A note on the concordance invariants Upsilon and phi

Dai, Hom, Stoffregen and Truong defined a family of concordance invariants [Formula: see text]. The example of a knot with zero Upsilon invariant but nonzero epsilon invariant previously given by Hom also has nonzero phi invariant. We show there are infinitely many such knots that are linearly independent in the smooth concordance group. In the opposite direction, we build infinite families of linearly independent knots with zero phi invariant but nonzero Upsilon invariant. We also give a recursive formula for the phi invariant of torus knots.

A note on the four-dimensional clasp number of knots

Among the knots that are the connected sum of two torus knots with cobordism distance 1, we characterise those that have 4-dimensional clasp number at least 2, and we show that their n-fold connected self-sum has 4-dimensional clasp number at least 2n. Our proof works in the topological category. To contrast this, we build a family of topologically slice knots for which the n-fold connected self-sum has 4-ball genus n and 4-dimensional clasp number at least 2n.

A combinatorial description of the knot concordance invariant epsilon

In this paper, we give a combinatorial description of the concordance invariant [Formula: see text] defined by Hom, prove some properties of this invariant using grid homology techniques. We compute the value of [Formula: see text] for [Formula: see text] torus knots and prove that [Formula: see text] if [Formula: see text] is a grid diagram for a positive braid. Furthermore, we show how [Formula: see text] behaves under [Formula: see text]-cabling of negative torus knots.

Colored HOMFLY-PT for hybrid weaving knot $$ {\hat{\mathrm{W}}}_3 $$(m, n)

Weaving knots W(p, n) of type (p, n) denote an infinite family of hyperbolic knots which have not been addressed by the knot theorists as yet. Unlike the well known (p, n) torus knots, we do not have a closed-form expression for HOMFLY-PT and the colored HOMFLY-PT for W(p, n). In this paper, we confine to a hybrid generalization of W(3, n) which we denote as $$ {\hat{W}}_3 $$ W ̂ 3 (m, n) and obtain closed form expression for HOMFLY-PT using the Reshitikhin and Turaev method involving $$ \mathrm{\mathcal{R}} $$ ℛ -matrices. Further, we also compute [r]-colored HOMFLY-PT for W(3, n). Surprisingly, we observe that trace of the product of two dimensional $$ \hat{\mathrm{\mathcal{R}}} $$ ℛ ̂ -matrices can be written in terms of infinite family of Laurent polynomials $$ {\mathcal{V}}_{n,t}\left[q\right] $$ V n , t q whose absolute coefficients has interesting relation to the Fibonacci numbers $$ {\mathrm{\mathcal{F}}}_n $$ ℱ n . We also computed reformulated invariants and the BPS integers in the context of topological strings. From our analysis, we propose that certain refined BPS integers for weaving knot W(3, n) can be explicitly derived from the coefficients of Chebyshev polynomials of first kind.

Four-dimensional aspects of tight contact 3-manifolds

We conjecture a four-dimensional characterization of tightness: A contact structure on a 3-manifold Y is tight if and only if a slice-Bennequin inequality holds for smoothly embedded surfaces in Y×[0,1]. An affirmative answer to our conjecture would imply an analogue of the Milnor conjecture for torus knots: If a fibered link L induces a tight contact structure on Y, then its fiber surface maximizes the Euler characteristic among all surfaces in Y×[0,1] with boundary L. We provide evidence for both conjectures by proving them for contact structures with nonvanishing Ozsváth–Szabó contact invariant.

Twisted torus knots T(mn + m + 1,mn + 1,mn + m + 2,−1) and T(n + 1,n,2n − 1,−1) are torus knots

A twisted torus knot [Formula: see text] is a torus knot [Formula: see text] with [Formula: see text] adjacent strands twisted fully [Formula: see text] times. In this paper, we determine the braid index of the knot [Formula: see text] when the parameters [Formula: see text] satisfy [Formula: see text]. If the last parameter [Formula: see text] additionally satisfies [Formula: see text], then we also determine the parameters [Formula: see text] for which [Formula: see text] is a torus knot.

Tidal surface states as fingerprints of non-Hermitian nodal knot metals

Non-Hermitian nodal knot metals (NKMs) contain intricate complex-valued energy bands which give rise to knotted exceptional loops and new topological surface states. We introduce a formalism that connects the algebraic, geometric, and topological aspects of these surface states with their parent knots. We also provide an optimized constructive ansatz for tight-binding models for non-Hermitian NKMs of arbitrary knot complexity and minimal hybridization range. Specifically, various representative non-Hermitian torus knots Hamiltonians are constructed in real-space, and their nodal topologies studied via winding numbers that avoid the explicit construction of generalized Brillouin zones. In particular, we identify the surface state boundaries as “tidal” intersections of the complex band structure in a marine landscape analogy. Beyond topological quantities based on Berry phases, we further find these tidal surface states to be intimately connected to the band vorticity and the layer structure of their dual Seifert surface, and as such provide a fingerprint for non-Hermitian NKMs.

Construction of Unknotted and Knotted Symmetric Developable Bands

We describe a method for constructing developable bands with N ≥ 3 half twists. Each band is formed by threading a flat rectangular strip through a scaffold made from identical circular cylinders and smoothly connecting its short ends. The N cylinders in a scaffold are arranged with N-fold rotational symmetry. The number of half twists in a band is equal to the number N of cylinders in its scaffold and each band inherits the symmetry of its scaffold. Each scaffold admits a family of bands of the same length but variable width up to a maximum value determined by the features of the scaffold. Apart from orientable and nonorientable unknots, our method allows for the construction of bands with the topology of torus knots. We detail the geometric properties of the construction, discuss certain fundamental restrictions that must be met to ensure constructability, and calculate the elastic bending energy of each band. The rotational symmetry underlying the construction is essential for obtaining the presented bands, as the general non-symmetric problem is even more complex and has not yet been investigated. The bands and their corresponding scaffolds can be used as structural elements in practical applications, one of which we describe and analyze. The construction serves as a basis for a general framework for building a large variety of scaffolds and the corresponding unstretchable bands. Together, these assemblies can be used in architectural, interior, and machine design. They also open new avenues for the layout of conveyor belts in factories, airports, and other settings.