Moves and transformations in knot theory

I INTRO 2
1 Introduction 3
1.1 INTRODUCTION 3
1.2 STRUCTURE OF THE THESIS 7
1.3 ACKNOWLEDGMENTS 7
2 Preliminaries 8
2.1 BASIC DEFINITIONS 8
2.2 RATIONAL TANGLES 10
2.3 GORDIAN GRAPHS 17
2.4 LOCAL MOVES 19
2.5 EXAMPLES OF LOCAL MOVES 21
2.6 BRANCHED COVERS 24
2.7 THE ALEXANDER POLYNOMIAL 27
II ENDS 30
3 Proofs of the Main Theorems 31
3.1 THE BASIC LEMMA 31
3.2 BU-ENDS OF RATIONAL MOVES 33
3.3 BU-ENDS OF C(n)-MOVES 34
3.4 BU-ENDS OF H(n)-MOVES 36
3.5 FI-ENDS AND FU-ENDS OF MOVES 36
III BIBLIOGRAPHY 38
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The present thesis refers to the classical theory of knots and is de­voted to the study of knot transformations. The knot transformation theory is a rapidly developing young line of investigation in knot the­ory that only began to take shape towards the end of the twentieth century. In a sense, the formation of this theory is still far from com­plete. However, knot transformations have often played a significant role in knot theory long before there was an understanding of their importance in their own right. This role was usually somewhat util­itarian. Transformations had its origin in connection with certain problems in a completely natural way, after which they were suc­cessfully investigated within the framework of these problems, but remained unnoticed as objects removed from the context. However, now, thanks to a new perspective, transformations are experienc­ing a “global renaissance”. In the course of it, knot transformation theory reveals its fundamental place in the perception of knot the­ory. New unusual methods and results appear, and old results are rethought in a new context, giving rise to far-reaching generaliza­tions.Our new results presented in this thesis are at the forefront of re­search on the global behavior of transformations and substantially generalize some previous results in the literature. But before we dis­cuss them in more detail, let us shed some light on the history of the knot transformation theory formation and the current state of affairs in this area. This allows the reader to delve deeper into the essence of our results. First, we need to define the notion of knot and link transformation. To clarify this notion, we first give its in­formal but natural description. A knot and link transformation is any geometric procedure that transforms a given link into some new link, possibly the same. Moreover, transformations in all interesting cases, of course, are not functions, in other words, we can obtain many (maybe infinitely many) distinct links from the same given link by choosing one or another method of applying the procedure. To make this abstract concept more tangible, we give as an example one of the earliest and most well-studied transformations, which is called the crossing change (here we give an easy-to-understand loose definition to avoid congestion, but a formal description can be found in Section 2.5). The crossing change (or X-move) can be imagined as follows. Let L be a link in S3, and let B be a 3-ball such that L П B looks like the one shown in Figure 1.1 on the left (note that the figure shows a planar projection, and for clarity we depict not only L П B, but also dB in the form of a circle). Let B* be a 3-ball, and let A be a subset of B such that A looks like the one shown in Figure 1.1 on the right. If we cut B from S3, and glue B' to S3B along the boundary so that the endpoints of A coincide with d(S3B) П L, then we obtain some new link L', which is called the result of apply­ing a crossing change to L. It is also allowed to perform the reverse operation, finding in S3 such a ball as shown in Figure 1.1 on the right and replacing it with such a ball as shown in Figure 1.1 on the left. Of course, the result depends both on the choice of B and on the choice of the embedding of L. The simplicity of the construction and its wide variability give rise to a non-trivial theory. For the first time the crossing change appears in [63] in 1898 (here the cross­ing change arises in connection with the ”beknottedness” concept and has a slightly different equivalent definition). Tait’s work an­ticipated the emergence of a knot invariant called the unknotting number. The unknotting number of a knot is the minimum number of successive X-moves needed to transform this knot into the un­knot (hereinafter we prefer to use the more modern term ”X-move” due to the connection to local moves, which are discussed below). The significance of this invariant is difficult to overestimate. A huge number of papers were subsequently devoted to the study of the un­knotting number and its relations with other objects of knot theory, for example, see [67, 8, 69, 70, 49, 1, 62]. But special attention was paid to unknotting number one knots, see [40, 36, 65, 41, 43, 54, 9]. A detailed analysis of the research made in this area is a subject for another paper, so we restrict ourselves to only one example. But on the other hand, this example is both a remarkable result in itself and fits well into the narrative. In 1985, Scharlemann proved that all un­knotting number one knots are prime, see [59] (and, see definition of a prime knot in [60]). This result can be perceived with some degree of visualization. To see this we turn to one of the basic concepts of the knot transformation theory, called the Gordian graph.
Let S be a knot transformation, and let M be a set of links. Then we denote by G(S, M) such a graph whose vertex set is in one-to-one correspondence with M, and two vertices are connected by an edge if and only if the corresponding links are obtained from each other (sic!) by a single application of S. Graph G(S, M) is called the Gordian graph for S and M . We consider this graph with a natural metric. The distance between two vertices in this metric is the number of edges in any shortest path connecting these vertices if they are in the same connected component. Further, we neglect the difference between a vertex of the Gordian graph and its corresponding link, identifying these objects and perceiving them as a single object, but nevertheless using both of these words to refer to it.
Note that it would certainly be more natural in the sense of gen­erality to assume that two vertices are connected by an edge if and only if at least one of the links can be obtained from the other by a single application of S. However, this would complicate the presen­tation too much without compensating for them by the (real) greater generality, since all transformations in our thesis are such that if one link is obtained from the other by a single application of this transformation, then the opposite is also true. Therefore, we use this symmetric definition of the Gordian graph.
Moreover, we say that two knot transformations S and Z are equiv­alent if G(S, L) coincides with G(Z, L) (we denote by L the set of all links). In other words, knot transformations are uniquely deter­mined by their Gordian graphs. Let us go back to the beginning and note that we could immediately formally define a knot trans­formation as an arbitrary graph whose vertex set is in one-to-one correspondence with some set of links. However, we are sure that such a definition out of context can give a misperception of trans­formations. This definition detracts from the importance of geomet­ric procedures, but our global goal is to study precisely "geometric” transformations. An arbitrary graph can be so complex that it is al­most impossible to reconstruct the corresponding ”geometric” trans­formation. In other words, we want to avoid shifting the focus from geometric transformations to abstract graphs. We propose to per­ceive Gordian graphs rather as an effective and convenient tool for studying transformations.....
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