The incompatibility of crossing number and bridge number for knot diagrams

We study methods for computing the bridge number of a knot from a knot diagram. We prove equivalence between a geometric and a combinatorial definition of the bridge number of a knot diagram. For each notion of diagrammatic bridge number considered, we find crossing number minimizing knot diagrams which fail to minimize bridge number. Furthermore, we construct a family of minimal crossing diagrams for which the difference between diagrammatic bridge number and the actual bridge number of the knot grows to infinity.     

Virtual knots undetected by 1- and 2-strand bracket polynomials

Kishino's knot is not detected by the fundamental group or the bracket polynomial. However, we can show that Kishino's knot is not equivalent to the unknot by applying either the 3-strand bracket polynomial or the surface bracket polynomial. In this paper, we construct two non-trivial virtual knot diagrams, KD and Km, that are not detected by the 1-strand or the 2-strand bracket polynomial. From these diagrams, we construct two infinite families of non-classical virtual knot diagrams that are not detected by the bracket polynomial. Additionally, these virtual knot diagrams are trivial as flats.     

Invariants of knot diagrams

We construct a new order 1 invariant for knot diagrams. We use it to determine the minimal number of Reidemeister moves needed to pass between certain pairs of knot diagrams. J. Hass was partially supported by an NSF grant.     

The Canonical Genus of a Classical and Virtual Knot

A diagram D of a knot defines the corresponding Gauss Diagram G _D . However, not all Gauss diagrams correspond to the ordinary knot diagrams. From a Gauss diagram G we construct closed surfaces F _G and S _G in two different ways, and we show that if the Gauss diagram corresponds to an ordinary knot diagram D, then their genus is the genus of the canonical Seifert surface associated to D. Using these constructions we introduce the virtual canonical genus invariant of a virtual knot and find estimates on the number of alternating knots of given genus and given crossing number.     

Unknot Diagrams Requiring a Quadratic Number of Reidemeister Moves to Untangle

Given any knot diagram E, we present a sequence of knot diagrams of the same knot type for which the minimum number of Reidemeister moves required to pass to E is quadratic with respect to the number of crossings. These bounds apply both in S ² and in ?².     

The determination of the pairs of two-bridge knots or links with Gordian distance one

We thoroughly determine the pairs of two-bridge knots or links with Gordian distance one. In addition, we examine the Gordian distance between a Montesinos knot (or link) and a two-bridge knot (or link).

     

Expansions for the Bollob��s-Riordan Polynomial of Separable Ribbon Graphs

We define 2-decompositions of ribbon graphs, which generalize 2-sums and tensor products of graphs. We give formulae for the Bollobás-Riordan polynomial of such a 2-decomposition, and derive the classical Brylawski formula for the Tutte polynomial of a tensor product as a (very) special case. This study was initially motivated from knot theory, and we include an application of our formulae to mutation in knot diagrams.     

The almost alternating diagrams of the trivial knot

Bankwitz characterized the alternating diagrams of the trivialknot. A non-alternating diagram is called almost alternatingif one crossing change makes the diagram alternating. We characterizethe almost alternating diagrams of the trivial knot. As a corollary,we determine the unknotting number one alternating knots withthe property that the unknotting operation can be done on itsalternating diagram. Received July 3, 2007. Revised September 29, 2008.     

Epstein–Penner Decompositions and Thrice Punctured Spheres in Hyperbolic 3-Manifolds

Let M be a cusped hyperbolic 3-manifold containing an incompressible thrice punctured sphere S. Suppose that M is not the Whitehead link complement. We prove that a certain arc on S is isotopic to an edge of a Euclidean decomposition of M. By using the above result, we relate alternating knot diagrams and the canonical decompositions. Let K be an alternating hyperbolic knot. On a reduced alternating knot diagram of K, if we replace one of the crossings with a large number of half twists, the polar axis of the crossing is isotopic to an edge of the canonical decomposition for the resulting knot.     

Topological realizations of ortho-projection graphs

We discuss the production of ortho-projection graphs from alternating knot diagrams, and introduce a more general construction of such graphs from “splittings” of closed, non-orientable surfaces. As our main result, we prove that this new topological construction generates all ortho-projection graphs. We present a minimal example of an ortho-projection graph that does not arise from a knot diagram, and provide a surface-splitting that realizes this graph.     

The Compressibility of Checkerboard Surfaces of Link Diagrams

Consider the checkerboard surfaces defined by some link diagrams. When they are not orientable, one considers the boundary surfaces of small regular neighborhoods of them. This article studies the compressibility problem of these kinds of surfaces in the link complements. The problem is solved by devising a normalization theory for the compressing discs, which brings up an algorithm to read out compressibility directly from the link diagrams. As an application of the algorithm, the compressibility changes under Reidermeister moves are studied. Diagrams from the knot tables are also studied, and surprisingly, some of them are shown to define completely compressible surfaces of this kind. Infinitely many examples of non-alternating knot diagrams with incompressible surfaces of this kind are also constructed.     

Closed incompressible surfaces in the complements of positive knots

We show that any closed incompressible surface in the complement of a positive knot is algebraically non-split from the knot, positive knots cannot bound non-free incompressible Seifert surfaces and that the splittability and the primeness of positive knots and links can be seen from their positive diagrams. Received: June 28, 2000     

Every Reidemeister move is needed for each knot type

We show that every knot type admits a pair of diagrams that cannot be made identical without using Reidemeister -moves. The proof is compatible with known results for the other move types, in the sense that every knot type admits a pair of diagrams that cannot be made identical without using all of the move types.

     

Knot graphs

We consider the equivalence classes of graphs induced by the unsigned versions of the Reidemeister moves on knot diagrams. Any graph that is reducible by some finite sequence of these moves, to a graph with no edges, is called a knot graph. We show that the class of knot graphs strictly contains the set of delta‐wye graphs. We prove that the dimension of the intersection of the cycle and cocycle spaces is an effective numerical invariant of these classes. © 2000 John Wiley & Sons, Inc. J Graph Theory 34: 100–111, 2000     

A lower bound for the number of Reidemeister moves of type III

We study the number of Reidemeister type III moves using Fox n-colorings of knot diagrams.     

Vanishing of 3-loop Jacobi diagrams of odd degree

We prove the vanishing of the space of 3-loop Jacobi diagrams of odd degree. This implies that no 3-loop Vassiliev invariant can distinguish between a knot and its inverse.     

Triple-loop networks with arbitrarily many minimum distance diagrams

Minimum distance diagrams are a way to encode the diameter and routing information of multi-loop networks. For the widely studied case of double-loop networks, it is known that each network has at most two such diagrams and that they have a very definite form (“L-shape”).In contrast, in this paper we show that there are triple-loop networks with an arbitrarily big number of associated minimum distance diagrams. For doing this, we build-up on the relations between minimum distance diagrams and monomial ideals.     

On region crossing change and incidence matrix

In a recent work of Ayaka Shimizu, she studied an operation named region crossing change on link diagrams, which was proposed by Kishimoto, and showed that a region crossing change is an unknotting operation for knot diagrams. In this paper, we prove that the region crossing change on a 2-component link diagram is an unknotting operation if and only if the linking number of the diagram is even. Besides, we define an incidence matrix of a link diagram via its signed planar graph and its dual graph. By studying the relation between region crossing change and incidence matrix, we prove that a signed planar graph represents an n-component link diagram if and only if the rank of the associated incidence matrix equals c n + 1, where c denotes the size of the graph.     

On the Alexander polynomials of knots with Gordian distance one

We consider a condition on a pair of the Alexander polynomials of knots which are realizable by a pair of knots with Gordian distance one. We show that there are infinitely many mutually disjoint infinite subsets in the set of the Alexander polynomials of knots such that every pair of distinct elements in each subset is not realizable by any pair of knots with Gordian distance one. As one of the subsets, we have an infinite set containing the Alexander polynomials of the trefoil knot and the figure eight knot. We also show that every pair of distinct Alexander polynomials such that one is the Alexander polynomial of a slice knot is realizable by a pair of knots of Gordian distance one, so that every pair of distinct elements in the infinite subset consisting of the Alexander polynomials of slice knots is realizable by a pair of knots with Gordian distance one. These results solve problems given by Y. Nakanishi and by I. Jong.