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On Milnor moves and Alexander polynomials of knots

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Ishikawa, T., Kobayashi K. and Shibuya, T. Osaka J. Math. 40 (2003), 845–855 ON MILNOR MOVES AND ALEXANDER POLYNOMIALS OF KNOTS TSUNEO ISHIKAWA, KAZUAKI KOBAYASHI and TETSUO SHIBUYA (Received April 11, 2002) 1. Introduction Recently, several local moves of knots and links were defined and studied actively in many papers, for example [2], [5], [7], and [8]. In this paper, we define a new local move on knot diagram called a Milnor move of order or simply an -move. Namely, let be an oriented knot in an ori- ented 3-space 3 and let 3 be a 3-ball in 3 such that ∩ 3 is the tangle illus- trated in Fig. 1. The transformation from Fig. 1(a) to 1(b) is called an +-move and that from Fig. 1(b) to 1(a) is called an −-move. Furthermore an -move means either an +-move or an −-move. For two knots , ′ in 3, is said to be -equivalent to ′ or and ′ are said to be -equivalent if can be transformed into ′ by a finite sequence of -moves, [5]. In [6], Milnor introduced the Milnor link. Namely a link is called the Milnor link if is transformed into a trivial link by an 2-move. Now we generalize this move to an -move for any positive integer (≥ 2). Almost local moves known up to the present change the knot cobordism, [1]. But we will see that an -move does not change the knot cobordism for any integer (≥ 2), see Proposition. In Section 2, we study a relation between the Alexander polynomials of -equivalent knots and a property of -equivalence of knots and prove Theorems 1 and 2. A relation of Alexander polynomials of cobordant knots was known in [1]. The result we obtain in Theorem 1 is more concrete than that of [1] for cobordant knots which are -equivalent. Theorems 1 and 2 give a classification of cobordant knots by an -move. For a knot , ( ) means the Alexander polynomial of . Theorem 1. For two knots , ′ and an integer ≥ 2, if is -equivalent to ′ , then ∏ =1 {(1− ) − (− ) }{(1− ) − (− ) } ( ) 846 T. ISHIKAWA, K. KOBAYASHI AND T. SHIBUYA 11 22 (a) (b) + − ≥ 2 Fig. 1.

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