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A lower bound for the number of Reidemeister moves for unknotting
 J. Knot Theory Ramif
, 2006
"... I would like to thank him for his encouragement, and letting me study anything I like when I was a student. Abstract. How many Reidemeister moves do we need for unknotting a given diagram of the trivial knot? Hass and Lagarias gave an upper bound. We give an upper bound for deforming a diagram of a ..."
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I would like to thank him for his encouragement, and letting me study anything I like when I was a student. Abstract. How many Reidemeister moves do we need for unknotting a given diagram of the trivial knot? Hass and Lagarias gave an upper bound. We give an upper bound for deforming a diagram of a split link to be disconnected. On the other hand, the absolute value of the writhe gives a lower bound of the number of Reidemeister I moves for unknotting. That of a complexity of knot diagram “cowrithe” works for Reidemeister II, III moves. We give an example of an infinite sequence of diagrams Dn of the trivial knot with an O(n) number of crossings such that the author expects the number of Reidemeister moves needed for unknotting it to be O(n2). However, writhe and cowrithe do not prove this. 1. An upper bound for the number of Reidemeister moves for unlinking A Reidemeister move is a local move of a link diagram as in Figure 1. Any such move does not change the link type. As Alexander and Briggs [1] and Reidemeister [7] showed that, for any pair of diagrams D1, D2 which represent the same link type, there is a finite sequence of Reidemeister moves which deforms D1 to D2. Let D be a diagram of the trivial knot. We consider sequences of Reidemeister moves which unknot D, i.e., deform D to have no crossing. Over all such sequences, we set ur(D) to be the minimal number of the moves in a sequence. Then let ur(n) denote the maximum ur(D) over all digrams of the trivial knot with n crossings. In [3], J. Hass and J. Lagarias gave an upper bound for ur(n), showing that ur(n) ≤ 2cn, where c = 1011. (See also [2].)
The size of spanning disks for polygonal knots
, 1999
"... For each integer n ≥ 1 we construct a closed unknotted Piecewise Linear curve Kn in R 3 having less than 11n edges with the property that any Piecewise Linear triangluated disk spanning the curve contains at least 2 n−1 triangles. 1 Introduction. We show the existence of a sequence of unknotted simp ..."
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For each integer n ≥ 1 we construct a closed unknotted Piecewise Linear curve Kn in R 3 having less than 11n edges with the property that any Piecewise Linear triangluated disk spanning the curve contains at least 2 n−1 triangles. 1 Introduction. We show the existence of a sequence of unknotted simple closed curves Kn in R 3 having the following properties: • The curve Kn is a polygon with at most 11n edges. • Any Piecewise Linear (PL) embedding of a triangulated disk into R 3 with
The size of spanning disks for PL Knots.
, 1998
"... For each integer n ? 1 we construct a closed unknotted PL curve Kn in R 3 having less than 33n edges with the property that any PL triangluated disk spanning the curve contains at least 2 n triangles. 1 Introduction. We show the existence of a sequence of unknotted simple closed curves Kn in ..."
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For each integer n ? 1 we construct a closed unknotted PL curve Kn in R 3 having less than 33n edges with the property that any PL triangluated disk spanning the curve contains at least 2 n triangles. 1 Introduction. We show the existence of a sequence of unknotted simple closed curves Kn in R 3 having the following properties: ffl The curve Kn is a polygon with at most 33n edges. ffl Any PL embedding of a triangulated disk into R 3 with boundary Kn contains at least 2 n triangular faces. The existence of such disks has implications to the complexity of geometric algorithms. For example, it shows that algorithms to test knot triviality that search for embedded disks in the complement need to deal with disks containing exponentially many triangles. Thus the exponential bounds on the size of the normal disks that are analyzed in [1],[3],[4],[5], and [6] cannot be replaced with polynomial bounds. Approaches to other problems, such as the word problem for 3manifold groups,...