Kai Ishihara

Yamaguchi University · Faculty of Education

Based on polymer scaling theory and numerical evidence, Orlandini, Tesi, Janse van Rensburg and Whittington conjectured in 1996 that the limiting entropy of knot-type K lattice polygons is the same as that for unknot polygons, and that the entropic critical exponent increases by one for each prime knot in the knot decomposition of K. This Knot Entr...

From polymer models, it has been conjectured that the exponential growth rate of the number of lattice polygons with knot-type $K$ is the same as that for unknot polygons, and that the entropic critical exponent increases by one for each prime knot factor in the knot decomposition of $K$. Here we prove this conjecture for any knot or non-split link...

In this chapter, we will discuss the mathematical method used in analyses of topological polymers. First, we apply graph theory to define a notation for multi-cyclic polymers. We also consider the types of polymers and study the construction method. Second, we apply knot theory to multi-cyclic polymers. We analyze topological isomers derived from k...

We introduce the concept of a handlebody decomposition of a 3-manifold, a generalization of a Heegaard splitting, or a trisection. We show that two handlebody decompositions of a closed orientable 3-manifold are stably equivalent. As an application to materials science, we consider a mathematical model of polycontinuous patterns and discuss a topol...

The control of topological chain properties is essential in biopolymer processes, including DNA transcription and replication promoted by topoisomerase enzymes. Cleavage and re-bonding of DNA chains enables transformation of the chain topologies between a trivial knot (simple ring) and higher knotted constructions, as well as linked counterparts[1,...

There are abundant examples in which the form of objects dictates their functions and properties at all dimensions and scales. In polymer chemistry and materials science, macromolecular structures have mostly been limited to linear or randomly branched forms. However, a variety of precisely controlled polymer topologies have been synthesized using...

As seen in Chapter 2, topology variety increases with increasing multicyclic graphs: e.g., dicyclic, tricyclic, tetracyclic. Multicyclic graphs are classified into spiro, bridge, fused, and hybrid types [1]. In this chapter, we present graph theory definitions of the various multicyclic graphs and characterize each type via construction and decompo...

In this chapter, we introduce polymers, long-chain molecules with diverse chemical compositions and structures. Topology can provide fundamental insights into the principle properties of polymers via their segment structures. We also present a brief description of the following chapters with respect to topological geometry and polymer chemistry.

In this chapter, we discuss topological isomers of multicyclic polymers by using knots, links, and spatial graphs. Essential references on knot theory are [1, 2, 3, 4]. Chemistry applications of knot theory and low-dimensional topology are widely discussed in [5, 6].

In this chapter, we describe the chemistry-based, hierarchical classification procedure in which a series of nonlinear, cyclic, and branched polymer architectures are classified from the molecular graph presentation of alkanes and cycloalkanes. We also discuss a systematic notation protocol for nonlinear polymer topologies, modified from the previo...

In this chapter, we introduce graph theory for analyzing structures of multicyclic polymers. An essential graph theory reference is [1].

In this monograph, we discussed and demonstrated ongoing developments in the unique collaboration of topological geometry and polymer chemistry. We described current topological polymer chemistry by highlighting the diverse nature of polymers with respect to both their chemistry and their line constructions. Topological analyses could provide funda...

A multibranched surface is a 2-dimensional polyhedron without vertices. We introduce moves for multibranched surfaces embedded in a 3-manifold, which connect any two multibranched surfaces with the same regular neighborhoods in finitely many steps.

A multibranched surface is a 2-dimensional polyhedron without vertices. We introduce moves for multibranched surfaces embedded in a 3-manifold, which connect any two multibranched surfaces with the same regular neighborhoods in finitely many steps.

Using a lattice model of polymers in a tube, we define one way to characterise different configurations of a given knot as either "local" or "non-local" and, for several ring polymer models, we provide both theoretical and numerical evidence that, at equilibrium, the non-local configurations are more likely than the local ones. These characterisati...

In Escherichia coli DNA replication yields interlinked chromosomes. Controlling topological changes associated with replication and returning the newly replicated chromosomes to an unlinked monomeric state is essential to cell survival. In the absence of the topoisomerase topoIV, the site-specific recombination complex XerCD- dif-FtsK can remove re...

In Escherichia coli DNA replication yields interlinked chromosomes. Controlling topological changes associated with replication and returning the newly replicated chromosomes to an unlinked monomeric state is essential to cell survival. In the absence of the topoisomerase topoIV, the site-specific recombination complex XerCD- dif -FtsK can remove r...

We show that the monodromy for a genus one, fibered knot can have at most two monodromy equivalence classes of once-unclean arcs. We use this to classify all monodromies of genus one, fibered knots that possess once-unclean arcs, all manifolds containing genus one fibered knots with generalized crossing changes resulting in another genus one fibere...

Knots are ubiquitous in nature and their analysis has important implications in a wide variety of fields including fluid dynamics, material science and molecular and structural biology. In many systems particles are found in crowded environments hence it is natural to rigorously characterize the properties of knots in confined volumes. In this work...

The tangle method, first introduced by Ernst and Sumners in the late 1980s, uses tools from knot theory and low-dimensional topology to analyze the topological changes induced by site-specific recombination on a circular DNA substrate. Often, a recombination reaction can be modeled by a band surgery. Here we provide a brief description of the tangl...

We categorise coherent band (aka nullification) pathways between knots and 2-component links. Additionally, we characterise the minimal coherent band pathways (with intermediates) between any two knots or 2-component links with small crossing number. We demonstrate these band surgeries for knots and links with small crossing number. We apply these...

Significance Newly replicated circular chromosomes are topologically linked. XerC/XerD- dif (XerCD- dif )–FtsK recombination acts in the replication termination region of the Escherichia coli chromosome to remove links introduced during homologous recombination and replication, whereas Topoisomerase IV removes replication links only. Based on gel m...

We characterize cutting arcs on fiber surfaces producing fiber surfaces. As corollaries we characterize band surgeries and generalized crossing changes between fibered links.

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 X...

Volume confinement is a key determinant of the topology and geometry of a polymer. However, the direct relationship between the two is not fully understood. For instance, recent experimental studies have constructed P4 cosmids, i.e. P4 bacteriophages whose genome sequence and length have been artificially engineered and have shown that upon extract...

A handlebody-knot is a handlebody embedded in the 3-sphere. We improve Luo’s result about markings on a surface, and show that an IH-move is sufficient to investigate handlebody-knots with spatial trivalent graphs without cut-edges. We also give fundamental moves with a height function for handlebody-tangles, which helps us to define operator invar...

Which chiral knots can be unknotted in a single step by a + to − (+−) crossing change, and which by a − to + (−+) crossing change? Numerical results suggest that if a knot with 6 or fewer crossings can be unknotted by a +− crossing change then it cannot be unknotted by a −+ one, and vice versa. However, we exhibit one chiral 8-crossing knot and one...

During site-specific recombination, the topology of circular DNA can change, e.g. unknotted molecules can become knotted or linked. We model Xer site-specific recombinations as the mathematical operation of band surgeries. In this paper, we consider band surgeries on knots with 7 and fewer crossings and links with 8 and fewer crossings.

Cho and McCullough gave a numerical parameterization of the collection of all tunnels of all tunnel number 1 knots and links in the 3-sphere. Here we give an algorithm for finding the parameter of a given tunnel by using its Heegaard diagram.

Gordon and Luecke showed that knots are determined by their complements. Therefore a non-trivial Dehn surgery on a non-trivial knot does not yield the 3-sphere. But the situation for links is different from that for knots. Berge constructed some examples of Dehn surgeries of 2-component links yielding the 3-sphere with interesting properties. By ex...

Knots are found in DNA as well as in proteins, and they have been shown to be good tools for structural analysis of these molecules. An important parameter to consider in the artificial construction of these molecules is the minimum number of monomers needed to make a knot. Here we address this problem by characterizing, both analytically and numer...

Let L be a 2-component link in the 3-sphere S 3 consisting of a knot K and its meridian c. Let b:[0,1]×[0,1]→S 3 be an embedding such that b([0,1]×[0,1])∩K=b([0,1]×{0}) and b([0,1]×[0,1])∩c=b([0,1]×{1}). Then we obtain a knot L b by replacing b([0,1]×{0,1}) in L with b({0,1}×[0,1]). We call L b a band sum of L with the band b. If K is a trivial kno...

The tunnel number of knots directly gives the Heegaard genus of their exteriors. For the link case, if we admit in addition splittings of link exteriors into two compression bodies, things become more complicated. In this paper we introduce a concept of types of Heegaard splittings for compact orientable 3-manifolds and give relations between these...