math.GT daily digest: 5 new submissions for 1 July 2026

Deeply Slice Knot Detection via Immersed Curves

• arXiv: 2606.30823
• Authors: Rob McConkey, Christopher St. Clair, Tristan Wells, Chen Zhang 2
• Subjects/tags: Geometric Topology (math.GT)
• Comments: 28 pages, 23 figures, comments welcome!

On the Kirby list, Akbulut poses the question of whether there exists a homology 3-sphere , other than , with the following property: Any knot , representing which is slice in some contractible 4-manifold which bounds, is already slice in . In this paper, we make progress on this question by producing a class of deeply slice knots. We construct these knots by first specifying a pair , where is a contractible 4-manifold with integral homology 3-sphere boundary and is slice in . Then, we show the knot is deeply slice using concordance invariants from Heegaard Floer homology. We employ immersed curve techniques to compute these invariants.

HOMFLY Polynomials of the Torus Links

• arXiv: 2606.31130
• Authors: Norihisa Takahashi 3
• Subjects/tags: Geometric Topology (math.GT)
• Comments: 30 pages, 7 figures

We derive explicit formulas for the HOMFLY polynomials of the torus links using braid groups and the skein relation. We first treat the case of and then derive a five-term linear recurrence for an auxiliary sequence associated with . By solving this recurrence using a generating function, we obtain an explicit formula for the HOMFLY polynomial of . The corresponding formula for is subsequently obtained from the mirror-image formula for the HOMFLY polynomial. As an application, we show that the HOMFLY polynomial distinguishes the links within this family and distinguishes from its mirror image for .

Exotic diffeomorphisms of reducible -manifolds with odd

• arXiv: 2606.31295
• Authors: David Baraglia 4
• Subjects/tags: Geometric Topology (math.GT); Differential Geometry (math.DG)
• Comments: 22 pages

A diffeomorphism of a -manifold is said to be exotic if it is continuously isotopic to the identity but not smoothly isotopic to the identity. Ruberman constructed the first examples of exotic diffeomorphisms on simply-connected closed -manifolds. His examples were reducible -manifolds that necessarily have even in order that they can be detected by the families Seiberg--Witten or Donaldson invariants. Later Konno and Baraglia produced exotic diffeomorphisms on irreducible -manifolds with odd . In this paper, we will construct exotic diffeomorphisms on reducible -manifolds with odd . Exoticness is detected using a families Bauer--Furuta invariant. In proving our results we need to work with families moduli spaces which are not framed and so do not give rise to framed cobordism invariants. We overcome this difficulty by considering a Bauer--Furuta type invariant valued in {\em pin-cobordism}. In addition to constructing exotic diffeomorphisms, we also find new examples of simply-connected -manifolds whose mapping class groups are not finitely generated.

Thurston norm, polytopes and splitting complexity

• arXiv: 2606.31774
• Cross-list from: math.GR
• Authors: Andrei Jaikin-Zapirain, Monika Kudlinska, Pablo Sánchez-Peralta 5
• Subjects/tags: Group Theory (math.GR); Geometric Topology (math.GT)
• Comments: 36 pages

We show that if is a finitely generated torsion-free group satisfying the Strong Atiyah Conjecture with vanishing first -Betti number, then the map that assigns to each surjective integral character the first -Betti number of the kernel extends to a seminorm on the first cohomology group of with real coefficients. We call this seminorm the Thurston norm. Moreover, we show that this norm is induced by a polytope in the first homology group with real coefficients. We also generalize this result to higher -Betti numbers of the kernels, thereby confirming a conjecture of Friedl, Lück and Tillmann.

In the case where is either a free-by-cyclic group or the fundamental group of an admissible -manifold, we show that the Thurston norm of admits a combinatorial interpretation that relates it to the splitting complexity of the character. This confirms a conjecture of Gardam and Kielak. As an application, we show that there exists an algorithm to compute the Bieri--Neumann--Strebel invariant of free-by-cyclic groups, and discuss connections to the isomorphism problem in free-by-cyclic groups.

Low-dimensional topology of deep neural networks

• arXiv: 2606.31856
• Cross-list from: cs.LG
• Authors: Junyu Ren, Lek-Heng Lim 6
• Subjects/tags: Machine Learning (cs.LG); Geometric Topology (math.GT)
• Comments: Accepted at ICML 2026

We study layered models, including feedforward networks, ResNets, and transformers, by limiting each layer to a width of , i.e., as representation space. This allows us to track how a neural network changes low-dimensional topological invariants through its layers. Just about any topological structure may be simplified or even trivialized by simply increasing dimension; e.g., any knot is equivalent to an unknot in . By restricting to , we not only isolate the effects of activation and depth from that of width, we work in a space that lends itself to easy visualization. We focus on linking number here, deferring other invariants like link groups, Milnor's -invariants, knot types, ambient cobordisms, to a sequel. We provide full proofs and empirical experiments to justify the following insights: When measured by their power to effect changes in linking numbers, the layer-skipping feature in ResNets is as powerful as the attention mechanism in transformers; both ResNets and transformers are strictly more powerful than feedforward neural networks with monotonic activations, which are in turn more powerful than invertible and flow-based models; but replacing monotonic activation with a nonmonotonic one elevates a feedforward network into the same expressivity class as ResNets and transformers. These results suggest that low-dimensional topology can be a useful tool to guide designs of AI architectures. We also generalize our results from to arbitrary .