# Algebraic and Geometric Topology In Dimensions Three and Four

Lead Research Organisation:
University of Leeds

Department Name: Pure Mathematics

### Abstract

Knot theory in dimension 3 concerns the classifications of knots (continuous embeddings of some finite number of copies of the unit circle into the 3-dimensional euclidean space, up to isotopy, namely connectedness via a continuous path of embeddings). This means that the properties of interest in the field of knot theory are those which are stable under continuous deformation, and also concerns the classification of all knot types. A key method in this classification is the use of knot invariants: quantities assigned to knots which remain constant as the knot is continuously deformed. The discovery of knew knot invariants is a challenging and intriguing task involving the use of tools from areas such as representation theory, category theory, differential geometry and mathematical physics.

The theory of knots in dimension 3 is well understood. Several tools have been developed for distinguishing inequivalent knots, such as the Jones and HOMFLY-PT polynomial, and the Kontsevich integral, the latter of which takes its values in an 'algebra of chord diagrams'. This is a diagrammatically defined algebra of 'chord diagrams', modulo some set of Lie theoretical relations called the 4-term relations., which ensure that the Kontsevich integral is a knot invariant.

The main subject of my thesis will be knot theory in dimension 4, investigating the properties of surfaces embedded in 4-dimensional euclidean space, considered up to isotopy. We aim to study these properties by constructing invariants of knotted surfaces, which raises deep problems in geometry, topology and representation theory.

Possible projects in this area include:

A) Investigate diagrammatic algebras as targets for an analogue of the Kontsevich integral for knotted surfaces in dimension 4. This area was explored during a summer research project which I undertook with the University of Leeds during the summer of 2018. Key references would include: Cirio & Faria Martins: Infinitesimal 2-Braidings(arXiv:1309.4070v3, 16 Mar 2015) and Moutier: A Kontsevich integral of order 1 ( arXiv:1810.05747v1 , 12 Oct 2018)

B) Investigate representations of 4-dimensional analogues of the braid group. One such analogue is the loop braid group, which can be

presented by generators and relations in a similar manner to the braid group itself. As in Celeste Damiani: A journey through loop braid groups ( arXiv:1605.02323v3, 30 Sep 2016) (the author of which is now a research fellow at the university of Leeds). An potential aim for the project could be to study and improve the representations recently found by Bullivant, Martin and Faria Martins (All of whom are now at the university of Leeds) in Representations of the Loop Braid Group and Aharonov-Bohm like effects in discrete (3+1)-dimensional higher gauge theory. ( arXiv:1807.09551v2, 19 Dec 2018)

C) Investigate invariants of knotted surfaces derived from higher gauge theory. This would have strong connections to algebraic topology.

D) Investigate the higher categorical structure formed by knotted surfaces. The category of tangles can be represented by generators and relations within the language of monoidal categories. (This is essentially due to the fact that tangles form a monoidal category). Since knotted surfaces can be composed in two different directions, the 4-dimensional analogue of tangles(2-tangles) is no longer a category but a 2-category. This 2-category can also be presented by generators and relations, which renders the theory of knotted surfaces combinatorial. A project could focus on investigating the 2-category of knotted surfaces. This would have strong connections to my current master's project on the monoidal category of tangle

The theory of knots in dimension 3 is well understood. Several tools have been developed for distinguishing inequivalent knots, such as the Jones and HOMFLY-PT polynomial, and the Kontsevich integral, the latter of which takes its values in an 'algebra of chord diagrams'. This is a diagrammatically defined algebra of 'chord diagrams', modulo some set of Lie theoretical relations called the 4-term relations., which ensure that the Kontsevich integral is a knot invariant.

The main subject of my thesis will be knot theory in dimension 4, investigating the properties of surfaces embedded in 4-dimensional euclidean space, considered up to isotopy. We aim to study these properties by constructing invariants of knotted surfaces, which raises deep problems in geometry, topology and representation theory.

Possible projects in this area include:

A) Investigate diagrammatic algebras as targets for an analogue of the Kontsevich integral for knotted surfaces in dimension 4. This area was explored during a summer research project which I undertook with the University of Leeds during the summer of 2018. Key references would include: Cirio & Faria Martins: Infinitesimal 2-Braidings(arXiv:1309.4070v3, 16 Mar 2015) and Moutier: A Kontsevich integral of order 1 ( arXiv:1810.05747v1 , 12 Oct 2018)

B) Investigate representations of 4-dimensional analogues of the braid group. One such analogue is the loop braid group, which can be

presented by generators and relations in a similar manner to the braid group itself. As in Celeste Damiani: A journey through loop braid groups ( arXiv:1605.02323v3, 30 Sep 2016) (the author of which is now a research fellow at the university of Leeds). An potential aim for the project could be to study and improve the representations recently found by Bullivant, Martin and Faria Martins (All of whom are now at the university of Leeds) in Representations of the Loop Braid Group and Aharonov-Bohm like effects in discrete (3+1)-dimensional higher gauge theory. ( arXiv:1807.09551v2, 19 Dec 2018)

C) Investigate invariants of knotted surfaces derived from higher gauge theory. This would have strong connections to algebraic topology.

D) Investigate the higher categorical structure formed by knotted surfaces. The category of tangles can be represented by generators and relations within the language of monoidal categories. (This is essentially due to the fact that tangles form a monoidal category). Since knotted surfaces can be composed in two different directions, the 4-dimensional analogue of tangles(2-tangles) is no longer a category but a 2-category. This 2-category can also be presented by generators and relations, which renders the theory of knotted surfaces combinatorial. A project could focus on investigating the 2-category of knotted surfaces. This would have strong connections to my current master's project on the monoidal category of tangle

### Organisations

### Studentship Projects

Project Reference | Relationship | Related To | Start | End | Student Name |
---|---|---|---|---|---|

EP/R513258/1 | 30/09/2018 | 29/09/2023 | |||

2267124 | Studentship | EP/R513258/1 | 30/09/2019 | 30/03/2023 | Benjamin Horton |