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Unknot diagrams requiring a quadratic number of Reidemeister moves to untangle
, 2007
"... We present a sequence of diagrams of the unknot for which the minimum number of Reidemeister moves required to pass to the trivial diagram is quadratic with respect to the number of crossings. These bounds apply both in S 2 and in R 2. 1 ..."
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We present a sequence of diagrams of the unknot for which the minimum number of Reidemeister moves required to pass to the trivial diagram is quadratic with respect to the number of crossings. These bounds apply both in S 2 and in R 2. 1
COMPLEXITY OF RELATIONS IN THE BRAID GROUP
, 906
"... Abstract. We show that for any given n, there exists a sequence of words {ak}k≥1 in the generators σ1,..., σn−1 of the braid group Bn, representing the identity element of Bn, such that the number of braid relations of the form σiσi+1σi = σi+1σiσi+1 needed to pass from ak to the empty word is quadra ..."
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Abstract. We show that for any given n, there exists a sequence of words {ak}k≥1 in the generators σ1,..., σn−1 of the braid group Bn, representing the identity element of Bn, such that the number of braid relations of the form σiσi+1σi = σi+1σiσi+1 needed to pass from ak to the empty word is quadratic with respect to the length of ak. 1.
MINIMAL SEQUENCES OF REIDEMEISTER MOVES ON DIAGRAMS OF TORUS KNOTS
"... Abstract. Let D(p, q) be the usual knot diagram of the (p, q)torus knot; that is, D(p, q) is the closure of the pbraid (σ −1 1 σ−1 2 ···σ −1 p−1)q. As is wellknown, D(p, q) andD(q, p) represent the same knot. It is shown that D(n +1,n) can be deformed to D(n, n + 1) by a sequence of {(n − 1)n(2n ..."
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Abstract. Let D(p, q) be the usual knot diagram of the (p, q)torus knot; that is, D(p, q) is the closure of the pbraid (σ −1 1 σ−1 2 ···σ −1 p−1)q. As is wellknown, D(p, q) andD(q, p) represent the same knot. It is shown that D(n +1,n) can be deformed to D(n, n + 1) by a sequence of {(n − 1)n(2n − 1)/6} +1 Reidemeister moves, which consists of a single RI move and (n − 1)n(2n − 1)/6 RIII moves. Using cowrithe, we show that this sequence is minimal over all sequences which bring D(n +1,n)toD(n, n +1). 1.
Unknotting number and number of Reidemeister moves . . .
 TOPOLOGY AND ITS APPLICATIONS
, 2012
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COMBINATORIAL DISTANCE BETWEEN BRAID WORDS
, 906
"... Abstract. We give a simple naming argument for establishing lower bounds on the combinatorial distance between (positive) braid words. 1 It is wellknown that, for n � 3, Artin’s braid group Bn, which is the group defined by the presentation σ 1,..., σ n−1 σiσj = σjσi 〉 for i − j  � 2 σiσjσi = σj ..."
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Abstract. We give a simple naming argument for establishing lower bounds on the combinatorial distance between (positive) braid words. 1 It is wellknown that, for n � 3, Artin’s braid group Bn, which is the group defined by the presentation σ 1,..., σ n−1 σiσj = σjσi 〉 for i − j  � 2 σiσjσi = σjσiσj for i − j  = 1 has a quadratic Dehn function, i.e., there exist constants Cn, C ′ n such that, if w is an nstrand braid word of length ℓ that represents the unit braid, then the number of braid relations needed to transform w into the empty word is at most Cnℓ 2 and, on the other hand, there exists for each ℓ at least one length ℓ word w such that the minimal number of such braid relations is at least C ′ n ℓ2 —see for instance [2]. In a recent posting [3], Hass, Kalka, and Nowik developed a knot theoretical