arXiv research

We show that a positive braid knot has maximal topological 4-genus exactly if it has maximal signature invariant. As an application, we determine all positive braid knots with maximal topological 4-genus and compute the topological 4-genus for all positive braid knots with up to 12 crossings.

Classifies 3-braid knots with maximal 4-genus using McCoy's method.

Sharp signature bound for 3-strand torus knots proved.

Under a simple assumption on Seifert surfaces, we characterise knots whose stable topological 4-genus coincides with the genus.

We show that the difference between the genus and the stable topological 4-genus of alternating knots is either zero or at least 1/3.

Researchers compute bounds and formulas for non-orientable 4-genus of torus knots.

New knots with specific properties have larger 4-genus than previously known bounds.

New method to untangle knots using null-homologous twists.

We prove that the topological locally flat slice genus of large torus knots takes up less than three quarters of the ordinary genus. As an application, we derive the best possible linear estimate of the topological slice genus for torus knots with non-maximal signature invariant.

New stabilizing number defined for knots, linking bounds in 4D.

The stable 4-genus of a knot K in 3-space is the limiting value of g_4(nK)/n, where g_4 denotes the 4-genus and n goes to infinity. This induces a seminorm on CQ, the concordance group tensored with the rational numbers. Basic properties of the stable genus are developed, as are examples focused on understanding the un…

Lower bound on stable 4-genus of knots using Casson-Gordon signatures.

This paper calculates the non-orientable 4-genus for knots with 10 crossings.

Study shows non-equivariant and equivariant non-orientable 4-genus of periodic knots can differ.

Computed the 4-genus for all 12-crossing prime knots.

Study equivariant 4-genus of knots in symmetric 4-manifolds.

Researchers refine the non-orientable $4$-genus of torus knots using Batson's surfaces.

New invariant defined for unoriented knots, proving no factorization through topological concordance.

Study non-orientable 4-genus for 11-crossing non-alternating knots.

The paper shows knots with specific properties have smaller 4-genus.

Study shows knots can have large genus difference from concordance.

Kearton observed that mutation can change the concordance class of a knot. A close examination of his example reveals that it is of 4-genus 1 and has a mutant of 4-genus 0. The first goal of this paper is to construct examples to show that for any pair of nonnegative integers m and n there is a knot of 4-genus m with a…

Study finds new knots that bound Möbius bands in 4D space.

We prove relative versions of the symplectic capping theorem and sufficiency of Giroux's criterion for Stein fillability and use these to study the 4-genus of knots.

Average signature of 2-bridge knots approximates sqrt(2c/π).

New knots found that are 4-genus minimal.

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 conject…

Study on nonorientable 4-genus of double twist knots.

Study on knot unknotting numbers and their behavior under connected sums.

We show that the difference between the Seifert genus and the topological 4-genus of a prime positive braid knot is bounded from below by an affine function of the minimal number of strands among positive braid representatives of the knot. We deduce that among prime positive braid knots, the property of having such a g…

To a Seifert matrix of a knot K one can associate a matrix w(K) with entries in the rational function field, Q(t). The Murasugi, Milnor, and Levine-Tristram knot signatures, all of which provide bounds on the 4-genus of a knot, are determined by w(K). More generally, the minimal rank of a representative of the class re…

Study on Whitehead doubles and their sliceness properties.

An oriented link is positive if it has a link diagram whose crossings are all positive. An oriented link is almost positive if it is not positive and has a link diagram with exactly one negative crossing. It is known that the Rasmussen invariant, $4$-genus and $3$-genus of a positive knot are equal. In this paper, we p…

We give a new proof that the Levine-Tristram signatures of a link give lower bounds for the minimal sum of the genera of a collection of oriented, locally flat, disjointly embedded surfaces that the link can bound in the 4-ball. We call this minimal sum the 4-genus of the link. We also extend a theorem of Cochran, Frie…

New findings on knot genera using advanced techniques.

Delta-unlinking number measures how to unlink algebraically split links.

Geography problem for nonorientable surfaces bounded by knots.

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 Conjec…

An open question asks if every knot of 4-genus g_s can be changed into a slice knot by g_s crossing changes. A counterexample is given.

The knot concordance invariant Upsilon, recently defined by Ozsvath, Stipsicz, and Szabo, takes values in the group of piecewise linear functions on the closed interval [0,2]. This paper presents a description of one approach to defining Upsilon and of proving its basic properties related to the knot 3-genus, 4-genus, …

This paper, to be regularly updated, lists those prime knots with the fewest possible number of crossings for which values of basic knot invariants, such as the unknotting number or the smooth 4-genus, are unknown. This list is being developed in conjunction with "KnotInfo" (www.indiana.edu/~knotinfo), a web-based tabl…

For a given knot, we study the minimal number of positive eigenvalues of the double branched cover over spanning surfaces for the knot. The value gives a lower bound for various genera, the dealternating number and the alternation number of knots, and we prove that Batson's bound for the non-orientable 4-genus gives an…

We refine prior bounds on how the multivariable signature and the nullity of a link change under link cobordisms. The formula generalizes a series of results about the 4-genus having their origins in the Murasugi-Tristram inequality, and at the same time extends previously known results about concordance invariance of …

Refines knot defect measurement in 3D and 4D.

Let $L$ be a oriented link such that $Σ_n(L)$, the $n$-fold cyclic cover of $S^3$ branched over $L$, is an L-space for some $n \geq 2$. We show that if either $L$ is a strongly quasipositive link other than one with Alexander polynomial a multiple of $(t-1)^{2g(L) + (|L|-1)}$, or $L$ is a quasipositive link other than …

For links with vanishing pairwise linking numbers, the link components bound pairwise disjoint surfaces in $B^{4}$. In this paper, we describe the set of genera of such surfaces in terms of the $h$-function, which is a link invariant from Heegaard Floer homology. In particular, we use the $h$-function to give lower bou…

Proves any three or more knots can form a genus-zero link in a 3-manifold.

A point in the $(N,q)$-torus knot in $\mathbb{R}^3$ goes $q$ times along a vertical circle while this circle rotates $N$ times around the vertical axis. In the Lissajous-toric knot $K(N,q,p)$, the point goes along a vertical Lissajous curve (parametrized by $t\mapsto(\sin(qt+φ),\cos(pt+ψ)))$ while this curve rotates $N…