2018 Special courses on advanced topics in Mathematics C

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Endo Hisaaki  Kamada Seiichi 
Class Format
Media-enhanced courses
Day/Period(Room No.)
Intensive ()  
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Syllabus updated
Lecture notes updated
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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


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


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

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