Math 389.60 – Introduction to Topology and Differential Geometry
Review for Midterm Test
The test will consist in 5 problems: 2 theoretical questions and 3 homework problems. One page of notes, front and back, is allowed, but without solutions to the homework problems.
Ch 1 Informal Topology
- Topological space, mapping, homeomorphism, embedding, isotopy.
Ch.2 Graphs
- Nodes, arcs, adjacency matrix, degree sequence of a graph, subgraphs; classes of graphs: simple graphs, complete graphs, bipartite and k-partite graphs;
- Traversability: walk, trail, circuit, paths, cycle; Eulerian graph & criterion (all nodes have even degrees), Hamiltonian graph; connected graphs and components of a graph;
- Colorings (of nodes); criterion for 2-colorable: no odd cycles.
- Planarity: definition, criterion (no K5 or K3,3), linking distance
- Graph homeomorphisms & graph morphisms (combinatorial/categorical definition), topological reduced form, reduced degree sequence, correspondence: homeomorphism <-> isomorphic (combinatorial).
Ch.3 Surfaces
- Polygonal presentations: presentations, skeleton, types of surfaces: closed/w. boundary, orientable or not, embeddable in R3 or not; examples.
- Closed surfaces: presentations and cut-and-paste equivalence, correspondence with homeomorphism classes of surfaces, invariants: Euler characteristic, orientability, classification of closed surfaces, normal presentations and correspondence with the pair of invariants.
- Operations on surfaces: Excision and gluing, connected sum (tubular joining), handle addition, properties: Sn=S0+n handles, X(S’+handle)=X(S’)+2, fattening a graph, cross-cap addition.
- Bordered surfaces: presentations, closure of a bordered surface, invariants (X,O,b) & classification, excision on bordered surfaces: separating or not.
- Riemann surfaces: presentations by a cycle of permutations acting transitively; relations with polynomial equations; the EP-characteristic formula of a RS-presentation.
Ch. 4 Graphs ands Surfaces
- Embeddings and their regions: definition, regions and their visualization, genus of a graph; all graphs are embeddable on an orientable surface;
- Polygonal embeddings: definition, EP-formula, rotation systems, embeddings and associated rotation systems (onto).
- Embedding a fixed graph: minimal embedding (it is polygonal), steps of the algorithm for determining the genus of a graph, estimating the genus of a graph, amalgamation of graphs.
- Voltage graphs and their coverings: voltage graph, cover graph, covering, isotopy group, branched cover, covering map.
Math 389.60 – Review for Final Exam
The final exam tests the second part of the course. It will consist in 5 problems: 2 theoretical questions and 3 homework problems. One page of notes, front and back, is allowed, without solutions to homework problems.
Ch. 5 Knots and Links
- Def. knot, link, 3D-isotopy, 2D-diagram, 2D-isotopy
- Rademeister moves and Theorem: 3D-isotopy 2D-isotopy + R-moves
- Def. Invariants of knots and links, complete invariants
- P-labelings of a link, the isotopy invariant: existence of a p-labeling
- Left/right crossing, linking number, it is an invariant, properties: it is an integer;
- Computing lk(K1,K2)
- Placements of graphs: def.
- A,B-connections, states, bracket polynomial
- Kauffman polynomial: specialization B=1/A, d=(A2+B2)
- Writhe: def. and properties (invariant to RM3, defining properties: 1) [O]=1, 2) [D]=A [DA]+B[DB], 3) [D O]=[D] d), computing it;
- Jones polynomial in terms of bracket polynomial; skein relation (relating L0, L+, L-);
- Alternating diagrams: def., regions, reduced alternating diagrams
- Number of crossings: def., invariant under RM3 => invariant of links for reduced alternating diagrams
- Spanning surfaces, 2-coloring
- Seifert surfaces: def., existence by construction (the idea)
Ch. 6 Differential Geometry of Surfaces
- Parameterized surfaces, tangent plane, normal: def., computing the normal for a given parameterization
- Orthogonal parameterizations
- Gaussian curvature: def., Gauss Theorem: formula in terms of E,F,G,L,M,N
- First fundamental form: def., computing it in a given parameterization
- Geodesics: def., inclination angle, equation of geodesics for orthogonal parameterizations.
Types of Problems
- Computing: the linking number of two disjoint knots, the crossing number for an alternating diagram, the bracket polynomial and the writhe, using the skein relation to find the Jones polynomial.
- Sketching the spanning surface using a 2-coloring, describing how to construct the Seifert surface in an example together with a rough sketch.
- Computing in a given parameterization: the principal normal to a surface, Gaussian curvature, first fundamental form.