Naoko Kamada
The notion of chckerboard colorability for virtual links and abstract links is introduced. We study the Jones polynomials of virtual links and abstruct links. It is proved that a certain property of the Jones polynomials of classical links is valid for virtual links which admit checkerboard colorings.
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In this paper we give two new criteria of detecting the checkerboard colorability of virtual links by using odd writhe and arrow polynomial of virtual links, respectively. By applying new criteria, we prove that 6 virtual knots are not checkerboard colorable, leaving only one virtual knot whose checkerboard colorability is unknown among all virtual knots up to four classical crossings.
On the determinant of checkerboard colorable virtual knots
研究了可棋盘着色虚拟结的行列式，通过升序多项式系数扩展Conway多项式，揭示行列式模8与Arf不变量的关系。
A New Polynomial for Checkerboard-Colorable 4-Valent Virtual Graphs
本文通过欧拉回路展开为棋盘可着色四价虚拟图定义新多项式，并据此给出相应虚拟链环 Kauffman-Jones 多项式的新组合刻画。
Checkerboard framings are an extension of checkerboard colorings for virtual links. According to checkerboard framings, in 2017, Dye obtained an independent invariant of virtual links: the cut point number. Checkerboard framings and cut points can be used as a tool to extend other classical invariants to virtual links. We prove that one of the conjectures in Dye's paper is correct. Moreover, we analyze the connection and difference between checkerboard framing obtained from virtual link diagram by adapting cut points and twisted link diagram obtained from virtual link diagram by introducing ba…
The $3$-colorable subgroup of Thompson's group and tricolorability of links
本文基于Jones关于Thompson群F表示的研究，探讨了F的3-可着色子群F与纽结理论中3-可着色链环的关系，并证明了F中所有元素均给出3-可着色链环。
This monograph derives direct and concrete relations between colored Jones polynomials and the topology of incompressible spanning surfaces in knot and link complements. Under mild diagrammatic hypotheses that arise naturally in the study of knot polynomial invariants (A- or B-adequacy), we prove that the growth of the degree of the colored Jones polynomials is a boundary slope of an essential surface in the knot complement. We show that certain coefficients of the polynomial measure how far this surface is from being a fiber in the knot complement; in particular, the surface is a fiber if and…
We extend the Gordon-Litherland pairing to links in thickened surfaces, and use it to define signature, determinant, and nullity invariants for links that bound (unoriented) spanning surfaces. The invariants are seen to depend only on the $S^*$-equivalence class of the spanning surface. We prove a duality result relating the invariants from one $S^*$-equivalence class of spanning surfaces to the restricted invariants of the other. Using Kuperberg's theorem, these invariants give rise to well-defined invariants of checkerboard colorable virtual links. The determinants can be applied to determin…
On Alternateness of the Jones-Kauffman Polynomial of Virtual Links
本文研究虚拟链环的Jones-Kauffman多项式的交错性，定义拟交错虚拟链环类，利用扭转技术构造无限族，并给出棋盘可着色链环的拟行列式的连通状态展开。
A virtual link diagram is called normal if the associated abstract link diagram is checkerboard colorable, and a virtual link is normal if it has a normal diagram as a representative.In this paper, we introduce a method of converting a virtual link diagram to a normal virtual link diagram by use of the double covering technique. We show that the normal virtual link diagrams obtained from two equivalent virtual link diagrams are related by generalized Reidemeister moves and Kauffman flypes.
We introduce stable equivalence classes of oriented links in orientable three-manifolds that are orientation $I$-bundles over closed but not necessarily orientable surfaces. We call these twisted links, and show that they subsume the virtual knots introduced by L. Kauffman, and the projective links introduced by Yu. Drobotukhina. We show that these links have unique minimal genus three-manifolds. We use link diagrams to define an extension of the Jones polynomial for these links, and show that this polynomial fails to distinguish two-colorable links over non-orientable surfaces from non-two-co…
The Jones Polynomial from a Goeritz Matrix
本文提出一种显式算法，从 Goeritz 矩阵计算链环图的 Kauffman 括号并恢复 Jones 多项式，同时发展余图拟阵理论将括号多项式推广至任意对称整数矩阵，置于增厚曲面链环背景下。
The $p$-colorable subgroup of Thompson's group
针对大于2的任意奇数$p$，研究者定义汤普森群$F$的$p$-可着色子群（其非平凡元素生成$p$-可着色纽结链环），并证明该子群与特定Brown--Thompson群同构。
We study the head and tail of the colored Jones polynomial while focusing mainly on alternating links. Various ways to compute the colored Jones polynomial for a given link give rise to combinatorial identities for those power series. We further show that the head and tail functions only depend on the reduced checkerboard graphs of the knot diagram. Moreover the class of head and tail functions of prime alternating links forms a monoid.
A virtual link diagram is called normal if the associated abstract link diagram is checkerboard colorable, and a virtual link is normal if it has a normal diagram as a representative. Normal virtual links have some properties similar to classical links.In this paper, we introduce a method of converting a virtual link diagram to a normal virtual link diagram. We show that the normal virtual link diagrams obtained by this method from two equivalent virtual link diagrams are equivalent. We discuss the relationship between this method and some invariants of virtual links.
The Jones polynomial of an alternating link is a certain specialization of the Tutte polynomial of the (planar) checkerboard graph associated to an alternating projection of the link. The Bollobas-Riordan-Tutte polynomial generalizes the Tutte polynomial of planar graphs to graphs that are embedded in closed oriented surfaces of higher genus. In this paper we show that the Jones polynomial of any link can be obtained from the Bollobas-Riordan-Tutte polynomial of a certain oriented ribbon graph associated to a link projection. We give some applications of this approach.
Asymptotically CAT(0) metrics, Z-structures, and the Farrell-Jones Conjecture
本文证明可着色层级双曲群具有渐近CAT(0)度量，由此构造Z-结构，并应用于证明包括额外大型Artin群在内的许多此类群满足Farrell-Jones猜想。
In this paper, we study the Khovanov homology of an alternating virtual link $L$ and show that it is supported on $g+2$ diagonal lines, where $g$ equals the virtual genus of $L$. Specifically, we show that $Kh^{i,j}(L)$ is supported on the lines $j=2i-σ_ξ+2k-1$ for $0\leq k\leq g+1$ where $σ_{ξ^*}(L)+2g= σ_ξ(L)$ are the signatures of $L$ for a checkerboard coloring $ξ$ and its dual $ξ^*$. Of course, for classical links, the two signatures are equal and this recovers Lee's $H$-thinness result for $Kh^{*,*}(L)$. Our result applies more generally to give an upper bound for the homological width o…
We show that the Kauffman bracket $[L]$ of a checkerboard colorable virtual link $L$ is an evaluation of the Bollobás-Riordan polynomial $R_{G_L}$ of a ribbon graph associated with $L$. This result generalizes Thistlethwaite's celebrated theorem relating the Kauffman bracket with the Tutte polynomial of planar graphs.
For a signed cyclic graph G, we can construct a unique virtual link L by taking the medial construction and convert 4-valent vertices of the medial graph to crossings according to the signs. If a virtual link can occur in this way then we say that the virtual link is graphical. In the article we shall prove that a virtual link L is graphical if and only if it is checkerboard colorable. On the other hand, we introduce a polynomial F[G] for signed cyclic graphs, which is defined via a deletion-marking recursion. We shall establish the relationship between F[G] of a signed cyclic graph G and the …
Remarks on some maximal subgroups of $F$ and on the $\vec{F}$-index of knots
本文研究了Thompson群$F$中某些极大子群的性质，并探讨了结的$\vec{F}$-指数在改变结方向时的变化上限。
The Jones polynomial can be expressed in terms of spanning trees of the graph obtained by checkerboard coloring a knot diagram. We show there exists a complex generated by these spanning trees whose homology is the reduced Khovanov homology. The spanning trees provide a filtration on the reduced Khovanov complex and a spectral sequence that converges to its homology. For alternating links, all differentials on the spanning tree complex are zero and the reduced Khovanov homology is determined by the Jones polynomial and signature. We prove some analogous theorems for (unreduced) Khovanov homolo…
Thompson's Group $V$ and Virtual Link Theory
该研究将琼斯关于汤普森群的纽结表示方案推广到汤普森群 $V$，定义了其到加厚曲面CC虚拟纽结的满射，构造了定向子群 $\vec{V}$ 及其酉表示，实现了所有定向几乎经典虚拟纽结。
Analogue of Goeritz matrices for computation of bipartite HOMFLY-PT polynomials
本文提出一种改进的Goeritz方法，用于计算双部链接的HOMFLY-PT多项式，通过纯代数运算实现高效计算。
Let $D$ be an oriented classical or virtual link diagram with directed universe $\vec{U}$. Let $C$ denote a set of directed Euler circuits, one in each connected component of $U$. There is then an associated looped interlacement graph $L(D,C)$ whose construction involves very little geometric information about the way $D$ is drawn in the plane; consequently $L(D,C)$ is different from other combinatorial structures associated with classical link diagrams, like the checkerboard graph, which can be difficult to extend to arbitrary virtual links. $L(D,C)$ is determined by three things: the structu…
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