brokencurves represent the outer connections.projection) as shown in Figure 1. An inverse-half-twisted splice does notpreserve an orientation of a spherical curve. Hence HS − is a different localtransformation from the splice called a “smoothing” in knot theory. [3] showsthat for every pair of two nontrivial reduced spherical curves P and P ′ , thereexists a finite sequence of HS − s and its inverses which transform P into P ′ such that a spherical curve at each step of the sequence is also reduced. Thisimplies that all nontrivial reduced spherical curves are connected by HS − sand its inverses. The reductivity of a nontrivial spherical curve P is definedto be the minimal number of inverse-half-twisted splices, HS − s, which arerequired to obtain a reducible spherical curve from P . The reductivity tellsus how reduced a spherical curve is like the connectivity in graph theory. In[6], it is shown that every nontrivial spherical curve has the reductivity fouror less. Also, in [6] and [5], it is mentioned that there are infinitely manyspherical curves with reductivity 0, 1, 2 and 3. At the moment, the followingproblem is open: Problem A. ([6]) . For any nontrivial spherical curve, is the reductivitythree or less?In other words, it is unknown if there exists a spherical curve whose reduc-tivity is four. An unavoidable set of configurations for a spherical curve in aclass is a set of configurations with the property that any spherical curve inthe class has at least one member of the set (see, for example, [2]). It is im-portant to find unavoidable sets for a spherical curve of reductivity four fromvarious viewpoints. In [6], 3-gons were classified into four types consideringouter connections as shown in Figure 2 and the unavoidable set U , shown inFigure 3, of configurations with outer connections for a spherical curve withreductivity four was given. The unavoidable set U was obtained by the Mathematics Subject Classification 2010: 57M25 U and U for a sphericalcurve of reductivity four. Broken curves represent outer connections.following two facts; the first one is that every nontrivial reduced sphericalcurve has a 2-gon or 3-gon ([1]). The second one is that if a spherical curvehas a 2-gon or a 3-gon of type A, B or C, then the reductivity is three or less([6]). The following problem was also posed in [6]: Problem B. ([6])
Is the set consisting of a 2-gon, 3-gons of type A, B andC an unavoidable set for a reduced spherical curve?If the answer to Problem B is “yes”, then the answer to Problem A is also“yes”. However, the following theorem gives the negative answer to ProblemB: 3igure 4: The 13 types of 4-gons.Figure 5: Unavoidable sets of configurations (with any outer connections) T and T for a reduced spherical curve. Theorem 1.1.
There exists a reduced spherical curve which has no 2-gonsand 3-gons of type A, B and C. (See Figures 7 and 8 in Section 2.) In [5], 4-gons were classified into 13 typesas shown in Figure 4 and the unavoidable set U , in Figure 3, for a sphericalcurve with reductivity four was given by combining 3-gons and 4-gons basedon an unavoidable set T in Figure 5 for a nontrivial reduced spherical curvewhich was obtained in [6] in the same way to the four-color-theorem. Notethat a necessary condition for a spherical curve with reductivity four wasalso given using the notion of the warping degree in [4]. In this paper, 5-gonsare classified in a systematic way which can be used for general n -gons (inSection 3) and another unavoidable set for a spherical curve with reductivityfour is given: Theorem 1.2.
The set U shown in Figure 6 is an unavoidable set for aspherical curve with reductivity four. In this section, Theorem 1.1 is shown.Proof of Theorem 1.1. The spherical curves depicted in Figure 7 are reduced,and have no 2-gons and 3-gons of type A, B and C. The point is that thereare no 2-gons, and all the 3-gons are of type D. (cid:3)
Note that the spherical curves shown in Figure 7 have the reductivity one,not four, because an inverse-half-twisted splice at a crossing at the middle4-gons with a star derives a reducible spherical curve. Another example is5igure 7: Reduced spherical curves without 2-gons, 3-gons of type A, B andC.Figure 8: Reduced spherical curve without 2-gons, 3-gons of type A, B andC.shown in Figure 8. The reductivity of the spherical curve in Figure 8 is notfour because it has a 4-gon, with a star in the figure, of type 4a; it is shownin [5] that if a spherical curve has a 4-gon of type 4a, then the reductivity isthree or less.In [6], a reduced spherical curve which has no 2-gons and 3-gons of typeA and B was given. Further spherical curves are shown in Figure 9.
In this section, the following lemma is shown:
Lemma 3.1. a, b, c, d andFigure 10: 5-gons of type 1 to 4 with relative orientations of the sides. e be the sides of a 5-gon as illustrated in Figure 11. The 5-gon of type 1 hasFigure 11: The rotation and reflections on the 5-gons. (One of the relativeorientations is shown by arrows at each type.)two types of symmetries: the (2 π/ ρ and the reflection ϕ definedby the following permutations ρ = (cid:18) a b c d eb c d e a (cid:19) , ϕ = (cid:18) a b c d ea e d c b (cid:19) . ϕ , ϕ and ϕ defined by the following permutations, respectively: ϕ = (cid:18) a b c d ed c b a e (cid:19) , ϕ = (cid:18) a b c d ec b a e d (cid:19) , ϕ = (cid:18) a b c d eb a e d c (cid:19) . Now let a 5-gon be a part of a spherical curve on S . Let a, b, c, d and e be sides of the 5-gon located as same as Figure 11. Fix the orientation of a as e to b . By reading the sides up as one passes the spherical curve, a cyclicsequence consisting of a, b, c, d and e is obtained. In particular, a sequencestarting with a is called a standard sequence . With the type of relativeorientations of the sides, a 5-gon with outer connections is represented by asequence uniquely. There are 4! = 24 standard sequences on each type, andwe remark that there are some multiplicity by symmetries as a 5-gon of aspherical curve. Type 1:
A 5-gon of type 1 has two symmetries ρ and ϕ . Two cyclicsequences which can be transformed into each other by some ρ s representthe same 5-gon with outer connections. For example, abced and aebcd rep-resent the same 5-gon because ρ ( abced ) = bcdae = aebcd . Since the ori-entation is fixed, two sequences represent the same 5-gon when they aretransformed into each other by a single ϕ and orientation reversing (de-noted by γ ). For example, abced and acbde represent the same 5-gon because γ ( ϕ ( abced )) = γ ( aedbc ) = cbdea = acbde . There are 8 equivalent classes ofstandard sequences up to some ρ s and a pair of ϕ and γ : abcde, abced = abdce = acbde = acdeb = aebcd,abdec = abecd = acdba = adbce = adebc,abedc = adcbe = adecb = aecdb = aedbc,acebd, acedb = acbed = adceb = aebdc = aecbd, adbec, aedcb. Type 2:
A 5-gon of type 2 has the reflection symmetry ϕ . Two cyclicsequences represent the same 5-gon when they are transformed into eachother by a single ϕ and orientation reversing γ . There are 16 equivalentclasses of standard sequences up to a pair of ϕ and γ : abcde, abced = aebcd, abdce = acdeb, abdec = acdbe, abecd,abedc = aecdb, acbde, acbed = aecbd, acebd, acedb = aebdc, adbce = adebc, adbec, adcbe = adecb, adceb, aedbc, aedcb .Figure 13: The 5-gons of type 2. Type 3:
Two cyclic sequences represent the same 5-gon when they aretransformed into each other by a single ϕ and orientation reversing γ . Thereare 16 equivalent classes of standard sequences up to a pair of ϕ and γ : abcde, abced, abdce = aebcd, abdec = adebc, abecd = adbce,abedc = aedbc, acbde = acdeb, acbed = acedb, acdbe, acebd, adbec,adcbe = aecdb, adceb = aecbd, adecb, aebdc, aedcb . Type 4:
Two cyclic sequences represent the same 5-gon when they aretransformed into each other by a single ϕ and orientation reversing γ . Thereare 16 equivalent classes of standard sequences up to a pair of ϕ and γ : abcde, abced = abdce, abdec = abecd, abedc, acbde = aebcd, acbed = aebdc, acdbe = adebc, acdeb, acebd, acedb = adceb,adbce, adbec, adcbe = aedbc, adecb = aecdb, aecbd, aedcb .Figure 15: The 5-gons of type 4.Thus, 5-gons are classified into the 56 types shown in Figure 21. (cid:3) In this section, Theorem 1.2 is proved.10roof of Theorem 1.2. Let P be a spherical curve with reductivity four.Since P is reduced, the set T in Figure 5 is also an unavoidable set for P .Here, P can not have the first and second configuration because they makereductivity three or less as discussed in [6] and [5]. The third one of T hasalready been discussed in Theorem 1 in [5]. Hence just the fourth one needsto be discussed here. Since the 3-gon should be of type D because 3-gonsof type A, B and C make reductivity three or less, the 5-gon should be oftype 2 or 4 with respect to the relative orientations of the sides (see Figure16). Let a, b, c, d and e be the sides of a 5-gon of type 2 and 4 as same asFigure 16: Type 2 and 4.Figures 13 and 15. When the 5-gon is of type 2, only the side e can beshared with the 3-gon. In this case, by considering the outer connections ofthe 3-gon of type D, the 5-gon should be the one whose sequence includes a , e , d with this cyclic order, which are the type of 2 abced , 2 abecd , 2 abedc ,2 acbed , 2 acebd , 2 acedb , 2 aedbc and 2 aedcb . Hence the eight configurationswith outer connections illustrated in Figure 17 are obtained. When the 5-gonis of type 4, the sides e , d and c can be shared with the 3-gon. When e isshared, the 5-gon should be the one whose sequence includes a , e , d with thiscyclic order, which are the type of 4 abced , 4 abedc , 4 acbed , 4 acebd , 4 acedb ,4 aecbd and 4 aedcb . Hence the seven configurations with outer connections inFigure 18 are obtained. When d is shared, the 5-gon should be the one whosesequence includes c , d , e with this cyclic order, which are the type of 4 abcde ,4 abdec , 4 acbde , 4 acdbe , 4 acdeb , 4 adbec , 4 adecb and 4 aecbd . Hence the eightconfigurations with outer connections in Figure 19 are obtained. When c isshared, the 5-gon should be the one whose sequence includes b , d , c with thiscyclic order, which are the type of 4 abdec , 4 abedc , 4 acbde , 4 acbed , 4 acebd ,4 adcbe , 4 adecb , 4 aecbd and 4 aedcb . Hence the nine configurations with outerconnections in Figure 20 are obtained.Hence, the set U is an unavoidable set for a spherical curve of reductivity11igure 17: The case that a 3-gon of type D and a 5-gon of type 2 share theside e .Figure 18: The case that a 3-gon of type D and a 5-gon of type 4 share theside e .four. (cid:3) A chord diagram of a spherical curve P is a preimage of P with each pair ofpoints corresponding to the same double point connected by a segment as P is assumed to be an image of an immersion of a circle to S . In Figure 22,all the 5-gons of a spherical curve on chord diagrams are listed.12igure 19: The case that a 3-gon of type D and a 5-gon of type 4 share theside d .Figure 20: The case a 3-gon of type D and a 5-gon of type 4 share the side c .ACKNOWLEDGMENTS. The authors are grateful to the members of COm-binatoric MAthematics SEMInar (COMA Semi) for helpful comments. Theyalso thank the timely help given by Yuki Miyajima in discovering reducedspherical curves without 2-gons and 3-gons of type A and B. The second au-thor was supported by Grant for Basic Science Research Projects from TheSumitomo Foundation (160154). 13 eferences [1] C. C. Adams, R. Shinjo and K. Tanaka: Complementary regions of knotand link diagrams , Ann. Comb. (2011), 549–563.[2] G. Chartrand and P. Zhang: A first course in graph theory, Dover Pub-lications, 2012.[3] N. Ito and A. Shimizu: The half-twisted splice operation on reduced knotprojections , J. Knot Theory Ramifications , 1250112 (2012) [10 pages].[4] A. Kawauchi and A. Shimizu: On the orientations of monotone knotdiagrams , J. Knot Theory Ramifications , 1750053 (2017) [15 pages].[5] Y. Onoda and A. Shimizu: The reductivity of spherical curves Part II:4-gons , to appear in Tokyo J. Math.[6] A. Shimizu:
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