### 2018　Special courses on advanced topics in Mathematics C

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Mathematics
Instructor(s)
Class Format
Lecture
Media-enhanced courses
Day/Period(Room No.)
Intensive ()
Group
-
Course number
ZUA.E333
Credits
2
2018
Offered quarter
2Q
Syllabus updated
2018/3/20
Lecture notes updated
-
Language used
Japanese
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### Course description and aims

The main subject of this course is basic concepts in knot theory and surface-knot theory, invariants using quandles, braids and 2-dimensional braids and their chart descriptions, and braid presentations of knots and surface-knots. First, we introduce basic concepts in knot theory and invariants using quandles. Generalizing these into a higher dimension, we discuss surface-knots and basic concepts including motion picture method and surface diagrams and then we introduce invariants of surface-knots using quandles. Finally we introduce braids and 2-dimensional braids, their chart descriptions and the relationship with knot and surface-knot theory.

As a knot is a curve in 3-space, a surface-knot is a surface in 4-space. Invariants and braid presentations used in knot theory are naturally generalized to surface-knots. The aims of this course is to understand this natural generalization.

### Student learning outcomes

* Be familiar with presentation of knots by diagrams and usage of invariants
* Understood the axioms of a quandle and their geometric interpretation in knot diagrams
* Be familiar with motion picture method and surface diagrams
* Understood Artin’s presentation of the braid group and chart description of 2-dimensional braids
* Understood the relationship between (classical/2-dimensional) braids and (classical/surface-) knot theory

### Keywords

knot, knot diagram, knot group, quandle, quandle coloring, cocycle invariant, surface-knot, motion picture method, surface diagram, Roseman move, trivial surface-knot, surface-knot group, normal Euler number, ch-diagram, marked graph diagram, braid group, Artin’s braid group presentation, 2-dimensional braid, chart description, monodromy, Alexander theorem, Markov theorem

### Competencies that will be developed

 ✔ Specialist skills Intercultural skills Communication skills Critical thinking skills Practical and/or problem-solving skills

### Class flow

This is a standard lecture course. There will be some assignments.

### Course schedule/Required learning

Course schedule Required learning
Class 1 The following topics will be covered in this order: * knot diagrams, connected sum, knot groups and Wirtinger presentation * quandles, quandle colorings, cocycle invariants * motion picture method, trivial surface-knots, surface-knot groups, normal Euler number * ch-diagrams, marked graph diagrams, computation of knot groups and quandle coloring numbers * surface diagrams, Roseman moves, quandle colorings, cocycle invariants * Artin’s braid group presentation, chart description for braid deformations * 2-dimensional braids and their chart descriptions * monodomy representations of 2-dimensional braids * braid presentation of surface-knots Details will be provided during each class session

### Textbook(s)

Seiichi Kamada “Kyokumen Musubime Riron” (in Japanese) Maruzen Publ., 2012.

### Reference books, course materials, etc.

Seiichi Kamada, Braid and Knot Theory in Dimension Four, American Mathematical Society, 2002.
Scott Carter, Seiichi Kamada and Masahico Saito, Surfaces in 4-Space, Springer-Verlag, 2004.
Seiichi Kamada, Surface-Knots in 4-Space, Springer Monographs in Mathematics, Springer, 2017.

### Assessment criteria and methods

Assignments (100%).

### Related courses

• MTH.B341 ： Topology

### Prerequisites (i.e., required knowledge, skills, courses, etc.)

fundamentals of topology