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Polygonal Approximation of Unknots by Quadrisecants

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Polygonal Approximation of Unknots by Quadrisecants Gyo Taek Jin Polygonal approximation of unknots by quadrisecants Abstract: If a knot K in R3 has only finitely many quadrisecants which meet K at finitely many points, we use these points to form a polygonal approximation called the quadrisecant approximation which is denoted by Kb. The quadrisecant approxima- tion conjecture states that Kb has the knot type of K if K is nontrivial. We give examples of unknots, some polygonal and some smooth, having finitely many but at least two quadrisecants, for which the conjecture holds. Keywords: knots, quadrisecants, polygonal approximation MSC: 57M25 8.1 Introduction A knot K is the image of a locally flat embedding of S1 in R3, i.e., we do not consider wild knots. A quadrisecant of a knot K is a straight line which meets K at four distinct places. The existence of quadrisecants for nontrivial knots was first announced by Pannwitz in 1933 for polygonal knots and later by Morton and Mond for smooth knots and by Kuperberg for tame knots. Fig. 8.1: A quadrisecant Theorem 8.1.1 ([10, 9, 8]). Every nontrivial knot has a quadrisecant. It is not hard to see that every knot can be perturbed to have infinitely many quadrise- cants. On the other hand, every knot can be perturbed to have only finitely many quadrisecants which meet the knot at finitely many points. Gyo Taek Jin, Korea Advanced Institute of Science and Technology, 291 Daehak-ro Yuseong-gu Dae- jeon Korea 34141, E-mail: [email protected] https://doi.org/10.1515/9783110571493-007 Open Access. © 2018 Gyo Taek Jin, published by De Gruyter. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivs 4.0 License. 160 | Gyo Taek Jin Theorem 8.1.2 ([1, 6]). A polygonal knot satisfying the following conditions has only finitely many quadrise- cants8.1. 1. No four vertices are coplanar. 2. No three edges are parallel to a single plane. 3. No four edges lie on a single hyperboloid of one sheet. Definition 8.1.3. A polygonal knot is said to be in general position if it satisfies the three conditions mentioned in Theorem 8.1.2. 8.2 Quadrisecant approximation of knots 3 Definition 8.2.1. Let α :[a, b] ! R be a parametrization of a knot K and let a ≤ t1 < ··· < tn = t0 < b, for some n ≥ 3. Then the polygonal closed curve ′ K = α(t0)α(t1) [ α(t1)α(t2) [ ··· [ α(tn−1)α(tn) is called a polygonal approximation, or simply, an approximation of the knot K. It may happen that K′ has one or more self intersection points. We say that the polygonal approximation K′ of K is a good approximation if K′ is a knot which has the knot type of K. Definition 8.2.2. Suppose that K is a knot having only finitely many quadrisecants which intersect K in finitely many points. The polygonal approximation of K using the points on the quadrisecants is called the quadrisecant approximation of K and is denoted by Kb. Fig. 8.2: A good approximation and a bad approximation 8.1 Possibly no quadrisecants in the case of an unknot Polygonal approximation of unknots by quadrisecants | 161 Fig. 8.3: The quadrisecant approximation of a smooth trefoil knot Figure 8.3 shows that the quadrisecant approximation of a smooth trefoil is a good approximation and Figure 8.4 shows that the quadrisecant approximations of a polyg- onal trefoil and a polygonal figure eight knot are good approximations. Conjecture 8.2.3 (Quadrisecant approximation conjecture8.2). Let K be a nontrivial knot having only finitely many quadrisecants which intersect K in finitely many points. Then 1. Kb is a good approximation of K. 2. The quadrisecants of Kb are those of K. Theorem 8.2.4 ([7, 11]). The quadrisecant approximation conjecture holds for 1. hexagonal trefoil knots and 2. heptagonal figure eight knots in general position. Fig. 8.4: The quadrisecant approximations of a polygonal 31 and a polygonal 41 8.2 In the original statement of the conjecture in [6], the word ‘nontrivial’ is missing. 162 | Gyo Taek Jin 8.3 Quadrisecants of Polygonal Unknots In this part, we examined the part 1 of the quadrisecant approximation conjecture for 133 polygonal unknots of which the vertex data were supplied by Claus Ernst [4]. It is explained in Section 8.5 how we obtained quadrisecants, and in Section 8.6 how we confirmed quadrisecant approximations are good. Example 8.3.1. We examined thirty 10-gonal unknots. They have 9.1 quadrisecants in average. Except the four cases of less than 2 quadrisecants, the quadrisecant ap- proximations are unknots. The quadrisecant approximation is not defined or not em- bedded if there exist less than two quadrisecants. Figure 8.6 shows two unknots which have no quadrisecant and only one quadrisecant. In each of Figures 8.7 and 8.8, the quadrisecants are shown on the left and the quadrisecant approximation is shown on the right. 0 2 28 0 10 20 30 Fig. 8.5: Distribution of quadrisecants of thirty 10-gonal unknots Fig. 8.6: 10-gonal unknots with 0 and 1 quadrisecant Fig. 8.7: A 10-gonal unknot with 9 quadrisecants and its quadrisecant approximation Polygonal approximation of unknots by quadrisecants | 163 Fig. 8.8: A 10-gonal unknot with 28 quadrisecants and its quadrisecant approximation Example 8.3.2. We examined thirty 20-gonal unknots. They have 305.5 quadrise- cants in average. All quadrisecant approximations are unknots. Figures 8.10 and 8.11 show two of them. 83 764 0 200 400 600 800 Fig. 8.9: Distribution of quadrisecants of thirty 20-gonal unknots Fig. 8.10: A 20-gonal unknot with 83 quadrisecants and its quadrisecant approximation 164 | Gyo Taek Jin Fig. 8.11: A 20-gonal unknot with 764 quadrisecants and its quadrisecant approximation Example 8.3.3. We examined thirty 30-gonal unknots. They have 1577.4 quadrise- cants in average. All quadrisecant approximations are unknots. Figure 8.13 shows one of them; the knot, its quadrisecants, and its quadrisecant approximation with those quadrisecants incident to the new vertices. 847 2395 0 500 1000 1500 2000 2500 Fig. 8.12: Distribution of quadrisecants of thirty 30-gonal unknots Fig. 8.13: A 30-gonal unknot with 1,431 quadrisecants and its quadrisecant approximation Polygonal approximation of unknots by quadrisecants | 165 Example 8.3.4. We examined thirty 40-gonal unknots. They have 5317.1 quadrise- cants in average. All quadrisecant approximations are unknots. Figure 8.15 shows one of them. 3560 7724 0 2000 4000 6000 8000 Fig. 8.14: Distribution of quadrisecants of thirty 40-gonal unknots Fig. 8.15: A 40-gonal unknot with 6,275 quadrisecants and its quadrisecant approximation Example 8.3.5. We examined thirteen 50-gonal unknots. They have 13161.4 quadrisecants in average. All quadrisecant approximations are unknots. Figure 8.17 shows one of them. 9175 15920 0 4000 8000 12000 16000 Fig. 8.16: Distribution of quadrisecants of thirteen 50-gonal unknots 166 | Gyo Taek Jin Fig. 8.17: A 50-gonal unknot with 15,920 quadrisecants and its quadrisecant approximation 8.4 Quadrisecants of Smooth Unknots We constructed seven smooth unknots to examine the part 1 of the quadrisecant ap- proximation conjecture. It is explained in Section 8.5 how we obtained quadrisecants, and in Section 8.6 how we confirmed quadrisecant approximations are good. Example 8.4.1. Figure 8.18 shows a smooth unknot, its quadrisecants and the quadrisecant approximation. With a = 1, 000 random choices of four points on this knot, we performed Newton’s method and obtained b = 184 quadrisecants. Reduc- ing duplicates, we obtained c = 99 quadrisecants as shown in the middle. Among the a−b = 816 failures, d = 27 cases diverged and e = 789 cases converged with less than four secant points. The approximation shown on the right is a good approximation. Fig. 8.18: A smooth unknot with 99 quadrisecants and its quadrisecant approximation Polygonal approximation of unknots by quadrisecants | 167 Example 8.4.2. Figure 8.19 shows a smooth unknot. We performed Newton’s method with a = 1, 000, b = 143, c = 107, d = 28, and e = 829 as in Example 8.4.1, and obtained a good approximation. Fig. 8.19: A smooth unknot with 107 quadrisecants and its quadrisecant approximation Example 8.4.3. Figure 8.20 shows a smooth unknot. We performed Newton’s method with a = 1, 000, b = 54, c = 31, d = 19, and e = 927 as in Example 8.4.1, and obtained a good approximation. Fig. 8.20: A smooth unknot with 31 quadrisecants and its quadrisecant approximation 168 | Gyo Taek Jin Example 8.4.4. Figure 8.21 shows the smooth unknot parametrized by 8 x (︀ 1 t t)︀ t > = 2 cos 3 + 4 + cos 2 cos 6 < (︀ )︀ y = 1 cos 3 t + 4 + cos 2 t sin 6 t 0 ≤ t ≤ 2π > 2 :>z t 1 t = 4 sin 2 − 2 sin 3 We performed Newton’s method with a = 1, 000, b = 129, c = 51, d = 6, and e = 865 as in Example 8.4.1, and obtained a good approximation. Fig. 8.21: A smooth unknot with 51 quadrisecants and its quadrisecant approximation Example 8.4.5. Figure 8.22 shows the smooth unknot parametrized by 8 x (︀ 1 t t)︀ t > = 2 cos 5 + 4 + cos 3 cos 9 < (︀ )︀ y = 1 cos 5 t + 4 + cos 3 t sin 9 t 0 ≤ t ≤ 2π > 2 :>z t 1 t = 4 sin 3 − 2 sin 5 We performed Newton’s method with a = 1, 000, b = 152, c = 121, d = 44, and e = 804 as in Example 8.4.1, and obtained a good approximation.
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