Definitions Definitions, II X and Y are homeomorphic if there is a one to one onto K and J are equivalent if there is a sequence of knots function from X to Y with a continuous inverse. K = K0,K1,...,Kn = J where each in the sequence is A closed polygonal curve is simple if each segment an elementary deformation of the preceding knot. intersects exactly two other segments, intersecting only at A projection of a knot K is a projection of the knot onto endpoints some plane. A knot is a simple closed polygonal curve in R3 A projection is regular if no three points on the knot project to the same point in the projection and if no vertex in the A is a finite collection of pairwise disjoint knots in R3. knot projects to the same point as any other point in the If (p ,...,p ) defines a knot and no proper subset of these 1 k knot. points defines the same knot, then the elements p are { i } A knot diagram for K is a regular projection for K with over called the vertices of the knot. and under crossings indicated. The is the knot determined by three nonlinear A composite knot ()is the composition J#K points. The is a union of , all lying in one of nontrivial knots. The knots J and K are called factor plane. knots. A knot is prime if it is not composite.

Mth 333 – Spring 2013 Midterm One Review 1/6 Mth 333 – Spring 2013 Midterm One Review 2/6

Definitions, III Definitions, IV An orientation for a knot is a consist orientation of the line A in a knot or link projection is a region in the segments in the knot. projection plane surrounded by a circle such that the knot A knot is invertible if it can be deformed back onto itself so or link projection crosses the circle exactly four times.

that a given orientation is taken to the opposite orientation. A rational tangle is a tangle of the form n1 n2 ... nk A knot is amphichiral if it is equivalent to its mirror image. where the ni are integers. A link is splittable if the components of the link can be Two tangles can be added by placing them next to each deformed so that the lie on different sides of a plane in other and joining strands. three dimensional space. An algebraic tangle is a tangle formed by by the operations A strand in a knot or link diagram is a part of the diagram of addition and multiplication on rational tangles. that goes from one under crossing to another with only over The of a knot is n if crossings in between. there is a projection of the knot such that changing n A projection of a knot or link is tricolorable if each strand of crossings in the projection results in the unknot, and the projection can be colored one of three colors so that: there is no projection such that fewer changes would result At each crossing, either one or three colors appear, and in the unknot. At least two colors are used.

Mth 333 – Spring 2013 Midterm One Review 3/6 Mth 333 – Spring 2013 Midterm One Review 4/6 Definitions, V Tasks: A k move is formed by replacing two untwisted strands in a Be able to draw a composition of two given knots. diagram by two strands that wind around each other k Be able to determine the Reidemeister moves to use in times. Two knots or links are k equivalent if we can get going from one diagram to another. from one projection to the other by a finite series of k and Be able to compute . k moves. − Be able to show a diagram either is or is not tricolorable. An overpass in a knot diagram is a subarc of the knot that Be able to list the Dowker notation for a knot diagram and goes over at least one crossing but never goes under a to draw a knot diagram from Dowker notation. crossing. A maximal overpass is an overpass that could not be made any longer Be able to list the Conway notation associated with a diagram and be able to draw a diagram given Conway The bridge number of a projection is the number of maximal notation. Be able to compute the continued fraction overpasses in the projection. The bridge number of a knot associated with Conway notation. K is the least bridge number of any projection of K . Be able to compute the bridge number of a projection.

Mth 333 – Spring 2013 Midterm One Review 5/6 Mth 333 – Spring 2013 Midterm One Review 6/6