math.GT daily digest: 3 new submissions for 30 July 2026

New submissions

Traceless characters and instanton gradings for two-bridge and -torus knots

• Authors: Bernd J. Wuebben
• arXiv: 2607.26095
• Subjects: Geometric Topology (math.GT); Symplectic Geometry (math.SG)
• Comments: 10 pages, 1 figure

We assemble, and where possible independently verify, the representation-theoretic data underlying the pillowcase (symplectic) side of the Atiyah-Floer conjecture for knots, for two-bridge knots and -torus knots. For a two-bridge knot we give a short self-contained proof that every irreducible traceless representation is binary-dihedral; these are the dihedral characters at meridian angles , independent of , and the traceless Riley polynomial is the explicit product , monic of degree with constant term . This gives a transparent account of the Hedden-Herald-Kirk theorem that pillowcase homology equals reduced singular instanton knot homology on this family, and of why the figure-eight bubbling and bounding cochains obstructing the general conjecture are structurally inert there. For the -torus knots we compute the full traceless character variety and prove a dichotomy: exactly characters are dihedral, so for odd every irreducible traceless character is non-dihedral. Passing to the double branched cover , we self-compute the spectral-flow gradings of the generators from the Fintushel-Stern index and the equivariant -invariant, calibrated against the Poudel-Saveliev and Anvari computations; for odd the gradings split evenly between and , giving the chain complex , . The homology equals this for (differential zero) but is smaller by for , where it is nonzero: rank throughout, and has rank , not . We reproduce the first nonzero pillowcase differential, for , and identify it as the corner figure-eight bigon absent on two-bridge knots.

The instanton homology of the pretzel knots and computed bounding cochains in the pillowcase

• Authors: Bernd J. Wuebben
• arXiv: 2607.26096
• Subjects: Geometric Topology (math.GT); Symplectic Geometry (math.SG)
• Comments: 11 pages, 1 figure

We prove that the reduced singular instanton knot homology of the pretzel knots has rank for every odd : the Alexander polynomials of the family, computed in closed form by a skein recursion (Hironaka's Lehmer-like polynomials), give the lower bound , and Manion's closed-form reduced Khovanov homology gives the matching upper bound. We then turn to the pillowcase (symplectic) side of the knot Atiyah-Floer program. In the immersed-curve combinatorial model of Herald-Kirk and Smith we reconstruct the pillowcase Lagrangians of the natural tangle decomposition of the family and compute the naive Lagrangian-Floer homology of its members through . The outcome is a sharp experimental law: the naive rank differs from by exactly one differential, the difference being , vanishing for the torus member and changing direction as the determinant crosses (equivalently, as binary-dihedral traceless characters appear). Finally we compute the bounding cochains conjectured by Cazassus-Herald-Kirk-Kotelskiy to repair the deficiency: for a unique two-crossing cochain acting through an immersed quadrilateral, for a unique single crossing acting through a triangle (each cancelling a bigon and raising the rank by ), and for single-crossing cochains acting in the opposite direction (creating a differential and lowering the rank by ). To our knowledge these are the first computed nonzero bounding cochains on Conway-sum tangles, and the first anywhere acting by cancellation; they realize both directions of the conjectured correction within one family, with a rigidity asymmetry: cancellation admits a unique minimal cochain, creation many. We separate throughout what is proved unconditionally, what is computed within the model, and what remains conjectural.

A topological interpretation of numbers

• Authors: Christoforos Neofytidis
• arXiv: 2607.26926
• Subjects: Geometric Topology (math.GT); Algebraic Topology (math.AT); Group Theory (math.GR); Number Theory (math.NT)
• Comments: 14 pages; survey chapter for the book "Nisyros 2025: Essays in Geometry, History and Philosophy", ed. A. Papadopoulos and S. Yamada

We survey recent advancements on the realisation problem for mapping degree sets. In particular, we explain that any finite set containing zero is the mapping degree set between some -manifolds, for each . Our building factors for the construction of the realising manifolds will be aspherical manifolds. Thus, along the way, we review a couple of open questions related to maps of non-zero degree for aspherical manifolds.