Lower bounds for the warping degree of a knot projection
aa r X i v : . [ m a t h . G T ] J a n Lower bounds for the warping degree of a knotprojection
Atsushi Ohya ∗ Ayaka Shimizu † January 25, 2021
Abstract
The warping degree of an oriented knot diagram is the minimal numberof crossings which we meet as an under-crossing first when we travel alongthe diagram from a fixed point. The warping degree of a knot projectionis the minimal value of the warping degree for all oriented alternatingdiagrams obtained from the knot projection. In this paper, we consider themaximal number of regions which share no crossings for a knot projectionwith a fixed crossing, and give lower bounds for the warping degree.
In this paper we assume that every knot diagram and knot projection has atleast one crossing. A based knot diagram is a knot diagram which is given a basepoint on the diagram avoiding crossings. We denote by D b a based diagram D with the base point b . The warping degree , d ( D b ), of an oriented based knotdiagram D b is the number of crossings such that we encounter the crossing asan under-crossing first when we travel along D with the orientation starting at b . We call such a crossing a warping crossing point of D b (see Figure 1). TheFigure 1: The oriented based knot diagram D b has warping degree one. Thecrossing p is the warping crossing point of D b . ∗ Department of Computer Science and Engineering, University of Yamanashi, 4-4-37,Takeda, Kofu-shi, Yamanashi, 400-8510, Japan. † Department of Mathematics, National Institute of Technology (KOSEN), Gunma Col-lege, 580 Toriba, Maebashi-shi, Gunma, 371-8530, Japan. Email: [email protected],[email protected] arping degree , d ( D ), of an oriented knot diagram D is the minimal value of d ( D b ) for all base points b of D ([4]). A knot diagram is said to be monotone , ordescending, if the warping degree is zero. Conversely, we can assume that thewarping degree represents a complexity of a diagram in terms of how distant aknot diagram is from a monotone diagram. Note that a monotone knot diagramis a diagram of the trivial knot. A knot diagram is said to be alternating if weencounter an over-crossing and an under-crossing alternatively when travelingthe diagram starting at any point on the diagram.Let P be an unoriented knot projection. The warping degree of P is definedto be the minimal value of the warping degree for all the oriented alternatingdiagrams obtained from P by giving the orientation and crossing information.In Figure 2, all the reduced knot projections with warping degree one and twoare shown ([6]). Further examples for warping degree three or four are listed inthe table in Section 5 in this paper. As we may see, the warping degree of a knotprojection shows somewhat complexity of a knot projection, like how “curly”a knot projection is, or how “quick” to back to a crossing when traveling theprojection.Figure 2: All the reduced knot projections of warping degree one or two. Thefirst two knot projections have warping degree one, and the others have two.The knot projections of warping degree two are determined in [6] by consider-ing all possibilities of connections of the unavoidable parts. Further explorationsin the same way for warping degree three or more would be difficult since thereare too many kinds of unavoidable parts and too many possibilities of theirconnections. In this paper, we introduce the maximal independent region num-ber , IR( P ), of a knot projection in Section 2, and show the following inequalitywhich is useful to estimate the warping degree. Theorem 1.1.
The inequality
IR( P ) ≤ d ( P ) ≤ c ( P ) − IR( P ) − holds for every reduced knot projection P , where c ( P ) denotes the crossing num-ber of P . As mentioned in Sections 2 and 3, the value of IR( P ) can be obtained withouttraveling along the knot projection, and also calculated just by solving simulta-2eous equations.Figure 3: Knot projections of warping degree three with 10, 11, 12 crossings.Since all the reduced knot projections with warping degree one and two are de-termined and we can find some knot projections with warping degree three (seeFigure 3 and Section 5), we obtain the following table about the minimal valueof the warping degree of reduced knot projections for each crossing number.Regions of a knot or link projection are independent if they share no crossings. c d min ( c ) 1 1 2 2 2 2 3 3 3 3Table 1: The crossing number c and the minimal value of warping degree d min ( c )for all reduced knot projections with c crossings.We also give the following lower bound for the warping degree which would behelpful to extend the above table. Theorem 1.2.
If all the connected link projections with n , n + 1 or n + 2 crossings have m or more independent regions, then d ( P ) ≥ m − holds for allreduced knot projections P with n or more crossings. The rest of the paper is organized as follows: In Section 2, we define the maximalindependent region number IR( P ) and prove Theorem 1.1. In Section 3, weintroduce the calculation for IR( P ) by simultaneous equations. In Section 4,we estimate IR( P ) and the warping degree d ( P ) and prove Theorem 1.2. InSection 5, we list and compare the values of IR( P ) and d ( P ). In this section, we define the maximal independent region number, and using itwe estimate the warping degree of a knot projection. Throughout this section,we assume that every knot diagram and knot projection is reduced. We havethe following lemma (cf. [6]). 3 emma 2.1.
Let D be an oriented alternating knot diagram. Let c be a crossingof D . Take a base point b just before an over-crossing of c . If D has a region R which does not incident to c , then one of the crossings on the boundary of R isa warping crossing point of D b , and one of that is a non-warping crossing pointof D b .Proof. Let e be the edge on the boundary of R such that we meet it first from b .Then e has an under-crossing and an over-crossing, that is, a warping crossingpoint and a non-warping crossing point.Figure 4: The edge e on the boundary of R which we meet first from the basepoint b has one under-crossing and one over-crossing, and they are a warpingcrossing point and non-warping crossing point of D b , respectively, regardless ofthe orientation.Similarly, we have the following. Corollary 2.2.
Let D be an oriented alternating knot diagram. Let c be acrossing of D . Take a base point b just before an over-crossing of c . If D has independent n regions which are not incident to c , then the inequality n ≤ d ( D ) ≤ c ( D ) − n − holds.Proof. By Lemma 2.1, D has at least n warping crossing points of D b . Also, D has at least n + 1 non-warping crossing points since the crossing c is a non-warping crossing point, too. Therefore we have n ≤ d ( D b ) ≤ c ( D ) − n −
1. Bythe location of the base point b , we have d ( D b ) = d ( D ) ([8]).By definition, we have the following corollary for knot projections. Corollary 2.3.
Let P be a knot projection, and c a crossing of P . If P hasindependent n regions which are not incident to c , then the inequality n ≤ d ( P ) ≤ c ( P ) − n − holds. P has 1 ≤ d ( P ) ≤ − −
1, and d ( P ) = 1. We also obtain d ( Q ) = 2 , d ( R ) = 3 from the inequalities.The strong point is that we can estimate the warping degree without travelingthe projection (see Figure 5). This would enable us to estimate the warpingdegree more combinatorically. We call the set of regions of a knot projection P which are independent and are not incident to a crossing c an independentregion set for P c . We call the crossing c a base crossing . We define the max-imal independent region number of P c , IR( P c ), to be the maximal cardinalityof an independent region set for P c . We define the maximal independent regionnumber of P , IR( P ), to be the maximal value of IR( P c ) for all base crossings c . We prove Theorem 1.1. Proof of Theorem 1.1.
It follows from Corollary 2.3. (cid:3)
In this section we explore how to find the independent region sets. A regionchoice matrix M , defined in [2], of a knot projection P of n crossings is thefollowing n × ( n + 2) matrix. (The transposition is known as an incidencematrix defined in [3].) If a crossing c i is on the boundary of a region R j , the( i, j ) component of M is 1, and otherwise 0 (see Figure 6).Figure 6: A region choice matrix M = ( m ij ), where m ij = 1 if R j is incident to c i , and otherwise m ij = 0. 5ow we find out all of the independent region sets for P c for the knotprojection P and the crossing c in Figure 6 by looking at its region choicematrix. Since the crossing c is involved with the four regions R , R , R and R , namely, the third row has 1 at the first, fourth, fifth and seventh column,we can not choose them as independent regions for P c . Hence we choose theregions from the rest regions R , R and R . Namely, we choose columns fromthe second, third and sixth so that there are no components with the value twoor more in the sum of the columns. Thus we obtain all the independent regionsets for P c , as { R } , { R } , { R } and { R , R } .More generally, we can find out all the independent region sets for a knotprojection P c for all base crossings c from the region choice matrix by solvingthe following simultaneous equations for x i ∈ { , } ( i = 1 , , . . . , x x x x x x x = b b b b b , for all b i ∈ { , } ( i = 1 , , . . . , b k = b l for some k and l ; If b i = 0 forall i , it implies that no regions are chosen. If b i = 1 for all i , it means all thecrossings are on the chosen regions, and we can not have a base crossing. As Theorem 1.1 implies, the warping degree is estimated by the maximal inde-pendent region numbers. In this section, we estimate the maximal independentregion number itself, and prove Theorem 1.2. First, we have the following:
Lemma 4.1.
The inequality
IR( P ) ≤ c ( P ) − holds for every reduced knot projection P . If it works on Z , it is known that the simultaneous equations have solutions for any b i ’sand any region choice matrix of a knot projection ([7], [3]). Besides, if x i ’s are permitted tohave the value for any integer, it is also known that the simultaneous equations have solutionsfor any region choice matrix of a knot projection even if b i ’s have the value for any integers([2]). In this case, however, the equation has no solutions for some b i ’s. In Lemma 4.2, wewill see that it definitely has solutions for some b i ’s. roof. By Theorem 1.1, we have IR( P ) ≤ c ( P ) − IR( P ) −
1, and then have2IR( P ) ≤ c ( P ) − P ). Lemma 4.2.
For any knot projection P with c ( P ) ≥ , we have IR( P ) ≥ .Proof. For the case that c ( P ) = 2, P has two independent bigons. By taking abase crossing at a crossing which belongs to one of the two bigons, we can takean independent region at the other bigon. For the case that c ( P ) ≥
3, thenthe number of regions is 3 + 2 = 5 or more by the Euler characteristic (see,for example, [2]). Take a base crossing c . Then either three or four regions areincident to c . This means there exists a region which is not incident to c . HenceIR( P ) ≥ P ), we show the following lemma for linkprojections. Lemma 4.3.
If all the connected link projections with n , n +1 or n +2 crossingshave m or more independent regions, then all the connected link projections with n or more crossings have m or more independent regions.Proof. Let P be a link projection with n + 3 crossings. If P is reducible, splice itat a reducible crossing as shown in Figure 7. Then we obtain a link projection,Figure 7: Splice P at a reducible crossing. P ′ , with n + 2 crossings. By assumption, P ′ has m independent regions. Takethe corresponding regions of P ; If the region R of P ′ created by the splice hasbeen chosen, take one of the parts R and R as a corresponding region (seeFigure 7). Thus, we obtain m independent regions of P .If P is reduced, it is shown in [1] that P has a bigon or trigon. Splice it ata bigon or a trigon as shown in Figure 8. Then P ′ and P ′′ have m independentregions. Similarly, take m regions of P properly from the corresponding regions,which are independent. 7igure 8: Splice P at a bigon or a trigon and obtain a link projection P ′ or P ′′ ,respectively. Note that P ′ has n + 2 and P ′′ has n crossings.We have the following corollary. Corollary 4.4.
If all the connected link projections with n , n + 1 or n + 2 crossings have m or more independent regions, then IR( P ) ≥ m − holds forall reduced knot projections P with n or more crossings.Proof. All reduced knot projections with n or more crossings have m indepen-dent regions by Lemma 4.3. Take a base crossing c at the boundary of one ofthe m regions. Then, the rest m − P c .We prove Theorem 1.2. Proof of Theorem 1.2.
From Corollary 4.4 we have IR( P ) ≥ m −
1, and fromTheorem 1.1 we have d ( P ) ≥ IR( P ). (cid:3) IR( P ) and d ( P ) For the knot projections P of prime alternating knots with up to nine crossingswhich are obtained from the knot diagrams in Rolfsen’s knot table ([5]), thevalues of IR( P ) and d ( P ) ([8]) are listed below. The values of IR( P ) wereobtained by the calculation using the SAT solver. Acknowledgment
The authors thank Yoshiro Yaguchi for helpful comments.
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