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28
Towards an implementation of the BH algorithm for recognizing the unknot
 In KNOTS2000
, 2001
"... In the manuscript [2] the rst author and Michael Hirsch presented a thennew algorithm for recognizing the unknot. The rst part of the algorithm required the systematic enumeration of all discs which support a `braid foliation' and are embeddable in 3space. The boundaries of these `foliated embe ..."
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In the manuscript [2] the rst author and Michael Hirsch presented a thennew algorithm for recognizing the unknot. The rst part of the algorithm required the systematic enumeration of all discs which support a `braid foliation' and are embeddable in 3space. The boundaries of these `foliated embeddable discs' (FED's) are the collection of all closed braid representatives of the unknot, up to conjugacy, and the second part of the algorithm produces a word in the generators of the braid group which represents the boundary of the previously listed FED's. The third part tests whether a given closed braid is conjugate to the boundary of a FED on the list. In this paper we describe implementations of the rst and second parts of the algorithm. We also give some of the data which we obtained. The data suggests that FED's have unexplored and interesting structure. Open questions are interspersed throughout the manuscript. The third part of the algorithm was studied in [3] and [4], and implemented by S.J. Lee [20]. At this writing his algorithm is polynomial for n 4 and exponential for n 5. 1
Virtual Knots and Links
, 2005
"... This paper is an introduction to the subject of virtual knot theory, combined with a discussion of some specific new theorems about virtual knots. The new results are as follows: We prove, using a 3dimensional topology approach that if a connected sum of two virtual knots K1 and K2 is trivial, then ..."
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This paper is an introduction to the subject of virtual knot theory, combined with a discussion of some specific new theorems about virtual knots. The new results are as follows: We prove, using a 3dimensional topology approach that if a connected sum of two virtual knots K1 and K2 is trivial, then so are both K1 and K2. We establish an algorithm, using HakenMatveev technique, for recognizing virtual knots. This paper may be read as both an introduction and as a research paper. For more about HakenMatveev theory and its application to classical knot theory, see [Ha, Hem, Mat, HL]. 1
NP and Mathematics  a computational complexity perspective
 Proc. of the ICM 06
"... “P versus N P – a gift to mathematics from Computer Science” ..."
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“P versus N P – a gift to mathematics from Computer Science”
COMPLEXITY OF PLANAR AND SPHERICAL CURVES
, 802
"... Abstract. We show that the maximal number of singular moves required to pass between any two regularly homotopic planar or spherical curves with at most n crossings, grows quadratically with respect to n. Furthermore, this can be done with all curves along the way having at most n + 2 crossings. 1. ..."
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Abstract. We show that the maximal number of singular moves required to pass between any two regularly homotopic planar or spherical curves with at most n crossings, grows quadratically with respect to n. Furthermore, this can be done with all curves along the way having at most n + 2 crossings. 1.
NONORIENTABLE FUNDAMENTAL SURFACES IN LENS SPACES
, 809
"... Abstract. We give a concrete example of an infinite sequence of (pn, qn)lens spaces L(pn, qn) with natural triangulations T(pn, qn) with pn taterahedra such that L(pn, qn) contains a certain nonorientable closed surface which is fundamental with respect to T(pn, qn) and of minimal crosscap number ..."
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Abstract. We give a concrete example of an infinite sequence of (pn, qn)lens spaces L(pn, qn) with natural triangulations T(pn, qn) with pn taterahedra such that L(pn, qn) contains a certain nonorientable closed surface which is fundamental with respect to T(pn, qn) and of minimal crosscap number among all closed nonorientable surfaces in L(pn, qn) and has n −2 parallel sheets of normal disks of a quadrilateral type disjoint from the pair of core circles of L(pn, qn). Actually, we can set p0 = 0, q0 = 1, pk+1 = 3pk + 2qk and qk+1 = pk + qk.
PACIFIC JOURNAL OF MATHEMATICS Vol. 208, No. 2, 2003 SIMPLIFYING TRIANGULATIONS OF S 3
"... In this paper we describe a procedure to simplify any given triangulation of S 3 using Pachner moves. We obtain an explicit exponentialtype bound on the number ofPachner moves needed for this process. This leads to a new recognition algorithm for the 3sphere. 1. Introduction. It has been known for ..."
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In this paper we describe a procedure to simplify any given triangulation of S 3 using Pachner moves. We obtain an explicit exponentialtype bound on the number ofPachner moves needed for this process. This leads to a new recognition algorithm for the 3sphere. 1. Introduction. It has been known for some time that any triangulation of a closed PL nmanifold can be transformed into any other triangulation of the same manifold by a finite sequence of moves [5]. We can describe the moves as follows.
Hard Unknots and Collapsing Tangles
"... Classical knot theory is about the classification, up to isotopy, of embedded closed curves in threedimensional space. Two closed curves embedded in threedimensional space are said to be isotopic if there is a continuous family of embeddings starting with one curve and ending with the other curve. ..."
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Classical knot theory is about the classification, up to isotopy, of embedded closed curves in threedimensional space. Two closed curves embedded in threedimensional space are said to be isotopic if there is a continuous family of embeddings starting with one curve and ending with the other curve. This
THE CROSSING NUMBER OF COMPOSITE KNOTS
"... One of the most basic questions in knot theory remains unresolved: is crossing number additive under connected sum? In other words, does the equality c(K1♯K2) = c(K1) + c(K2) always hold, where c(K) denotes the crossing number of a knot K and K1♯K2 is the connected sum of two (oriented) knots K1 and ..."
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One of the most basic questions in knot theory remains unresolved: is crossing number additive under connected sum? In other words, does the equality c(K1♯K2) = c(K1) + c(K2) always hold, where c(K) denotes the crossing number of a knot K and K1♯K2 is the connected sum of two (oriented) knots K1 and K2? The inequality
SIMPLIFYING TRIANGULATIONS OF S 3
, 2000
"... Abstract. In this paper we describe a procedure to simplify any given triangulation of S 3 using Pachner moves. We obtain an explicit exponentialtype bound on the number of Pachner moves needed for this process. This leads to a new recognition algorithm for the 3sphere. 1 ..."
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Abstract. In this paper we describe a procedure to simplify any given triangulation of S 3 using Pachner moves. We obtain an explicit exponentialtype bound on the number of Pachner moves needed for this process. This leads to a new recognition algorithm for the 3sphere. 1