Casson-Whitney unknotting, Deep slice knots and Group trisections of knotted surface type

In this thesis, we study knotted surfaces in 4-dimensional manifolds from three different but interconnected perspectives. In the first part, we introduce a measure of 'distance', or rather 'length', between regularly homotopic knotted surfaces in 4-manifolds via the minimal length of a regular homotopy. For knotted surfaces in the 4-sphere, it is natural to consider the smallest number of finger moves and Whitney moves in such a regular homotopy to an unknotted surface, which is defined to be the Casson-Whitney number. This number is closely related, but not equal, to the stabilization number of a surface. We give lower bounds on this unknotting number coming from the fundamental group of the complements and construct explicit regular homotopies for ribbon surfaces. The Casson-Whitney number of surfaces arising from rim surgery is related to the classical unknotting number of the pattern knot. We close the section by considering the Casson-Whitney number of a family of knotted surfaces, each with trefoil knot group, that was originally constructed by Suciu. In the second part, we move to the relative setting and study surfaces with knotted boundaries properly embedded in non-closed 4-manifolds. We then distinguish such knots in the boundary which have slice disks embedded in a collar (so called shallow slice knots) from knots which are slice in the 4-manifold, but the slice disks always need to go far away from the 3-manifold boundary (so called deep slice knots). In 4-dimensional 1-handlebodies, every slice knot is automatically shallow slice. On the other hand, every 4-dimensional 2-handlebody which is not the 4-ball contains deep slice knots in the boundary. In addition, we compare topological and smooth shallow and deep sliceness. We construct infinitely many non-local null-homotopic knots in the boundary of the (-1)-trace of the left-handed trefoil which are topologically shallow slice, but smoothly deep slice. For this result, topological constructions employing Casson towers and a result of Freedman are contrasted with smooth sliceness obstructions in products of a 3-manifold with an interval. We recall the immersed curve language for bordered Heegaard Floer homology, and use it to reprove special cases of the formulas for the tau-invariants for Whitehead doubles and cables. Various generalizations to links are discussed as well, in particular constructing examples of links for which every proper sublink is shallow slice, but the link as a whole is deep slice. Then we consider 4-manifolds where every knot in the boundary bounds an embedded disk in the interior. On the contrary, we show that every compact oriented topological 4-manifold V with boundary a 3-sphere contains a knot in its boundary that does not bound a null-homologous embedded disk in V, i.e., there exists a knot which is not topologically H-slice in V. In the third part, we define group trisection of knotted surface type for bridge trisected surfaces in the 4-sphere. This is a decomposition of the fundamental group of the complement of the surface into three group epimorphisms from a punctured sphere group onto three free groups, with a pairwise compatibility condition. This datum not only allows recovering the group, but also the whole bridge trisection of the knotted surface, and we give many examples together with a computer implementation

Deep and shallow slice knots in 4-manifolds

We consider slice disks for knots in the boundary of a smooth compact 4-manifold $X^{4}$. We call a knot $K \subset \partial X$ deep slice in $X$ if there is a smooth properly embedded 2-disk in $X$ with boundary $K$, but $K$ is not concordant to the unknot in a collar neighborhood $\partial X \times I$ of the boundary. We point out how this concept relates to various well-known conjectures and give some criteria for the nonexistence of such deep slice knots. Then we show, using the Wall self-intersection invariant and a result of Rohlin, that every 4-manifold consisting of just one 0- and a nonzero number of 2-handles always has a deep slice knot in the boundary. We end by considering 4-manifolds where every knot in the boundary bounds an embedded disk in the interior. A generalization of the Murasugi-Tristram inequality is used to show that there does not exist a compact, oriented 4-manifold $V$ with spherical boundary such that every knot $K \subset S^3 = \partial V$ is slice in $V$ via a null-homologous disk.Comment: 14 pages, 5 figures; v3 is the final draft which has been accepted for publication in Proceedings of the AMS, Series B; v3 includes improvements to the exposition thanks to the anonymous refere

Homotopy classification of 4-manifolds whose fundamental group is dihedral

We show that the homotopy type of a finite oriented Poincar\'{e} 4-complex is determined by its quadratic 2-type provided its fundamental group is finite and has a dihedral Sylow 2-subgroup. By combining with results of Hambleton-Kreck and Bauer, this applies in the case of smooth oriented 4-manifolds whose fundamental group is a finite subgroup of SO(3). An important class of examples are elliptic surfaces with finite fundamental group.Comment: 23 pages. Final version, to appear in Algebraic & Geometric Topolog

Unknotting via null-homologous twists and multi-twists

The untwisting number of a knot K is the minimum number of null-homologous twists required to convert K to the unknot. Such a twist can be viewed as a generalization of a crossing change, since a classical crossing change can be effected by a null-homologous twist on 2 strands. While the unknotting number gives an upper bound on the smooth 4-genus, the untwisting number gives an upper bound on the topological 4-genus. The surgery description number, which allows multiple null-homologous twists in a single twisting region to count as one operation, lies between the topological 4-genus and the untwisting number. We show that the untwisting and surgery description numbers are different for infinitely many knots, though we also find that the untwisting number is at most twice the surgery description number plus 1.Comment: 14 pages, 6 figure

ChatGPT for Zero-shot Dialogue State Tracking: A Solution or an Opportunity?

Recent research on dialogue state tracking (DST) focuses on methods that allow few- and zero-shot transfer to new domains or schemas. However, performance gains heavily depend on aggressive data augmentation and fine-tuning of ever larger language model based architectures. In contrast, general purpose language models, trained on large amounts of diverse data, hold the promise of solving any kind of task without task-specific training. We present preliminary experimental results on the ChatGPT research preview, showing that ChatGPT achieves state-of-the-art performance in zero-shot DST. Despite our findings, we argue that properties inherent to general purpose models limit their ability to replace specialized systems. We further theorize that the in-context learning capabilities of such models will likely become powerful tools to support the development of dedicated and dynamic dialogue state trackers.Comment: 13 pages, 3 figures, accepted at ACL 202

From Chatter to Matter: Addressing Critical Steps of Emotion Recognition Learning in Task-oriented Dialogue

Emotion recognition in conversations (ERC) is a crucial task for building human-like conversational agents. While substantial efforts have been devoted to ERC for chit-chat dialogues, the task-oriented counterpart is largely left unattended. Directly applying chit-chat ERC models to task-oriented dialogues (ToDs) results in suboptimal performance as these models overlook key features such as the correlation between emotions and task completion in ToDs. In this paper, we propose a framework that turns a chit-chat ERC model into a task-oriented one, addressing three critical aspects: data, features and objective. First, we devise two ways of augmenting rare emotions to improve ERC performance. Second, we use dialogue states as auxiliary features to incorporate key information from the goal of the user. Lastly, we leverage a multi-aspect emotion definition in ToDs to devise a multi-task learning objective and a novel emotion-distance weighted loss function. Our framework yields significant improvements for a range of chit-chat ERC models on EmoWOZ, a large-scale dataset for user emotion in ToDs. We further investigate the generalisability of the best resulting model to predict user satisfaction in different ToD datasets. A comparison with supervised baselines shows a strong zero-shot capability, highlighting the potential usage of our framework in wider scenarios.Comment: Accepted by SIGDIAL 202

EmoUS: Simulating User Emotions in Task-Oriented Dialogues

Existing user simulators (USs) for task-oriented dialogue systems only model user behaviour on semantic and natural language levels without considering the user persona and emotions. Optimising dialogue systems with generic user policies, which cannot model diverse user behaviour driven by different emotional states, may result in a high drop-off rate when deployed in the real world. Thus, we present EmoUS, a user simulator that learns to simulate user emotions alongside user behaviour. EmoUS generates user emotions, semantic actions, and natural language responses based on the user goal, the dialogue history, and the user persona. By analysing what kind of system behaviour elicits what kind of user emotions, we show that EmoUS can be used as a probe to evaluate a variety of dialogue systems and in particular their effect on the user's emotional state. Developing such methods is important in the age of large language model chat-bots and rising ethical concerns.Comment: accepted by SIGIR202

CAMELL: Confidence-based Acquisition Model for Efficient Self-supervised Active Learning with Label Validation

Supervised neural approaches are hindered by their dependence on large, meticulously annotated datasets, a requirement that is particularly cumbersome for sequential tasks. The quality of annotations tends to deteriorate with the transition from expert-based to crowd-sourced labelling. To address these challenges, we present \textbf{CAMELL} (Confidence-based Acquisition Model for Efficient self-supervised active Learning with Label validation), a pool-based active learning framework tailored for sequential multi-output problems. CAMELL possesses three core features: (1) it requires expert annotators to label only a fraction of a chosen sequence, (2) it facilitates self-supervision for the remainder of the sequence, and (3) it employs a label validation mechanism to prevent erroneous labels from contaminating the dataset and harming model performance. We evaluate CAMELL on sequential tasks, with a special emphasis on dialogue belief tracking, a task plagued by the constraints of limited and noisy datasets. Our experiments demonstrate that CAMELL outperforms the baselines in terms of efficiency. Furthermore, the data corrections suggested by our method contribute to an overall improvement in the quality of the resulting datasets

Unknotting numbers of 2-spheres in the 4-sphere

We compare two naturally arising notions of unknotting number for 2-spheres in the 4-sphere: namely, the minimal number of 1-handle stabilizations needed to obtain an unknotted surface, and the minimal number of Whitney moves required in a regular homotopy to the unknotted 2-sphere. We refer to these invariants as the stabilization number and the Casson-Whitney number of the sphere, respectively. Using both algebraic and geometric techniques, we show that the stabilization number is bounded above by one more than the Casson-Whitney number. We also provide explicit families of spheres for which these invariants are equal, as well as families for which they are distinct. Furthermore, we give additional bounds for both invariants, concrete examples of their non-additivity, and applications to classical unknotting number of 1-knots.Comment: 29 pages, 22 figures; v2 is the final draft which has been accepted for publication in Journal of Topology; v2 includes improvements to the exposition, the numbering of the theorems in the introduction and in some of the subsequent sections has change