Crossing Number Sections 4.1 and 4.2 Read Section 3.3 Read 4.1 and 4.2 (Surfaces) Definition: The crossing number of a knot K , Continue on homework due Friday May 10th denoted c(K ), is the least number of crossings Definition: A surface in R3 is a subset of R3 in that occur in any projection of the knot. which each point has an open neighborhood Definition: A projection of a knot is reduced if homeomorphic to an open disc. there are no crossings that can be eliminated by Definition: A surface with boundary in R3 is a a type I Reidemeister move. subset of R3 obtained from a surface by Theorem: A reduced alternating projection of a removing the interiors of a finite number of discs. knot has the minimum number of crossing for that knot.

Mth 333 – Spring 2013 Crossing Number 1/8 Mth 333 – Spring 2013 Surfaces 2/8

Triangulations Euler Characteristic Definition: A triangulation of a surface or a Definition: The Euler characteristic of a surface with boundary is a division of the surface triangulated surface or surface with boundary S into a finite number of triangular regions so that is the number of triangles minus the number of if two triangular regions intersect, they intersect edges plus the number of vertices in the in a common edge or common vertex. triangulation. Theorem: Every surface or surface with Theorem: Any two triangulations of a surface or boundary in R3 has a triangulation. Any two surface with boundary have the same Euler triangulations of a surface have a common characteristic. subdivision. Note: The Euler characteristic of S is denoted χ(S).

Mth 333 – Spring 2013 Surfaces 3/8 Mth 333 – Spring 2013 Surfaces 4/8 Orientability Surfaces with Boundary Definition: A surface or surface with boundary Theorem: A surface or surface with boundary in 3 in R3 is orientable if it is two sided. It is R is completely determined up to homeomorphism by non-orientable if it is one sided. Euler characteristic, 3 Theorem: Every surface without boundary in R Number of boundary curves, and is a 2-sphere or an n-holed torus. All of these Orientability. are orientable. Computations: Theorem: The Euler characteristic a sphere is Cutting a surface with boundary along an arc with endpoints in the boundary increases χ by 1. 2. The Euler characteristic of an n-holed torus is Cutting a surface along a simple closed curve does 2 2n. The Euler characteristic of a disc is 1. − not change χ. Adding a disc to a surface with boundary along a boundary curve increases χ by 1.

Mth 333 – Spring 2013 Surfaces 5/8 Mth 333 – Spring 2013 Surfaces 6/8

Other Definitions Other Definitions II Definition: Two surfaces or surfaces with Definition: A surface S in the complement of a boundary in R3 are isotopic if one can be knot K is compressible if there is a disc D in R3 K so that deformed into the other by a continuous − 3 D intersects S exactly in the boundary of D, and deformation of R . the boundary of D does not bound another disc in S. Note: surfaces can be homeomorphic without S is incompressible if it is not compressible. being isotopic. A compression of a surface S in the complement Definition: The genus of a sphere is 0. The of a knot K is the operation of cutting it open genus of an n-holed torus is n. along a compressing disc and attaching two copies of the disc to the resulting curves.

Mth 333 – Spring 2013 Surfaces 7/8 Mth 333 – Spring 2013 Surfaces 8/8