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Min Hoon Kim

Publications and source records attributed to Min Hoon Kim.

At least 19 recordsLinked to original sources
arXiv2026

Soohak: A Mathematician-Curated Benchmark for Evaluating Research-level Math Capabilities of LLMs

Following the recent achievement of gold-medal performance on the IMO by frontier LLMs, the community is searching for the next meaningful and challenging target for measuring LLM reasoning. Whereas olympiad-style problems measure step-by-step reasoning alone, research-level problems use such reasoning to advance the frontier of mathematical knowledge itself, emerging as a compelling alternative. Yet research-level math benchmarks remain scarce because such problems are difficult to source (e.g., Riemann Bench and FrontierMath-Tier 4 contain 25 and 50 problems, respectively). To support reliable evaluation of next-generation frontier models, we introduce Soohak, a 439-problem benchmark newly authored from scratch by 64 mathematicians. Soohak comprises two subsets. On the Challenge subset, frontier models including Gemini-3-Pro, GPT-5, and Claude-Opus-4.5 reach 30.4%, 26.4%, and 10.4% respectively, leaving substantial headroom, while leading open-weight models such as Qwen3-235B, GPT-OSS-120B, and Kimi-2.5 remain below 15%. Notably, beyond standard problem solving, Soohak introduces a refusal subset that probes a capability intrinsic to research mathematics: recognizing ill-posed problems and pausing rather than producing confident but unjustified answers. On this subset, no model exceeds 50%, identifying refusal as a new optimization target that current models do not directly address. To prevent contamination, the dataset will be publicly released in late 2026, with model evaluations available upon request in the interim.

cs.CL↗
arXiv2024

Calegari's homotopy 4-spheres from fibered knots are standard

In 2009, Calegari constructed smooth homotopy 4-spheres from monodromies of fibered knots. We prove that all these are diffeomorphic to the standard 4-sphere. Our method uses 5-dimensional handlebody techniques and results on mapping class groups of 3-dimensional handlebodies. As an application, we present potential counterexamples to the smooth 4-dimensional Schoenflies conjecture which are related to the work of Casson and Gordon on fibered ribbon knots.

math.GT↗
arXiv2023

A note on rational band moves

We introduce an oriented rational band move, a generalization of an ordinary oriented band move, and show that if a knot $K$ in the three-sphere can be made into the $(n+1)$-component unlink by $n$ oriented rational band moves, then $K$ is rationally slice.

math.GT↗
arXiv2021

The bipolar filtration of topologically slice knots

The bipolar filtration of Cochran, Harvey and Horn presents a framework of the study of deeper structures in the smooth concordance group of topologically slice knots. We show that the graded quotient of the bipolar filtration of topologically slice knots has infinite rank at each stage greater than one. To detect nontrivial elements in the quotient, the proof simultaneously uses higher order amenable Cheeger-Gromov $L^2$ $ρ$-invariants and infinitely many Heegaard Floer correction term $d$-invariants.

math.GT↗
arXiv2021

Non-slice 3-stranded pretzel knots

Greene-Jabuka and Lecuona confirmed the slice-ribbon conjecture for 3-stranded pretzel knots except for an infinite family $P(a,-a-2,-\frac{(a+1)^2}{2})$ where $a$ is an odd integer greater than $1$. Lecuona and Miller showed that $P(a,-a-2,-\frac{(a+1)^2}{2})$ are not slice unless $a\equiv 1, 11, 37, 47, 59 \pmod{60}$. In this note, we show that four-fifths of the remaining knots in the family are not slice.

math.GT↗
arXiv2020

Non-slice linear combinations of iterated torus knots

In 1976, Rudolph asked whether algebraic knots are linearly independent in the knot concordance group. This paper uses twisted Blanchfield pairings to answer this question in the affirmative for new large families of algebraic knots.

math.GT↗
arXiv2020

Pretzel links, mutation, and the slice-ribbon conjecture

Let p and q be distinct integers greater than one. We show that the 2-component pretzel link P(p,q,-p,-q) is not slice, even though it has a ribbon mutant, by using 3-fold branched covers and an obstruction based on Donaldson's diagonalization theorem. As a consequence, we prove the slice-ribbon conjecture for 4-stranded 2-component pretzel links.

math.GT↗
arXiv2020

The round handle problem

We present the Round Handle Problem, proposed by Freedman and Krushkal. It asks whether a collection of links, which contains the Generalised Borromean Rings, are slice in a 4-manifold R constructed from adding round handles to the four ball. A negative answer would contradict the union of the surgery conjecture and the s-cobordism conjecture for 4-manifolds with free fundamental group.

math.GT↗
arXiv2019

One-bipolar topologically slice knots and primary decomposition

Let {T_n} be the bipolar filtration of the smooth concordance group of topologically slice knots, which was introduced by Cochran, Harvey, and Horn. It is known that for each n not equal to 1 the quotient group T_n/T_{n+1} has infinite rank and T_1/T_2 has positive rank. In this paper, we show that T_1/T_2 also has infinite rank. Moreover, we prove that there exist infinitely many Alexander polynomials p(t) such that there exist infinitely many knots in T_1 with Alexander polynomial p(t) whose nontrivial linear combinations are not concordant to any knot with Alexander polynomial coprime to p(t), even modulo T_2. This extends the recent result of Cha on the primary decomposition of T_n/T_{n+1} for all n greater than 1 to the case n=1. To prove our theorem, we show that the surgery manifolds of satellite links of $ν^+$-equivalent knots with the same pattern link have the same Ozsváth-Szabó $d$-invariants, which is of independent interest. As another application, for each g greater than 0, we give a topologically slice knot of concordance genus g that is $ν^+$-equivalent to the unknot.

math.GT↗
arXiv2019

Primary decomposition of knot concordance and von Neumann rho-invariants

We address the primary decomposition of the knot concordance group in terms of the solvable filtration and higher-order von Neumann $ρ$-invariants by Cochran, Orr, and Teichner. We show that for a nonnegative integer n, if the connected sum of two n-solvable knots with coprime Alexander polynomials is slice, then each of the knots has vanishing von Neumann $ρ$-invariants of order n. This gives positive evidence for the conjecture that nonslice knots with coprime Alexander polynomials are not concordant. As an application, we show that if K is one of Cochran-Orr-Teichner's knots which are the first examples of nonslice knots with vanishing Casson-Gordon invariants, then K is not concordant to any knot with Alexander polynomial coprime to that of K.

math.GT↗
arXiv2019

Homology spheres and property R

We present infinitely many homology spheres which contain two distinct knots whose 0-surgeries are $S^1 \times S^2$. This resolves a question posed by Kirby and Melvin in 1978.

math.GT↗
arXiv2019

Irreducible 3-manifolds that cannot be obtained by 0-surgery on a knot

We give two infinite families of examples of closed, orientable, irreducible 3-manifolds $M$ such that $b_1(M)=1$ and $π_1(M)$ has weight 1, but $M$ is not the result of Dehn surgery along a knot in the 3-sphere. This answers a question of Aschenbrenner, Friedl and Wilton, and provides the first examples of irreducible manifolds with $b_1=1$ that are known not to be surgery on a knot in the 3-sphere. One family consists of Seifert fibered 3-manifolds, while each member of the other family is not even homology cobordant to any Seifert fibered 3-manifold. None of our examples are homology cobordant to any manifold obtained by Dehn surgery along a knot in the 3-sphere.

math.GT↗
arXiv2019

A family of freely slice good boundary links

We show that every good boundary link with a pair of derivative links on a Seifert surface satisfying a homotopically trivial plus assumption is freely slice. This subsumes all previously known methods for freely slicing good boundary links with two or more components, and provides new freely slice links.

math.GT↗
arXiv2018

Ideal classes and Cappell-Shaneson homotopy 4-spheres

Gompf proposed a conjecture on Cappell-Shaneson matrices whose affirmative answer implies that all Cappell-Shaneson homotopy 4-spheres are diffeomorphic to the standard 4-sphere. We study Gompf conjecture on Cappell-Shaneson matrices using various algebraic number theoretic techniques. We find a hidden symmetry between trace $n$ Cappell-Shaneson matrices and trace $5-n$ Cappell-Shaneson matrices which was suggested by Gompf experimentally. Using this symmetry, we prove that Gompf conjecture for the trace $n$ case is equivalent to the trace $5-n$ case. We confirm Gompf conjecture for the special cases that $-64\leq \textrm{trace}\leq 69$ and corresponding Cappell-Shaneson homotopy 4-spheres are diffeomorphic to the standard 4-sphere. We also give a new infinite family of Cappell-Shaneson spheres which are diffeomorphic to the standard 4-sphere.

math.GT↗
arXiv2016

An infinite-rank summand of knots with trivial Alexander polynomial

We show that there exists a $\mathbb{Z}^\infty$-summand in the subgroup of the knot concordance group generated by knots with trivial Alexander polynomial. To this end we use the invariant Upsilon $Υ$ recently introduced by Ozsváth, Stipsicz and Szabó using knot Floer homology. We partially compute $Υ$ of $(n,1)$-cable of the Whitehead double of the trefoil knot. For this computation of $Υ$, we determine a sufficient condition for two satellite knots to have identical $Υ$ for any pattern with nonzero winding number.

math.GT↗