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Michal Jablonowski

Publications and source records attributed to Michal Jablonowski.

At least 19 recordsLinked to original sources
arXiv2026

Squarefree Matrix Formulas for the CWR Invariant of Alternating Knots and Links

We give weighted-matrix formulas for the components of the $CWR$ invariant of oriented non-split alternating links. After recalling the known trace formulas for $CWR_{2}$ and $CWR_{3}$, we give a construction uniform in $k$: attaching an independent commuting variable to each vertex of a consolidated Tait graph and extracting the squarefree part of the resulting trace isolates simple cycles from closed walks. This yields a formula for $CWR_k$ for every $k\ge 3$, a log-determinant generating polynomial for each of the two Tait graphs, and an equivalent Moebius-inversion formula over principal submatrices. Specializing the uniform formula, we obtain explicit closed weighted formulas for $CWR_{4}$ and $CWR_{5}$. We also record a bipartiteness criterion for the vanishing of all odd components and a characteristic-polynomial formula for the unweighted specialization of the first nonvanishing odd component. The graph-theoretic constructions apply to arbitrary finite simple loopless weighted graphs; the alternating-link hypothesis enters through the invariance theorem for $CWR$.

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arXiv2026

Fundamental Quandles Do Not Determine the First Postnikov Invariant of 2-Knots

For every admissible pair of Brieskorn parameters in the Plotnick--Suciu construction, we obtain a pair of oriented $2$-knots whose knot groups are isomorphic, whose second homotopy modules are semilinearly isomorphic under a suitable group isomorphism, and whose fundamental quandles are isomorphic, while no compatible group and module isomorphisms carry one first Postnikov invariant to the other. Consequently, their exteriors are not homotopy equivalent. Thus, the knot group, the second homotopy module up to semilinear equivalence, and the fundamental quandle do not determine the homotopy type of an oriented $2$-knot exterior. Using an alternative construction from Suciu's thesis based on punctured lens spaces, the same peripheral argument yields, for every $N\geq2$, a family of $N$ oriented $2$-knots with these properties. We additionally show that the Tanaka--Taniguchi examples with isomorphic knot groups and distinct fundamental quandles have pairwise inequivalent second homotopy modules: no isomorphism between two of the knot groups makes the corresponding second homotopy modules semilinearly isomorphic.

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arXiv2026

Integer Knot Invariants: Inequalities, Computations, and Open Problems

We study inequalities between integer-valued knot invariants arising from classical knot theory, four-dimensional topology, knot homologies, and knot polynomials. We present a directed graph consisting of $47$ inequalities between $33$ knot invariants. Propagating known values through this graph for all $12965$ non-trivial prime knots up to $13$ crossings produces numerous strengthened bounds. The current \textsf{NewDB} version records, in particular, the $36$ knots whose unknotting number is pinned to the value $2$ and the $103$ knots whose doubly slice genus is pinned. We retain $10$ transitivity-irredundant candidate inequalities, establish family cases for nine of them---covering alternating, torus, fibered, homogeneous, signature-thin, quasi-alternating and positive knots (the last with equality)---and show that the remaining one is equivalent to the smooth Slice--Ribbon Conjecture. We further verify that the graph is acyclic, that no displayed arrow is a transitive consequence of the others, and that the ten candidates stay independent even when added jointly. We also show that the propagation is confluent, so that the fixed point to which all numerical statements refer is independent of the order in which the rules are applied. Finally, we report a complete structural audit of the distributed workbook against the rule set, including consistency check and a fixed-point verification of the interval propagation.

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arXiv2026

Unity of Jones polynomials in the unit circle and the plane

In this note, we study solutions of the equation $J_K(t)=1$ for the Jones polynomial of knots and links. For the family $K_n$ of double-twist knots, we show that every root of unity (except $-1$) satisfies $J_{K_n}(ζ)=1$ for some $n$. Consequently, the set of solutions to $J_{K_n}(t)=1$ arising from this family is dense in the unit circle. We further show that there exists a family of links for which the zeros of $J_L(t)-1$ are dense in the complex plane, adapting the density mechanism of Jin--Zhang--Dong--Tay for Jones polynomial zeros.

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arXiv2026

A resistance invariant of special alternating links

We introduce a Laplacian trace invariant for special, reduced, alternating diagrams of oriented knots and links and show that it is determined by two consecutive coefficients at an extreme of the Alexander polynomial. The invariant also admits an interpretation as one half of the sum of the directed resistances associated with the normalized balanced Tait-graph Laplacian. Explicit flype-related examples show that the Laplacian characteristic polynomial need not be preserved although the invariant is preserved.

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arXiv2025

On 4-dimensional 3-handle attachments

Kirby diagrams for smooth four-dimensional manifolds typically depict only the 1- and 2-handles, omitting the 3-handles. In this work, we investigate 3-handle attachments and provide tools to explicitly include them in handle diagrams. We show a set of moves involving 3-handles to extend the classical Kirby calculus. Under assumptions and the condition that the number of 3-handles equals the rank of the spherical part of the specific boundary's second homology group, we establish a homological criterion that identifies a geometric basis of disjoint embedded spheres in the boundary corresponding to 3-handle attachments, yielding a uniqueness theorem for 3-handle attachments.

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arXiv2025

On a computation of the skein tree depth of knots and links

The maximum length of the shortest path from a leaf to the root of a skein tree for knots and links gives a measure of the complexity of computing link polynomials by the skein relation (the Jones polynomial, the Alexander-Conway polynomial, and more generally HOMFLY-PT polynomial). In this paper, we prove the new upper bound on the skein tree depth of a link and give examples of links where the new bound is stronger than the known bound. We also give the new lower bound. Moreover, we derive tables of knots and links with their skein tree depth that were up to now undetermined (for some of them, we give their range of possible values).

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arXiv2025

Integer inequalities between knot invariants, skein tree depth and delta-crossing numbers

The maximum length of the shortest path from a leaf to the root of a skein tree for knots and links gives a measure of the complexity of computing link polynomials by the skein relation (the Jones polynomial, the Alexander-Conway polynomial, and more generally the HOMFLY-PT polynomial). We combine theoretical and computational results on the skein tree depth of knots and links. We prove the new upper bound on the skein tree depth of a link and give examples of links where the new bound is stronger than the known bound. We also give the new lower bound. Moreover, we derive tables of knots and links with their skein tree depth that were up to now undetermined (for some of them, we give their range of possible values). The paper surveys known (and new) inequalities between integer-valued classical knot invariants. It features a visual graph of the relations.

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arXiv2025

Biquandle cocycle condition for invariants of immersed surface-links in the four-space

We consider a biquandle-cohomological framework for invariants of oriented immersed surface-links in the four-space. After reviewing projections and Roseman moves for immersed surfaces, we prove that the move types (a, b, c, e, f, g, h) form a minimal generating set, showing in particular that the singular move (h) is independent of the embedded-case set (a, b, c, e, f, g). We extend biquandle colorings to broken surface diagrams with singular points and establish that coloring sets are in bijection for diagrams related by these moves, yielding a coloring number invariant for immersed surface-links. We introduce singular biquandle 3-cocycles: biquandle 3-cocycles satisfying an additional antisymmetry when the singular relations hold. Using such cocycles, we define a triple-point state-sum with Boltzmann weights and prove its invariance under all generating moves, including (h), thereby obtaining a state-sum invariant for immersed surface-links. The theory is illustrated on the Fenn-Rolfsen link example, where a computation yields a non-trivial integer value, demonstrating the nontriviality of the invariant in the immersed setting. These results unify and extend biquandle cocycle invariants from embedded to immersed surface-links.

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arXiv2025

On biquandle-based invariant of immersed surface-links, Yoshikawa oriented fifth move, and ribbon 2-knots

We resolve an open problem by showing that the Yoshikawa's fifth oriented move in his list cannot be reproduced by any finite sequence of the other nine moves and planar isotopies. Our proof introduces a link-type semi-invariant that remains unchanged under all moves except the fifth, highlighting its necessity in generating the full move set. Second, we extend the algebraic toolkit for immersed surface-links. After revisiting the banded-unlink description of immersed surfaces and the twelve local moves that relate their diagrams, we develop a biquandle-based coloring theory. By assigning elements of a biquandle to diagram arcs according to local rules, we obtain a counting invariant of immersed surfaces up to isotopy. Third, we show that there are infinitely many pairs of ribbon $2$-knots with isomorphic groups but different knot quandles.

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arXiv2024

CWR sequence of invariants of alternating links and its properties

We present the $CWR$ invariant, a new invariant for alternating links, which builds upon and generalizes the $WRP$ invariant. The $CWR$ invariant is an array of two-variable polynomials that provides a stronger invariant compared to the $WRP$ invariant. We compare the strength of our invariant with the classical HOMFLYPT, Kauffman $3$-variable, and Kauffman $2$-variable polynomials on specific knot examples. Additionally, we derive general recursive "skein" relations, and also specific formulas for the initial components of the $CWR$ invariant using weighted adjacency matrices of modified Tait graphs.

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arXiv2024

Rigid and shaky hard link diagrams

In this study of the Reidemeister moves within the classical knot theory, we focus on hard diagrams of knots and links, categorizing them as either rigid or shaky based on their adaptability to certain moves. We establish that every link possesses a diagram that is a rigid hard diagram and we provide an upper limit for the number of crossings in such diagrams. Furthermore, we investigate rigid hard diagrams for specific knots or links to determine their rigid hard index. In the topic of shaky hard diagrams, we demonstrate the existence of such diagrams for the unknot and unlink, regardless of the number of components, and present examples of shaky hard diagrams.

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arXiv2024

A polynomial pair invariant of alternating knots and links

We introduce an invariant of alternating knots and links (called here WRP), namely a pair of integer polynomials associated with their two checkerboard planar graphs from their minimal diagram. We prove that the invariant is well-defined and give its values obtained from calculations for some knots in the tables. This invariant is strong enough to distinguish all knots in the tables with up to 10 crossings (including their mirror images). We compare the strength of the new invariant with classical invariants, including the three-variable Kauffman bracket.

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arXiv2023

Minimal generating sets of moves for surfaces immersed in the four-space

For immersed surfaces in the four-space, we have a generating set of the Swenton--Hughes--Kim--Miller spatial moves that relate singular banded diagrams of ambient isotopic immersions of those surfaces. We also have Yoshikawa--Kamada--Kawauchi--Kim--Lee planar moves that relate marked graph diagrams of ambient isotopic immersions of those surfaces. One can ask if the former moves form a minimal set and if the latter moves form a generating set. In this paper, we derive a minimal generating set of spatial moves for diagrams of surfaces immersed in the four-space, which translates into a generating set of planar moves. We also show that the complements of two equivalent immersed surfaces can be transformed one another by a Kirby calculus not requiring the 1-1-handle or 2-1-handle slides. We also discuss the fundamental group of the immersed surface-link complement in the four-space and a quandle coloring invariant of an oriented immersed surface-link.

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arXiv2022

Upper and lower bound on delta-crossing number and tabulation of knots

We will strengthen the known upper and lower bounds on the delta-crossing number of knots in therms of the triple-crossing number. The latter bound turns out to be strong enough to obtain (unknown values of) triple-crossing numbers for a few knots. We also prove that we can always find at least one tangle from the set of four tangles, in any triple-crossing projections of any non-trivial knot or non-split link. In the last section, we enumerate and generate tables of minimal delta-diagrams for all prime knots up to the delta-crossing number equal to four. We also give a concise survey about known inequalities between integer-valued classical knot invariants.

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arXiv2020

Triple-crossing number, the genus of a knot or link and torus knots

We show that the triple-crossing number of any knot is greater or equal to twice its (canonical) genus and we show an even stronger bound in the case of links. As an application we show that this bound is strong enough to obtain the triple-crossing numbers of all torus knots, and of many more knots and their connected sums.

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arXiv2020

Triple-crossing projections, moves on knots and links and their minimal diagrams

In this paper we present a systematic method to generate prime knot and prime link minimal triple-point projections, and then classify all classical prime knots and prime links with triple-crossing number at most four. We also extend the table of known knots and links with triple-crossing number equal to five. By introducing a new type of diagrammatic move, we reduce the number of generating moves on triple-crossing diagrams, and derive a minimal generating set of moves connecting triple-crossing diagrams of the same knot.

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