Carmen Caprau, Katherine Urabe
We provide a formula for the Dubrovnik polynomial of a rational knot in terms of the entries of the tuple associated with a braid-form diagram of the knot. Our calculations can be easily carried out using a computer algebra system.
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
In this survey we summarize results regarding the Kauffman bracket, HOMFLYPT, Kauffman 2-variable and Dubrovnik skein modules, and the Alexander polynomial of links in lens spaces, which we represent as mixed link diagrams. These invariants generalize the corresponding knot polynomials in the classical case. We compare the invariants by means of the ability to distinguish between some difficult cases of knots with certain symmetries.
The genus of satellite tunnel number one knots and torti-rational knots is computed using the tools introduced by Floyd and Hatcher. An implementation of an algorithm is given to compute genus and slopes of minimal genus Seifert surfaces for such knots.
We prove that certain fibered, $-$amphicheiral knots are rationally slice. Moreover, we show that the concordance invariants $ν^+$ and $Υ(t)$ from Heegaard Floer homology vanish for a class of knots that includes rationally slice knots.
Real rational knots in the quadric of signature~$(3,2)$
研究低次实有理曲线在三维二次曲面中的空间，分类实有理结与节点曲线，并探讨其与三维实射影空间中实有理结的关系，给出度数不超过5的实有理结代表构造。
We show that every rational knot $K$ of crossing number $N$ admits a polynomial parametrization $x=T_a(t), y = T_b(t), z = C(t)$ where $T_k(t)$ are the Chebyshev polynomials, $a=3$ and $b+ °C = 3N.$ We show that every rational knot also admits a polynomial parametrization with $a=4$. If $C (t)= T_c(t)$ is a Chebyshev polynomial, we call such a knot a harmonic knot. We give the classification of harmonic knots for $a \le 4.$
We introduce new skein invariants of links based on a procedure where we first apply the skein relation only to crossings of distinct components, so as to produce collections of unlinked knots. We then evaluate the resulting knots using a given invariant. A skein invariant can be computed on each link solely by the use of skein relations and a set of initial conditions. The new procedure, remarkably, leads to generalizations of the known skein invariants. We make skein invariants of classical links, $H[R]$, $K[Q]$ and $D[T]$, based on the invariants of knots, $R$, $Q$ and $T$, denoting the reg…
The following is a long-standing open question: "If the zero-framed surgeries on two knots in the 3-sphere are integral homology cobordant, are the knots themselves concordant?" We show that an obvious rational version of this question has a negative answer. Namely, we give examples of knots whose zero-framed surgeries are rational homology cobordant 3-manifolds, wherein the knots are not rationally concordant (that is not concordant in any rational homology S^3 x [0,1]). Specifically, we prove that, for any positive integer p and any knot K, the zero framed surgery on K is Z[1/p]-homology cob…
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。
In this paper we classify, up to rigid isotopy, non-singular real rational curves of degrees less than or equal to 6 in a quadric homeomorphic to the 3-sphere. We also study their connections with rigid isotopy classes of real rational knots in $\mathbb{RP}^3$.
We prove that loose Legendrian knots in a rational homology contact 3-sphere, satisfying some additional hypothesis, are Legendrian isotopic if and only if they have the same classical invariants. The proof requires a result of Dymara on loose Legendrian knots and Eliashberg's classification of overtwisted contact structures on 3-manifolds.
We give an infinite family of knots that are not rationally concordant to their reverses. More precisely, if R denotes the involution of the rational knot concordance group QC induced by string reversal and Fix(R) denotes the subgroup of knots fixed under R in QC, then QC/Fix(R) contains an infinite rank subgroup. As a corollary, we show that there exists a knot K such that for every pair of coprime integers p and q, the (p,q)-cable of K is not concordant to the reverse of the (p,q)-cable of K.
We classify which positive integral surgeries on positive torus knots bound rational homology balls. Additionally, for a given knot K we consider which cables K(p,q) admit integral surgeries that bound rational homology balls. For such cables, let S(K) be the set of corresponding rational numbers q/p. We show that S(K) is bounded for each K. Moreover, if n-surgery on K bounds a rational homology ball then n is an accumulation point for S(K).
We construct a Kirby diagram of the rational homology ball used in "generalized rational blow-down" developed by Jongil Park. The diagram consists of a dotted circle and a torus knot. The link is simpler, but the parameters are a little complicate. Euclidean Algorithm is used three times in the construction and the proof.
We show that the problem of constructing a real rational knot of a reasonably low degree can be reduced to an algebraic problem involving the pure braid group: expressing an associated element of the pure braid group in terms of the standard generators of the pure braid group. We also predict the existence of a real rational knot in a degree that is expressed in terms of the edge number of its polygonal representation.
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.
Constructing knots with low rational genera
该研究通过构造方法展示了某些结在不同同调球中具有不同的亏群，并证明了所有结在有理同调球中可界定了莫比乌斯带。
Ozsváth and Szabó used the knot filtration on $\widehat{CF}(S^3)$ to define the $τ$-invariant for knots in the 3-sphere. In this article, we generalize their construction and define a collection of $τ$-invariants associated to a knot $K$ in a rational homology sphere $Y$. We then show that some of these invariants provide lower bounds for the genus of a surface with boundary $K$ properly embedded in a negative definite 4-manifold with boundary $Y$..
We introduce the notion of rational links in the solid torus. We show that rational links in the solid torus are fully characterized by rational tangles, and hence by the continued fraction of the rational tangle. Furthermore, we generalize this by giving an infinite family of ambient isotopy invariants of colored diagrams in the Kauffman bracket skein module of an oriented surface.
In the complement of a hyperbolic Montesinos knot with 4 rational tangles, we investigate the number of closed, connected, essential, orientable surfaces of a fixed genus $g$, up to isotopy. We show that there are exactly 12 genus 2 surfaces and $8ϕ(g - 1)$ surfaces of genus greater than 2, where $ϕ(g - 1)$ is the Euler totient function of $g - 1$. Observe that this count is independent of the number of crossings of the knot. Moreover, this class of knots form an infinite class of hyperbolic 3-manifolds and the result applies to all such knot complements.
Rational concordance of double twist knots
本文证明了双扭结有理切片条件的逆命题，利用 Donaldson 对角化定理构造有理球阻碍，表明除已知情形外无穷多个素数幂重循环分支覆盖不界定有理球。
We compute the nonabelian $\mathrm{SL_2}(\mathbb{C})$-character varieties of the rational knots $C(2n+1,2m,2)$ in the Conway notation, where $m$ and $n$ are non-zero integers. By studying real points on these varieties, we determine the left orderability of the fundamental groups of the cyclic branched covers of $C(2n+1,2m,2)$.
Recently Iltgen, Lewark and Marino introduced the concept of a proper rational tangle replacement and the corresponding notion of the proper rational unknotting number. In this note we derive a version of the Montesinos trick for proper rational tangle replacement and use it to study knots with proper rational unknotting number one. We prove that knots with proper rational unknotting number one are prime and classify the alternating knots with proper rational unknotting number one. We also study Montesinos knots with proper rational unknotting number one.
In his pioneering work from 1969, Jerry Levine introduced a complete set of invariants of algebraic concordance of knots. The evaluation of these invariants requires a factorization of the Alexander polynomial of the knot, and is therefore in practice often hard to realize. We thus propose the study of an alternative set of invariants of algebraic concordance - the rational Witt classes of knots. Though these are rather weaker invariants than those defined by Levine, they have the advantage of lending themselves to quite manageable computability. We illustrate this point by computing the ratio…
A torti-rational knot, denoted by K(2a,b|r), is a knot obtained from the 2-bridge link B(2a,b) by applying Dehn twists an arbitrary number of times, r, along one component of B(2a,b). We determine the genus of K(2a,b|r) and solve a question of when K(2a,b|r) is fibred. In most cases, the Alexander polynomials determine the genus and fibredness of these knots. We develop both algebraic and geometric techniques to describe the genus and fibredness by means of continued fraction expansions of b/2a. Then, we explicitly construct minimal genus Seifert surfaces. As an application, we solve the same …
We study the group of rational concordance classes of codimension two knots in rational homology spheres. We give a full calculation of its algebraic theory by developing a complete set of new invariants. For computation, we relate these invariants with limiting behaviour of the Artin reciprocity over an infinite tower of number fields and analyze it using tools from algebraic number theory. In higher dimensions it classifies the rational concordance group of knots whose ambient space satisfies a certain cobordism theoretic condition. In particular, we construct infinitely many torsion element…
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 …
Introducing a way to modify knots using $n$-trivial rational tangles, we show that knots with given values of Vassiliev invariants of bounded degree can have arbitrary unknotting number (extending a recent result of Ohyama, Taniyama and Yamada). The same result is shown for 4-genera and finite reductions of the homology group of the double branched cover. Closer consideration is given to rational knots, where it is shown that the number of $n$-trivial rational knots of at most $k$ crossings is for any $n$ asymptotically at least $C^{(\ln k)^2}$ for any $C<\sqrt[2\ln 2]{e}$.
A Chebyshev knot ${\cal C}(a,b,c,ϕ)$ is a knot which has a parametrization of the form $ x(t)=T_a(t); y(t)=T_b(t) ; z(t)= T_c(t + ϕ), $ where $a,b,c$ are integers, $T_n(t)$ is the Chebyshev polynomial of degree $n$ and $ϕ\in \R.$ We show that any two-bridge knot is a Chebyshev knot with $a=3$ and also with $a=4$. For every $a,b,c$ integers ($a=3, 4$ and $a$, $b$ coprime), we describe an algorithm that gives all Chebyshev knots $\cC(a,b,c,ϕ)$. We deduce a list of minimal Chebyshev representations of two-bridge knots with small crossing number.
Instanton knot invariants with rational holonomy parameters and an application for torus knot groups
本文推广了 Daemi-Scaduto 的等变奇异瞬子 Floer 理论，研究带有理完整参数的瞬子纽结不变量，并确定了除有限多个参数外环面纽结的不可约奇异瞬子同调，为 Slice-Ribbon 猜想的环面纽结版本提供了新证据。
A rational knot or link can be put into a standard alternating format which has horizontal and vertical twist sites (double helices). The number and type of these twist sites are determined by terms of next-to-highest $z$-degree in Kauffman's regular isotopy invariant $Λ(a,z)$. In particular, for a knot or link with $c$ crossings, the coefficient of the $z^{c-2}$ term is equal to the number of twist sites in its standard diagram. Furthermore, the coefficients of the $a^{-2}z^{c-2}$ and $a^2z^{c-2}$ terms count the number of left-turning and right-turning twist sites, respectively.
Obstructing two-torsion in the rational knot concordance group
本文利用局部化von Neumann $ρ$-不变量，给出代数有理协调群中阶2纽结在有理纽结协调群中具有有限阶的阻碍，并构造生成无限秩子群的纽结族。
Let $F$ be an incompressible, meridionally incompressible and not boundary-parallel surface with boundary in the complement of an algebraic tangle $(B,T)$. Then $F$ separates the strings of $T$ in $B$ and the boundary slope of $F$ is uniquely determined by $(B,T)$ and hence we can define the slope of the algebraic tangle. In addition to the Conway's tangle sum, we define a natural product of two tangles. The slopes and binary operation on algebraic tangles lead an algebraic structure which is isomorphic to the rational numbers. We introduce a new knot and link class, algebraically alternating …
The protein recombinase can change the knot type of circular DNA. The action of a recombinase converting one knot into another knot is normally mathematically modeled by band surgery. Band surgeries on a 2-bridge knot N((4mn-1)/(2m)) yielding a (2,2k)-torus link are characterized. We apply this and other rational tangle surgery results to analyze Xer recombination on DNA catenanes using the tangle model for protein-bound DNA.
Surgeries on Iterated Torus Knots Bounding Rational Homology 4-Balls
本文证明代数纽结上足够大的正积分手术界定具有负定管道树的4-流形，并利用唐纳森定理的格嵌入障碍分类了特定形式的迭代环面纽结手术，确定哪些界定有理同调4-球体。
A knot K is the resultant of a knot H if there exists a minimal crossing diagram D of K such that some crossings of D can be altered to produce H. K is fertile if every prime knot H with crossing number less than c(K) is a resultant of K. K is n-fertile if every prime knot H with crossing number less than n is a resultant of K. We classify the fertility and fertility number of all rational links. This requires the introduction of the analogous concept of link fertility.
This paper determines the minimal degree sequence for two compact rational knots, namely the trefoil and figure-eight knots. We find explicit projections with the minimal degree sequence of each knot. This is done by modifying a non-compact rational minimal-degree parameterization of the trefoil and figure-eight knots to make it compact. This research was conducted at Ramapo College of New Jersey during the summer of 2011 with Dr. Donovan McFeron.
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…
We consider the question of when a rational homology 3-sphere is rational homology cobordant to a connected sum of lens spaces. We prove that every rational homology cobordism class in the subgroup generated by lens spaces is represented by a unique connected sum of lens spaces whose first homology embeds in any other element in the same class. As a first consequence, we show that several natural maps to the rational homology cobordism group have infinite rank cokernels. Further consequences include a divisibility condition between the determinants of a connected sum of 2-bridge knots and any …
We study the rational Kontsevich integral of torus knots. We construct explicitely a series of diagrams made of circles joined together in a tree-like fashion and colored by some special rational functions. We show that this series codes exactly the unwheeled rational Kontsevich integral of torus knots, and that it behaves very simply under branched coverings. Our proof is combinatorial. It uses the results of Wheels and Wheeling and various spaces of diagrams.
The 3-strand pretzel knots and links are a well-studied source of examples in knot theory. However, while there have been computations of the Khovanov homology of some sub-families of 3-strand pretzel knots, no general formula has been given for all of them. We give a general formula for the unreduced Khovanov homology of all 3-strand pretzel links, over the rational numbers.
Equivariant rational sliceness and Klein amphichirality of odd-stranded Turk's head knots
研究奇数股土耳其头结的等变有理切片性和克莱因两性性，通过构造环境对合证明q为奇数时结的性质，利用等变福克斯 - 米尔诺平方条件和相关计算证明q为偶数时结不具有相应性质。
We study the linking numbers in a rational homology 3-sphere and in the infinite cyclic cover of the complement of a knot. They take values in $\Bbb Q$ and in ${Q}({\Bbb Z}[t,t^{-1}])$ respectively, where ${Q}({\Bbb Z}[t,t^{-1}])$ denotes the quotient field of ${\Bbb Z}[t,t^{-1}]$. It is known that the modulo-$\Bbb Z$ linking number in the rational homology 3-sphere is determined by the linking matrix of the framed link and that the modulo-${\Bbb Z}[t,t^{-1}]$ linking number in the infinite cyclic cover of the complement of a knot is determined by the Seifert matrix of the knot. We eliminate `…
We investigate coincidences of the (one-variable) Jones polynomial amongst rational knots, what we call `Jones rational coincidences'. We provide moves on the continued fraction expansion of the associated rational which we prove do not change the Jones polynomial and conjecture (based on experimental evidence from all rational knots with determinant $<900$) that these moves are sufficient to generate all Jones rational coincidences. These coincidences are generically not mutants, as is verified by checking the HOMFLYPT polynomial. In the process we give a new formula for the Jones polynomial …
In this paper, we enumerate the number of oriented rational knots and the number of oriented rational links with any given crossing number and minimum genus. This allows us to obtain a precise formula for the average minimal genus of oriented rational knots and links with any given crossing number.
We show that the knot lattice homology of a knot in an L-space is equivalent to the knot Floer homology of the same knot (viewed these invariants as filtered chain complexes over the polynomial ring Z/2Z [U]). Suppose that G is a negative definite plumbing tree which contains a vertex w such that G-w is a union of rational graphs. Using the identification of knot homologies we show that for such graphs the lattice homology HF(G)$ is isomorphic to the Heegaard Floer homology HF^-(Y_G) of the corresponding rational homology sphere Y_G.
We study a theory of finite type invariants for null-homologous knots in rational homology 3-spheres with respect to null Lagrangian-preserving surgeries. It is an analogue in the setting of the rational homology of the Goussarov-Rozansky theory for knots in integral homology 3-spheres. We give a partial combinatorial description of the graded space associated with our theory and determine some cases when this description is complete. For null-homologous knots in rational homology 3-spheres with a trivial Alexander polynomial, we show that the Kricker lift of the Kontsevich integral and the Le…
We consider the question of which Dehn surgeries along a given knot bound rational homology balls. We use Ozsváth and Szabó's correction terms in Heegaard Floer homology to obtain general constraints on the surgery coefficients. We then turn our attention to the case of integral surgeries, with particular emphasis on positive torus knots. Finally, combining these results with a lattice-theoretic obstruction based on Donaldon's theorem, we classify which integral surgeries along torus knots of the form $T_{kq\pm 1,q}$ bound rational homology balls.
Constructing Rational Homology 3-Spheres That Bound Rational Homology 4-Balls
本文通过两种新操作构造了能包围有理同调4球的有理同调3球，并探讨了相关猜想的验证。
Rational genus and Heegaard Floer homology
本文重新证明有理Seifert亏格的$d$不变量下界，并将Floer简单纽结的最小亏格结论推广到$Y\ imes I$中的有理切片亏格问题。
Prime power fold cyclic branched covers along smoothly slice knots all bound rational homology balls. This phenomenon, however, does not characterize slice knots. In this paper, we give a new construction of non-slice knots that have the above property. The sliceness obstruction comes from computing twisted Alexander polynomials, and we introduce new techniques to simplify their calculation.
Branched covers and rational homology balls
本文证明三维球面纽结合痕群中存在由二阶元素生成的无限子群，其纽结的分支覆盖界定有理同调球，并揭示典范同态核包含无限生成的二挠子群。
This paper gives new and elementary combinatorial topological proofs of the classification of unoriented and oriented rational knots and links. These proofs are based on the known classification of alternating knots through flyping, and the calculus of continued fractions. We characterize the class of strongly invertible rational links. Rational links are of fundamental importance in the study of DNA recombination.
We study two homomorphisms to the rational homology sphere group. If $ψ$ denotes the inclusion homomorphism from the integral homology sphere group, then using work of Lisca we show that the image of $ψ$ intersects trivially with the subgroup of the rational homology sphere group generated by lens spaces. As corollaries this gives a new proof that the cokernel of $ψ$ is infinitely generated, and implies that a connected sum $K$ of 2-bridge knots is concordant to a knot with determinant 1 if and only if $K$ is smoothly slice. Furthermore, if $β$ denotes the homomorphism from the knot concordanc…
We describe rational knots with any of the possible combinations of the properties (a)chirality, (non-)positivity, (non-)fiberedness, and unknotting number one (or higher), and determine exactly their number for a given number of crossings in terms of their generating functions. We show in particular how Fibonacci numbers occur in the enumeration of fibered achiral and unknotting number one rational knots. Then we show how to enumerate rational knots by crossing number and genus and/or signature. This allows to determine the distribution of these invariants among rational knots. We give also a…
We apply Donaldson's theorem on the intersection forms of definite 4--manifolds to characterize the lens spaces which smoothly bound rational homology 4--dimensional balls. Our result implies, in particular, that every smoothly slice 2--bridge knot is ribbon, proving the ribbon conjecture for 2--bridge knots.
We consider the recently introduced knotting-unknotting game, in which two players take turns resolving crossings in a knot diagram which initially is missing all its crossing information. Once the knot is fully resolved, the winner is decided by whether the knot is equivalent to the unknot. In this paper we determine which player wins under perfect play in the initial positions (knot shadows) that are guaranteed to produce rational knots or sums of rational knots. This is accomplished in part by an analysis of the effects of pseudo Reidemeister moves on positions, and a computer verification …
Let $K$ be a rationally null-homologous knot in a three-manifold $Y$. We construct a version of knot Floer homology in this context, including a description of the Floer homology of a three-manifold obtained as Morse surgery on the knot $K$. As an application, we express the Heegaard Floer homology of rational surgeries on $Y$ along a null-homologous knot $K$ in terms of the filtered homotopy type of the knot invariant for $K$. This has applications to Dehn surgery problems for knots in $S^3$. In a different direction, we use the techniques developed here to calculate the Heegaard Floer homolo…
A rational number $r$ is called a left orderable slope of a knot $K \subset S^3$ if the 3-manifold obtained from $S^3$ by $r$-surgery along $K$ has left orderable fundamental group. In this paper we consider the double twist knots $C(k,l)$ in the Conway notation. For any positive integers $m$ and $n$, we show that if $K$ is a double twist knot of the form $C(2m,-2n)$, $C(2m+1, 2n)$ or $C(2m+1, -2n)$ then there is an explicit unbounded interval $I$ such that any rational number $r \in I$ is a left orderable slope of $K$.
We study the Fox coloring invariants of rational knots. We express the propagation of the colors down the twists of these knots and ultimately the determinant of them with the help of finite increasing sequences whose terms of even order are even and whose terms of odd order are odd.
Integer surgeries rational homology cobordant to lens spaces
本文研究S³中给定纽结上的正整数手术产生与透镜空间有理同伦配边流形的数量，借助Heegaard Floer d-不变量相关工具开展研究。
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。