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More precisely, if K is a smooth knot, then for almost every unit vector v giving the direction, orthogonal projection onto the plane perpendicular to v gives a knot diagram, and we can compute the crossing number, denoted n(v). The average crossing number is then defined as the integral over the unit sphere: [1]
Knots can be described in various ways. Using different description methods, there may be more than one description of the same knot. For example, a common method of describing a knot is a planar diagram called a knot diagram, in which any knot can be drawn in many different ways.
Any such diagram (given that the vertices are labeled) uniquely determines a partial order, and any partial order has a unique transitive reduction, but there are many possible placements of elements in the plane, resulting in different Hasse diagrams for a given order that may have widely varying appearances. Knot diagram.
In the mathematical area of knot theory, a Reidemeister move is any of three local moves on a link diagram. Kurt Reidemeister () and, independently, James Waddell Alexander and Garland Baird Briggs (), demonstrated that two knot diagrams belonging to the same knot, up to planar isotopy, can be related by a sequence of the three Reidemeister moves.
In knot theory, the writhe is a property of an oriented link diagram. The writhe is the total number of positive crossings minus the total number of negative crossings. A direction is assigned to the link at a point in each component and this direction is followed all the way around each component. For each crossing one comes across while traveling in this direction, if the strand un
A reduced diagram is one in which all the isthmi are removed. Tait came up with his conjectures after his attempt to tabulate all knots in the late 19th century. As a founder of the field of knot theory, his work lacks a mathematically rigorous framework, and it is unclear whether he intended the conjectures to apply to all knots, or just to alternating knots.
Given a knot or link, draw a diagram of the link using the convention that a gap in the line denotes an undercrossing. Call an unbroken arc in this diagram a bridge if it includes at least one overcrossing. Then the bridge number of a knot can be found as the minimum number of bridges required for any diagram of the knot. [1]
In the mathematical area of knot theory, the unknotting number of a knot is the minimum number of times the knot must be passed through itself (crossing switch) to untie it. If a knot has unknotting number n {\displaystyle n} , then there exists a diagram of the knot which can be changed to unknot by switching n {\displaystyle n} crossings. [ 1 ]