The stable 4-genus of alternating knots

Crossing Change Alternating Knots

In this paper we define Crossing Change Alternating Knots (CCA knots) and their generalization: $k$-CCA knots.

Composite tunnel number one genus two handlebody-knots

We characterize composite tunnel number one genus two handlebody-knots.

A topological characterization of toroidally alternating knots

We extend Howie's characterization of alternating knots to give a topological characterization of toroidally alternating knots, which were defined by Adams. We provide necessary and sufficient conditions for a knot to be toroidally alternating. We also give a topological characterization of almost-alternating knots which is different from Ito's recent characterization.

Knots with unique minimal genus Seifert surface and depth of knots

We describe a procedure for creating infinite families of hyperbolic knots having unique minimal genus Seifert surface. A large subset of these knots have the further property that the surface cannot be the sole compact leaf of a depth one foliation of the knot exterior.

Knots of genus two

We classify all knot diagrams of genus two and three, and give applications to positive, alternating and homogeneous knots, including a classification of achiral genus 2 alternating knots, slice or achiral 2-almost positive knots, a proof of the 3- and 4-move conjectures, and the calculation of the maximal hyperbolic volume for weak genus two knots. We also study the values of the link polynomials at roots of unity, extending denseness results of Jones. Using these values, examples of knots with unsharp Morton (weak genus) inequality are found. Several results are generalized to arbitrary weak…

A higher-order genus invariant and knot Floer homology

It is known that knot Floer homology detects the genus and Alexander polynomial of a knot. We investigate whether knot Floer homology of $K$ detects more structure of minimal genus Seifert surfaces for $K$. We define an invariant of algebraically slice, genus one knots and provide examples to show that knot Floer homology does not detect this invariant. Finally, we remark that certain metabelian $L^2$-signatures bound this invariant from below.

纽结迹的三分亏格

Trisection genus of knot traces

本文分类了三分亏格至多为2的纽结迹，并构造了无限多具有三分亏格3和4的纽结迹，证明了存在无限族纽结迹具有任意大的三分亏格，同时给出了若干著名纽结迹的三分亏格精确值或紧界。

The almost alternating diagrams of the trivial knot

Bankwitz characterized an alternating diagram representing the trivial knot. A non-alternating diagram is called almost alternating if one crossing change makes the diagram alternating. We characterize an almost alternaing diagram representing the trivial knot. As a corollary we determine an unknotting number one alternating knot with a property that the unknotting operation can be done on its alternating diagram.

The non-orientable 4-genus for knots with 10 crossings

Given a knot in the 3-sphere, the non-orientable 4-genus or 4-dimensional crosscap number of a knot is the minimal first Betti number of non-orientable surfaces, smoothly and properly embedded in the 4-ball, with boundary the knot. In this paper, we calculate the non-orientable 4-genus of knots with crossing number 10.

Knots with many minimal genus Seifert surfaces

Roberts proved that a family of alternating, arborescent, prime knots each have at least $2^{2n-1}$ distinct minimal genus Seifert surfaces, where $n$ is the genus of the knot in question. We give a subfamily of these knots that have exactly this many minimal genus Seifert surfaces.

Almost alternating knots with 12 crossings and Turaev genus

Among non-alternating knots with $n\le 12$ crossings given in \cite{1} Turaev genus is not known for 191 knot. For 154 of them we show that they are almost alternating, so their Turaev genus is 1.

Genus two mutant knots with the same dimension in knot Floer and Khovanov homologies

We exhibit an infinite family of knots with isomorphic knot Heegaard Floer homology. Each knot in this infinite family admits a nontrivial genus two mutant which shares the same total dimension in both knot Floer homology and Khovanov homology. Each knot is distinguished from its genus two mutant by both knot Floer homology and Khovanov homology as bigraded groups. Additionally, for both knot Heegaard Floer homology and Khovanov homology, the genus two mutation interchanges the groups in $δ$-gradings $k$ and $-k$.

Positive braid knots of maximal topological 4-genus

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.

Cosmetic crossings of genus one knots

We show that for genus one knots the Alexander polynomial and the homology of the double cover branching over the knot provide obstructions to cosmetic crossings. As an application we prove the nugatory crossing conjecture for the negatively twisted, positive Whitehead doubles of all knots. We also verify the conjecture for several families of pretzel knots and all genus one knots with up to 10 crossings.

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.

Alternating Knots

This is a short expository article on alternating knots and is to appear in the Concise Encyclopedia of Knot Theory.

Free genus one knots with large volume

A Seifert surface F for a knot K is free if the complement of F is a handlebody (i.e., has free fundamental group). The free genus of K is the minimum genus among all free Seifert surfaces for K. In this paper we show that there exist families of hyperbolic knots with arbitrarily large volume, which each have free genus one. This implies that there are knots with free genus one but arbitrarily large canonical genus, and that there exist knots admitting incompressible free Seifert surfaces which cannot be built by applying Seifert's algorithm to a projection of the knot.

Adjoint twisted Alexander polynomials of genus one two-bridge knots

We give explicit formulas for the adjoint twisted Alexander polynomial and the nonabelian Reidemeister torsion of genus one two-bridge knots.

The support genus of certain Legendrian knots

In this paper, the support genus of all Legendrian right handed trefoil knots and some other Legendrian knots is computed. We give examples of Legendrian knots in the three-sphere with the standard contact structure which have positive support genus with arbitrarily negative Thurston-Benniquin invariant. This answers a question in Onaran.

最大亏格缺陷的分隔结

Divide knots of maximal genus defect

该研究构造了具有任意光滑四维亏格但拓扑四维亏格为一的分隔结，揭示了强拟正则纤维化结的四维亏格比值可趋近于零的特性。

A survey on the Turaev genus of knots

The Turaev genus of a knot is a topological measure of how far a given knot is from being alternating. Recent work by several authors has focused attention on this interesting invariant. We discuss how the Turaev genus is related to other knot invariants, including the Jones polynomial, knot homology theories, and ribbon-graph polynomial invariants.

Cosmetic crossing conjecture for genus one knots with non-trivial Alexander polynomial

We prove the cosmetic crossing conjecture for genus one knots with non-trivial Alexander polynomial. We also prove the conjecture for genus one knots with trivial Alexander polynomial, under some additional assumptions.

Genus one fibered knots in 3-manifolds with reducible genus two Heegaard splittings

We give a necessary and sufficient condition for a simple closed curve on the boundary of a genus two handlebody to decompose the handlebody into (torus with one boundary component times [0,1]. We use this condition to decide whether a simple closed curve on a genus two Heegaard surface is a GOF-knot (genus one fibered knot) which induces the Heegaard splitting. By using this, we determine the number and the positions with respect to the Heegaard splittings of GOF-knots in the 3-manifolds with reducible genus two Heegaard splittings. This is another proof of results of Morimoto and Baker.

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.

A characterization of almost alternating knot

Generalizing Howie and Greene's characterization of alternating knots, we give a topological characterization of almost alternating knots.

The smooth 4-genus of (the rest of) the prime knots through 12 crossings

We compute the smooth 4-genera of the prime knots with 12 crossings whose values, as reported on the KnotInfo website, were unknown. This completes the calculation of the smooth 4-genus for all prime knots with 12 or fewer crossings.

The concordance genus of a knot, II

The concordance genus of a knot K is the minimum three-genus among all knots concordant to K. For prime knots of 10 or fewer crossings there have been three knots for which the concordance genus was unknown. Those three cases are now resolved. Two of the cases are settled using invariants of Levine's algebraic concordance group. The last case depends on the use of twisted Alexander polynomials, viewed as Casson-Gordon invariants.

Braid ordering and knot genus

The genus of knots is a one of the fundamental invariant and can be seen as a complexity of knots. In this paper, we give a lower bound of genus using Dehornoy floor, which is a measure of complexity of braids in terms of braid ordering.

Knot concordances and alternating knots

There is an infinitely generated free subgroup of the smooth knot concordance group with the property that no nontrivial element in this subgroup can be represented by an alternating knot. This subgroup has the further property that every element is represented by a topologically slice knot.

An obstruction of Gordian distance one and cosmetic crossings for genus one knots

We give an obstruction for genus one knots $K$, $K'$ to have the Gordian distance one by using the $0$th coefficient of the HOMFLT polynomials. As an application, we give a new constraint for genus one knot to admit a (generalized) cosmetic crossing. Combining known results, we prove the (generalized) cosmetic crossing conjecture for genus one pretzel knots.

Twisted Alexander polynomials of genus one two-bridge knots

Morifuji computed the twisted Alexander polynomial of twist knots for nonabelian representations. In this paper we compute the twisted Alexander polynomial and the Reidemeister torsion of genus one two-bridge knots, a class of knots which includes twist knots. As an application, we give a formula for the Reidemeister torsion of the 3-manifold obtained by a Dehn surgery on a genus one two-bridge knot.

Cyclic branched covers of alternating knots and $L$-spaces

For any alternating knot, it is known that the double branched cover of the $3$-sphere branched over the knot is an $L$-space. We show that the three-fold cyclic branched cover is also an $L$-space for any genus one alternating knot.

Crossing number of an alternating knot and canonical genus of its Whitehead double

A conjecture proposed by J. Tripp in 2002 states that the crossing number of any knot coincides with the canonical genus of its Whitehead double. In the meantime, it has been established that this conjecture is true for a large class of alternating knots including $(2, n)$ torus knots, $2$-bridge knots, algebraic alternating knots, and alternating pretzel knots. In this paper, we prove that the conjecture is not true for any alternating $3$-braid knot which is the connected sum of two torus knots of type $(2, m)$ and $(2, n)$. This results in a new modified conjecture that the crossing number …

Uniqueness of higher genus bridge surfaces for torus knots

We show that a torus knot which is not 2-bridge has a unique irreducible bridge splitting of positive genus.

Closed incompressible surfaces of genus two in 3-bridge knot complements

In this paper, we characterize closed incompressible surfaces of genus two in the complements of 3-bridge knots and links. This characterization includes that of essential 2-string tangle decompositions for 3-bridge knots and links.

Genus two Heegaard splittings of exteriors of 1-genus 1-bridge knots II

A knot K is called a 1-genus 1-bridge knot in a 3-manifold M if (M,K) has a Heegaard splitting (V_1,t_1)\cup (V_2,t_2) where V_i is a solid torus and t_i is a boundary parallel arc properly embedded in V_i. If the exterior of a knot has a genus 2 Heegaard splitting, we say that the knot has an unknotting tunnel. Naturally the exterior of a 1-genus 1-bridge knot K allows a genus 2 Heegaard splitting, i.e., K has an unknotting tunnel. But, in general, there are unknotting tunnels which are not derived form this procedure. Some of them may be levelled with the torus \partial V_1=\partial V_2, who…

Classification of alternating knots with tunnel number one

This paper gives a complete classification of all alternating knots with tunnel number one, and all their unknotting tunnels. We prove that the only such knots are two-bridge knots and certain Montesinos knots.

On the genus defect of positive braid knots

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 genus difference less than any fixed constant is characterised by finitely many forbidden surface minors.

On Jones polynomials of alternating pretzel knots

We show that there are infinitely many pairs of alternating pretzel knots whose Jones polynomials are identical.

Toroidal Seifert fibered surgeries on alternating knots

We give a complete classification of toroidal Seifert fibered surgeries on alternating knots. Precisely, we show that if an alternating knot admits a toroidal Seifert fibered surgery, then the knot is either the trefoil knot and the surgery slope is zero, or the connected sum of a (2,p)-torus knot and a (2,q)-torus knot and the surgery slope is 2(p+q) with |p|, |q| at least three.

局部结、ν⁺-尖结和有理切 genus

Local Knots, $ν^+$-Sharp Knots, and Rational Slice Genus

本文研究了局部结和ν⁺-尖结的性质，证明了这些结在有理同调3-球体中切 genus 等于原结的切 genus，利用Heegaard Floer不变量ν⁺得出相关结论。

Alternating Montesinos knots and Conjecture $\mathbb{Z}$

Conjecture $\mathbb{Z}$ is a knot theoretical equivalent form of the Kervaire Conjecture. We say that a knot have property $\mathbb{Z}$ if it satisfies Conjecture $\mathbb{Z}$ for that specific knot. In this work, we show that alternating Montesinos knots with three tangles have property $\mathbb{Z}$. We also show that all the pretzel knots of the form $P(p,q,r)$ (not necessarily alternating) have property $\mathbb{Z}$.

几乎正结的Rasmussen不变量、4亏格和3亏格相等

The Rasmussen invariant, four-genus and three-genus of an almost positive knot are equal

本文证明了几乎正结的Rasmussen不变量、4亏格和3亏格相等，并基于其几乎正结图确定了Rasmussen不变量，得出任何几乎正结均非齐次且不存在4亏格为1的几乎正结。

A lower bound on the stable 4-genus of knots

We present a lower bound on the stable $4$-genus of a knot based on Casson-Gordon $τ$-signatures. We compute the lower bound for an infinite family of knots, the twist knots, and show that a twist knot is torsion in the knot concordance group if and only if it has vanishing stable $4$-genus.

Knot mutation: 4-genus of knots and algebraic 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 mutant of 4-genus n. A second result of this paper is a crossing change formula for the algebraic concordance class of a knot, which is then applied to prove the invariance of the algebraic concordance class under mutation. The paper concludes with an application of crossing cha…

On sums of torus knots concordant to alternating knots

We consider the question, asked by Friedl, Livingston and Zentner, of which sums of torus knots are concordant to alternating knots. After a brief analysis of the problem in its full generality, we focus on sums of two torus knots. We describe some effective obstructions based on Heegaard Floer homology.

Construction of non-alternating knots

We investigate the behaviour of Rasmussen's invariant $s$ under the sharp operation on knots and obtain a lower bound for the sharp unknotting number. This bound leads us to an interesting move that transforms arbitrary knots into non-alternating knots.

Unknotting number and genus of 3-braid knots

Let $u(K)$ and $g(K)$ denote the unknotting number and the genus of a knot $K$, respectively. For a 3-braid knot $K$, we show that $u(K)\le g(K)$ holds, and that if $u(K)=g(K)$ then $K$ is either a 2-braid knot, a connected sum of two 2-braid knots, the figure-eight knot, a strongly quasipositive knot or its mirror image.

All exceptional surgeries on alternating knots are integral surgeries

We show that all exceptional surgeries on hyperbolic alternating knots in the 3-sphere are integral surgeries.

On the topological 4-genus of torus knots

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.

Prime knot complements with meridional essential surfaces of arbitrarily high genus

We show the existence of infinitely many prime knots each of which having in their complements meridional essential surfaces with two boundary components and arbitrarily high genus.

结迹上的交替结和扩展交替结的Lefschetz纤维

Lefschetz Fibrations on Knot Traces of Alternating and Extended Alternating Knots

本文基于之前的工作，将构造正允许Lefschetz纤维的方法应用于交替结和扩展交替结的结迹，证明其纤维的亏格等于平面图中的白区域数，并得出正预结结和正扭转结的结迹的纤维亏格结果。

特殊交错纽结是带素纽结

Special Alternating Knots are Band Prime

本文证明特殊交错纽结不能分解为非平凡带和，由此限制其协合关系，并提出和验证特殊交错纽结是带状协合极小的猜想，受相关向下封闭猜想启发。

Heegaard genus and complexity of fibered knots

We prove that if a fibered knot $K$ with genus greater than one in a three-manifold $M$ has a sufficiently complicated monodromy, then $K$ induces a minimal genus Heegaard splitting $P$ that is unique up to isotopy, and small genus Heegaard splittings of $M$ are stabilizations of $P$. We provide a complexity bound in terms of the Heegaard genus of $M$. We also provide global complexity bounds for fibered knots in the three-sphere and lens spaces.