Definitions Definitions II X and Y are homeomorphic if there is a one to one onto The unknotting number of a knot is n if continuous function from X to Y with a continuous inverse. there is a projection of the knot such that changing n crossings in the projection results in the unknot, and A knot is a simple closed polygonal curve in R3. there is no projection such that fewer changes would result in the 3 A link is a finite collection of pairwise disjoint knots in R . unknot. Knots K and J are equivalent if there is a sequence of knots An overpass in a knot diagram is a sub-arc of the knot that goes K = K0,K1,...,Kn = J where each knot in the sequence is an over at least one crossing but never goes under a crossing. A elementary deformation of the preceding knot. maximal overpass is an overpass that could not be made any A knot projection is regular if no three points on the knot project longer to the same point in the projection and if no vertex in the knot The bridge number of a projection is the number of maximal projects to the same point as any other point in the knot. overpasses in the projection. The bridge number of a knot K is A composite knot (connected sum)is the composition J#K of the least bridge number of any projection of K . nontrivial knots. The knots J and K are called factor knots. A knot The crossing number of a knot K , denoted c(K ), is the least is prime if it is not composite. number of crossings that occur in any projection of the knot.

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

Definitions III Definitions IV A surface in R3 is a subset of R3 in which each point has an open The genus of a sphere is 0. The genus of an n-holed torus is n. neighborhood homeomorphic to an open disc. A surface with A Seifert surface for a knot K , is an orientable surface with one 3 3 boundary in R is a subset of R obtained from a surface by boundary component that is the knot K . removing the interiors of a finite number of discs. The genus of a knot K , g(K ), is the minimum genus of any A triangulation of a surface or a surface with boundary is a Seifert surface for the knot. division of the surface into a finite number of triangular regions so A meridian curve on a torus is a curve that bounds a disc in the that if two triangular regions intersect, they intersect in a common interior of the torus, but does not bound a disc on the surface of edge or common vertex. the torus. A longitude curve on a torus is a curve that intersects a The Euler characteristic of a triangulated surface or surface with meridian exactly one. boundary S, χ(S), is the number of triangles minus the number A (p,q) torus knot is a knot on the surface of a torus that goes of edges plus the number of vertices in the triangulation. around p times in the meridian direction and q times in the 3 A surface or surface with boundary in R is orientable if it is two longitude direction. sided. It is non-orientable if it is one sided.

Mth 333 – Spring 2013 Review 3/6 Mth 333 – Spring 2013 Review 4/6 Definitions V Tasks: Rules for Bracket Polynomial: 1: = 1 For a surface or surface with boundary, be able to determine the !"# Euler characteristic, number of boundary components, and = + 1 = + 1 2 A A− A A− orientability. ! " ! " 2 ! "!2 " ! 2" 2 ! " 3: L =( A A− ) L =( 1)(A + A− ) L Be able to determine the Seifert surface associated with a knot ! "# − − a(S) ! b(#S) −2 2 S 1 ! # projection and the genus of a knot or (including the genus of Alternately,! L = ∑S A A− ( A A− )| |− ! # − − composite knots). The writhe of an oriented knot or link projection L, w(L), is the sum of the crossing numbers over all crossings. Be able to compute bracket or X polynomial of a knot or link from X(L)=( A3) w(L) L the definition and by using states. − − ! # A state of a projection L is a choice of how to split all the Be able to compute the writhe of an oriented knot or link crossings in a projection. S represents the number of loops projection. | | associated with a state. Be able to use the results we went over in class on crossing An alternating projection is reduced if there are no crossings in numbers of alternating knots and torus knots. the projection where two of the four adjacent regions can be Be able to compute the bridge number of a projection. joined in the complement of the projection.

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