James F. Davis
Four-dimensional surgery is used to show that a two component link with Alexander polynomial one is topologically concordant to the Hopf link.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
We give infinitely many $2$-component links with unknotted components which are topologically concordant to the Hopf link, but not smoothly concordant to any $2$-component link with trivial Alexander polynomial. Our examples are pairwise non-concordant.
It was shown by Jim Davis that a 2-component link with Alexander polynomial one is topologically concordant to the Hopf link. In this paper, we show that there is a 2-component link with Alexander polynomial one that has unknotted components and is not smoothly concordant to the Hopf link, answering a question of Jim Davis. We construct infinitely many concordance classes of such links, and show that they have the stronger property of not being smoothly concordant to the Hopf link with knots tied in the components.
We give new Casson-Gordon style obstructions for a two-component link to be topologically concordant to the Hopf link.
We establish homotopy ribbon concordance obstructions coming from the Blanchfield form and Levine-Tristram signatures. Then, as an application of twisted Alexander polynomials, we show that for every knot K with nontrivial Alexander polynomial, there exists an infinite family of knots that are all concordant to K and have the same Blanchfield form as K, such that no pair of knots in that family is homotopy ribbon concordant.
We give new examples of 2-component links with linking number one and unknotted components that are topologically concordant to the positive Hopf link, but not smoothly so - in fact they are not smoothly concordant to the positive Hopf link with a knot tied in the first component. Such examples were previously constructed by Cha-Kim-Ruberman-Strle; we show that our examples are distinct from theirs.
We show that if a link J in the 3-sphere is homotopy ribbon concordant to a link L then the Alexander polynomial of L divides the Alexander polynomial of J.
Fox-Milnor condition for concordant knots in homology 3-spheres
本文证明定向同调3-球中切片型纽结的Alexander多项式满足Fox-Milnor条件，建立了该类空间中合痕纽结的Alexander多项式关系。
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 f…
For a knot K, the concordance crosscap number, c(K), is the minimum crosscap number among all knots concordant to K. Building on work of G. Zhang, which studied the determinants of knots with c(K) < 2, we apply the Alexander polynomial to construct new algebraic obstructions to c(K) < 2. With the exception of low crossing number knots previously known to have c(K) < 2, the obstruction applies to all but four prime knots of 11 or fewer crossings.
Let C_T be the subgroup of the smooth knot concordance group generated by topologically slice knots and let C_D be the subgroup generated by knots with trivial Alexander polynomial. We prove the quotient C_T/C_D is infinitely generated, and uncover similar structure in the 3-dimensional rational spin bordism group. Our methods also lead to the construction of links that are topologically, but not smoothly, concordant to boundary links.
For a knot in the 3-sphere, the Upsilon invariant is a piecewise linear function defined on the interval [0,2]. For an L-space knot, the Upsilon invariant is determined only by the Alexander polynomial of the knot. We exhibit infinitely many pairs of hyperbolic L-space knots such that two knots of each pair have distinct Alexander polynomials, so they are not concordant, but share the same Upsilon invariant. Conversely, we examine the restorability of the Alexander polynomial of an L-space knot from the Upsilon invariant through the Legendre-Fenchel transformation.
We use the Bar-Natan Zh-correspondence to identify the generalized Alexander polynomial of a virtual knot with the Alexander polynomial of a two component welded link. We show that the Zh-map is functorial under concordance, and also that Satoh's Tube map (from welded links to ribbon knotted tori in $S^4$) is functorial under concordance. In addition, we extend classical results of Chen, Milnor, and Hillman on the lower central series of link groups to links in thickened surfaces. Our main result is that the generalized Alexander polynomial vanishes on any knot in a thickened surface which is …
We show that there are links whose individual components are concordant to the unknot, but which are not concordant to any link with unknotted components. We give examples in the topological category, and examples in the smooth category which are topologically slice. We also give generalizations regarding components of prescribed Alexander polynomials. The main tools are covering link calculus, algebraic invariants of rational knot concordance theory, and the correction term of Heegaard Floer homology.
We construct links of arbitrarily many components each component of which is slice and yet are not concordant to any link with even one unknotted component. The only tool we use comes from the Alexander modules.
We show that for each Seifert form of an algebraically slice knot with nontrivial Alexander polynomial, there exists an infinite family of knots having the Seifert form such that the knots are linearly independent in the knot concordance group and not concordant to any knot with coprime Alexander polynomial. Key ingredients for the proof are Cheeger-Gromov-von Neumann rho-invariants for amenable groups developed by Cha and Orr and polynomial splittings of metabelian rho-invariants.
It follows from earlier work of Silver-Williams and the authors that twisted Alexander polynomials detect the unknot and the Hopf link. We now show that twisted Alexander polynomials also detect the trefoil and the figure-8 knot, that twisted Alexander polynomials detect whether a link is split and that twisted Alexander modules detect trivial links.
Shake concordance in the topological concordance class of the Hopf link
该研究在霍普夫链的拓扑配边类中，基于光滑0-晃动配边的等价关系构造出无穷多类，每类含无穷多链的光滑配边类，其第二个分量的光滑切片亏格无界。
Given a fibered link, consider the characteristic polynomial of the monodromy restricted to first homology. This generalizes the notion of the Alexander polynomial of a knot. We define a construction, called iterated plumbing, to create a sequence of fibered links from a given one. The resulting sequence of characteristic polynomials has the same form as those arising in work of Salem and Boyd in their study of distributions of Salem and P-V numbers. From this we deduce information about the asymptotic behavior of the large roots of the generalized Alexander polynomials, and define a new poset…
Oleg Viro studied in arXiv:math/0204290 two interpretations of the (multivariable) Alexander polynomial as a quantum link invariant: either by considering the quasi triangular Hopf algebra associated to $U_q sl(2)$ at fourth roots of unity, or by considering the super Hopf algebra $U_q gl(1|1)$. In this paper, we show these Hopf algebras share properties with the $-1$ specialization of $U_q gl(n|1)$ leading to the proof of a conjecture of David De Wit, Atsushi Ishii and Jon Links on the Links-Gould invariants.
The second coefficient of the Alexander polynomial as a satellite obstruction
本研究引入含正辫子链环等的链环集合，证明P中链环的Alexander多项式首项与第二系数符号相反，以此作为障碍证明部分卫星链环不在P中。
In a recent paper, Allen and Swenberg investigated which link polynomials are capable of detecting causality in (2+1)-dimensional globally hyperbolic spacetimes. They ultimately suggested it is likely that the Jones Polynomial accomplishes this, while the Alexander-Conway polynomial, independently, is insufficient. As the Alexander-Conway polynomial on its own is not likely to detect causality an additional piece of information must be supplemented to the link polynomial. This paper aims to examine the ability of Alexander quandles, to distinguish the connected sum of two Hopf links and Allen-…
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(…
We construct a 2-variable link polynomial, called $W_L$, for classical links by considering simultaneously the Kauffman state models for the Alexander and for the Jones polynomials. We conjecture that this polynomial is the product of two 1-variable polynomials, one of which is the Alexander polynomial. We refine $W_L$ to an ordered set of 3-variable polynomials for those links in 3-space which contain a Hopf link as a sublink.
Extended signatures and link concordance
本文研究了扩展Levine-Tristram签名作为连通性不变量，并提出了一个无法被传统方法检测的3链环族。
Characterization and Further Applications of the Bar-Natan Zh-Construction
该研究给出了Bar-Natan Zh构造的近经典链环刻画定理，由此得到其纯几何描述、Dye-Kauffman-Miyazawa多项式的简单推导，并证明广群类着色不变量可通过该构造扩展为新不变量。
We develop an explicit skein theoretical algorithm to compute the Alexander polynomial of a 3-manifold from a surgery presentation employing the methods used in the construction of quantum invariants of 3-manifolds. As a prerequisite we establish and prove a rather unexpected equivalence between the topological quantum field theory constructed by Frohman and Nicas using the intersection homology of U(1)-representation varieties on the one side and the combinatorially constructed Hennings-TQFT based on the quasitriangular Hopf algebra ${\cal N}=\Z/2\ltimes \ext *\R^2$ on the other side. We find…
Detecting Causality with Conjugation Quandles over Dihedral Groups
针对(2+1)维时空链环因果检测难题，发现二面体群D₅的共轭广群可区分因果相关与无关的链环，仅靠计数不变量就能实现简单可计算的因果检测。
Multivariable knot polynomials from braided Hopf algebras with automorphisms
本文通过带自同构的辫子 Hopf 代数上的 Yetter--Drinfel'd 模求解 Yang--Baxter 方程，构造出包含 ADO、着色 Jones 及新 2 变量多项式在内的多元纽结不变量，新多项式可检测手性与突变并给出紧致亏格界。
Detecting Causality with Symplectic Quandles
本文证明辛Quandle结合Alexander-Conway多项式可区分Allen-Swenberg 2-sky类链环与两个Hopf链环的连通和，从而为检测(2+1)维全局双曲时空中的因果性提供了新的拓扑不变量工具。
Causality Detection via Symplectic Quandles
本研究探讨辛圈着色能否揭示“天空链环”的因果结构，通过比较霍普夫链环连通和与艾伦-斯温伯格链环，发现增强辛圈计数不变量可有效区分因果关系，弥补经典多项式的不足。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。