arXiv subjects

Stanislav Jabuka

Publications and source records attributed to Stanislav Jabuka.

At least 19 recordsLinked to original sources

arXiv2025

NP-completeness of the $\ell_1$-embedding problem for simple graphs as sphere-of-influence graphs

In graph theory an interesting question is whether for a fixed choice of $p\in [0,\infty]$, all simple graphs appear as sphere-of-influence graphs in some Euclidean space with respect to the $\ell_p$ metric. The answer is affirmative for $p=\infty$, negative for any $p\in (0,\infty)$, and unknown for $p=1$. The result of this work shows that for the case of $p=1$, this embeddability question is a (Promise) NP-Complete problem.

math.MG↗

arXiv2025

On $\ell_1$ embeddings of finite metric spaces, and sphere-of-influence graphs

We introduce the {\em pair-cut cone $PCUT_n$} of metrics on sets with $n\ge 3$ elements, that correspond to linear combinations with non-negative coefficients of the cut-metrics resulting from cuts that are pairs. Given a metric, we fully characterize membership in the pair-cut cone in terms of quantities computed from the metric directly. We also prove a new result by which a metric $d$ that satisfies a system of inequalities, lies in the (full) cut cone of metrics, making it $\ell_1$-embeddable into Euclidean space. We give applications of our results to the $\ell_1$-embeddability of simple graphs into Euclidean space as {\em sphere-of-influence graphs}. We exhibit an example of a simple graph that admits no such $\ell_1$-metric in the pair-cut cone.

math.MG↗

arXiv2025

Band-unknotting numbers and connected sums of knots

We study the band-unknotting number $u_{nb}(K)$ of a knot $K$, and how it behaves with respect to connect sums. We show that this sub-additive function is not additive under connected sums, by finding infinitely many examples of knots $K_1, K_2$ with $u_{nb}(K_1\#K_2) < u_{nb}(K_1) + u_{nb}(K_2)$. Even more surprisingly, there are infinitely many examples of knots $K_1, K_2$ such that $u_{nb}(K_1\#K_2) < u_{nb}(K_i)$, $i=1,2$. Our work is motivated by the recent analogous results for the Gordian unknotting number by Brittenham and Hermiller \cite{BrittenhamHermiller}. We also prove new lower and upper bounds on the topological and smooth non-orientable 4-genus of a knot $K$.

math.GT↗

arXiv2021

Knot Graphs and Gromov Hyperbolicity

We define a broad class of graphs that generalize the Gordian graph of knots. These knot graphs take into account unknotting operations, the concordance relation, and equivalence relations generated by knot invariants. We prove that overwhelmingly, the knot graphs are not Gromov hyperbolic, with the exception of a particular family of quotient knot graphs. We also investigate the property of homogeneity, and prove that the concordance knot graph is homogeneous. Finally, we prove that that for any $n$, there exists a knot $K$ such that the ball of radius $n$ in the Gordian graph centered at $K$ contains no connected sum of torus knots.

math.GT↗

arXiv2021

On the periodic non-orientable 4-genus of a knot

We show that the equivariant and non-equivariant non-orientable 4-genus of p-periodic knots may differ, for any choice of p>1. Similar results have previously been obtained for the smooth 4-genus and non-orientable 3-genus of a periodic knot. These stand in contrast to Edmods' acclaimed result by which the equivariant and non-equivariant Seifert genus of a periodic knot agree.

math.GT↗

arXiv2021

Periodic spanning surfaces of periodic knots

Edmonds famously proved that every periodic knot of genus g possesses an equivariant Seifert surface of genus g. We show that this is not true if one instead considers nonorientable spanning surfaces of a periodic knot. We demonstrate by example that the difference between the first Betti number of an equivariant and a nonequivariant nonorientable spanning surface of a periodic knot, can be arbitrarily large.

math.GT↗

arXiv2020

The concordance crosscap number and rational Witt span of a knot

The concordance crosscap number $γ_c(K)$ of a knot $K$ is the smallest crosscap number $γ_3(K')$ of any knot $K'$ concordant to $K$ (and with $γ_3(K')$ defined as the least first Betti number of any nonorientable surface $Σ$ embedded in $S^3$ with boundary $K'$). This invariant has been introduced and studied by Zhang using knot determinants and signatures, and has further been studied by Livingston using the Alexander polynomial. We show in this work that the rational Witt class of a knot can be used to obtain a lower bound on the concordance crosscap number, by means of a new integer-valued invariant we call the Witt span of the knot.

math.GT↗

arXiv2020

The non-orientable 4-genus for knots with 8 or 9 crossings

The non-orientable 4-genus of a knot in the 3-sphere is defined as the smallest first Betti number of any non-orientable surface smoothly and properly embedded in the 4-ball, with boundary the given knot. We compute the non-orientable 4-genus for all knots with crossing number 8 or 9. As applications we prove a conjecture of Murakami's and Yasuhara's, and give a new lower bound for the slicing number of knot.

math.GT↗

arXiv2020

Comparing nonorientable three genus and nonorientable four genus of torus knots

We compare the values of the nonorientable three genus (or, crosscap number) and the nonorientable four genus of torus knots. In particular, let T(p,q) be any torus knot with p even and q odd. The difference between these two invariants on T(p,q) is at least k/2, where p = qk + a and 0 < a < q and $k\geq 0$. Hence, the difference between the two invariants on torus knots T(p,q) grows arbitrarily large for any fixed odd q, as p ranges over values of a fixed congruence class modulo q. This contrasts with the orientable setting. Seifert proved that the orientable three genus of the torus knot T(p,q) is (p-1)(q-1)/2, and Kronheimer and Mrowka later proved that the orientable four genus of T(p,q) is also this same value.

math.GT↗

arXiv2019

On a Nonorientable Analogue of the Milnor Conjecture

The nonorientable 4-genus $γ_4(K)$ of a knot $K$ is the smallest first Betti number of any nonorientable surface properly embedded in the 4-ball, and bounding the knot $K$. We study a conjecture proposed by Batson about the value of $γ_4$ for torus knots, which can be seen as a nonorientable analogue of Milnor's Conjecture for the orientable 4-genus of torus knots. We prove the conjecture for many infinite families of torus knots, by relying on a lower bound for $γ_4$ formulated by Ozsváth, Stipsicz, and Szabó. As a side product we obtain new closed formulas for the signature of torus knots.

math.GT↗

arXiv2014

Periodic knots and Heegaard Floer correction terms

We derive new obstructions to periodicity of classical knots by employing the Heegaard Floer correction terms of the finite cyclic branched covers of the knots. Applying our results to two fold covers, we demonstrate through numerous examples that our obstructions are successful where many existing periodicity obstructions fail. A combination of previously known periodicity obstructions and the results presented here, leads to a nearly complete (with the exception of a single knot) classification of alternating, periodic, 12-crossing knots with odd prime periods. For the case of alternating knots with 13, 14 and 15 crossings, we give a complete listing of all periodic knots with odd prime periods greater than 3.

math.GT↗

arXiv2014

Heegaard Floer groups of Dehn surgeries

We use an algorithm by Ozsvath and Szabo to find closed formulae for the ranks of the hat version of the Heegaard Floer homology groups for non-zero Dehn surgeries on knots in the 3-sphere. As applications we provide new bounds on the number of distinct ranks of the Heegaard Floer groups a Dehn surgery can have. These in turn give a new lower bound on the rational Dehn surgery genus of a rational homology 3-sphere. We also provide novel obstructions for a knot to be a potential counterexample to the Cabling Conjecture.

math.GT↗

arXiv2013

Heegaard Floer genus bounds for Dehn surgeries on knots

We provide a new obstruction for a rational homology 3-sphere to arise by Dehn surgery on a given knot in the 3-sphere. The obstruction takes the form of an inequality involving the genus of the knot, the surgery coefficient, and a count of L-structures on the 3-manifold, that is spin-c structures with the simplest possible associated Heegaard Floer group. Applications include an obstruction for two framed knots to yield the same 3-manifold, an obstruction that is particularly effective when working with families of framed knots. We introduce the rational and integral Dehn surgery genera for a rational homology 3-sphere, and use our inequality to provide bounds, and in some cases exact values, for these genera. We also demonstrate that the difference between the integral and rational Dehn surgery genera can be arbitrarily large.

math.GT↗

arXiv2012

The signature of an even symmetric form with vanishing associated linking form

We prove that the signature of an even, symmetric form on a finite rank integral lattice, has signature divisible by 8, provided its associated linking form vanishes in the Witt group of linking forms. Our result generalizes the well know fact that an even, unimodular form has signature divisible by 8. We give applications to signatures of 4n-dimensional manifolds, signatures of classical knots, and provide new restrictions to solutions of certain Diophantine equations.

math.GT↗

arXiv2011

Heegaard Floer correction terms and Dedekind-Rademacher sums

We derive a closed formula for the Heegaard Floer correction terms of lens spaces in terms of the classical Dedekind sum and its generalization, the Dedekind-Rademacher sum. Our proof relies on a reciprocity formula for the correction terms established by Ozsvath and Szabo. A consequence of our result is that the Casson-Walker invariant of a lens space equals the average of its Heegaard-Floer correction terms. Additionally, we find an obstruction for the equality and equality with opposite sign, of two correction terms of the same lens space. Using this obstruction we are able to derive an optimal upper bound on the number of vanishing correction terms of lens spaces with square order second cohomology.

math.GT↗

arXiv2011

When are two Dedekind sums equal?

A natural question about Dedekind sums is to find conditions on the integers $a_1, a_2$, and $b$ such that $s(a_1,b) = s(a_2, b)$. We prove that if the former equality holds then $ b \ | \ (a_1a_2-1)(a_1-a_2)$. Surprisingly, to the best of our knowledge such statements have not appeared in the literature. A similar theorem is proved for the more general Dedekind-Rademacher sums as well, namely that for any fixed non-negative integer $n$, a positive integer modulus $b$, and two integers $a_1$ and $a_2$ that are relatively prime to $b$, the hypothesis $r_n (a_1,b)= r_n (a_2,b)$ implies that $b \ | \ (6n^2+1-a_1a_2)(a_2-a_1)$.

math.NT↗

arXiv2009

The Goeritz matrix and signature of a two bridge knot

According to a formula by Gordon and Litherland, the signature of a knot K can be computed as the signature of a Goeritz matrix of K minus a suitable correction term, read off from the diagram of K. In this article, we consider the family of two bridge knots K(p/q) and compute the signature of their Goeritz matrices in terms of the coefficients of the continued fraction expansion of p/q. In many cases we also compute the value of the correction term. We show that for every two bridge knot K(p/q), there are "even continued fraction expansions" of p/q, for which the correction term vanishes, thereby fully computing the signature of K(p/q). We provide an algorithm for finding even continued fraction expansions. This article is the result of an REU study conducted by the first author under the direction of the second.

math.GT↗

arXiv2009

The rational Witt class and the unknotting number of a knot

We use the rational Witt class of a knot in the 3-sphere as a tool for addressing questions about its unknotting number. We apply these tools to several low crossing knots (151 knots with 11 crossing and 100 knots with 12 crossings) and to the family of n-stranded pretzel knots for various values of n>2. In many cases we obtain new lower bounds and in some cases explicit values for their unknotting numbers. Our results are mainly concerned with unknotting number one but we also address, somewhat more marginally, the case of higher unknotting numbers.

math.GT↗