Narrow Results

Bleiler and Eudave-Muñoz showed that composite ribbon number one knots have two-bridge summands. In this paper, we show that there exists an infinite family of composite ribbon number one knots which have arbitrary large bridge numbers.

We show that every p-fold strictly-cyclic branched covering of a b-bridge link in S³ admits a p-symmetric Heegaard splitting – in the sense of Birman and Hilden – of genus g = (b - 1)(p - 1). This gives a complete converse of one of the results of the two authors. Moreover, we introduce the concept of weakly p-symmetric Heegaard splittings and prove similar results in this more general context.

There are obvious inequalities relating the Nakanishi index of a knot, the bridge number, the degree 2n of the Alexander polynomial and the length of the chain of Alexander ideals. We give examples for every positive value of n to show that these bounds are sharp.

In this paper, we explore the issue of minimizing the degree sequence for torus knots. We find the minimal degree sequence for torus knots of type (p, 2p-1) for any integer p ≥2. We use some results from algebraic geometry to prove our main result.

In this paper we prove the following result: for coprime positive integers p and q with p < q, if r is the least positive integer such that 2p-1 and q + r are coprime, then the minimal degree sequence for a torus knot of type (p, q) is the triple (2p - 1, q + r, d) or (q + r, 2p - 1, d) where q + r + 1 ≤ d ≤ 2q - 1.

We introduce a new numerical invariant of knots and links from the descending diagrams. It is considered to live between the unknotting number and the bridge number. Some fundamental results and an incomplete table of the invariant for knots with 8-crossings or less are given.

We compare the path-width of a planar graph and the bridge number of the corresponding link. Is there a relationship between these two numbers? This study is an attempt to find a graph theoretical method of finding the bridge number of a link.

We show that the bridge number of a tunnel number t knot in S³ with respect to an unknotted genus t surface is bounded below by a function of the distance of the Heegaard splitting induced by the t tunnels. It follows that for any natural number n, there is a tunnel number one knot in S³ that is not (1, n).

In the previous paper, the representativity of a non-trivial knot K was defined as

In this paper, we will show that a (p, q, r)-pretzel knot has the representativity 3 if and only if (p, q, r) is either ±(-2, 3, 3) or ±(-2, 3, 5). We also show that an algebraic knot has the representativity less than or equal to 3.

The 2-bridge number of knots was introduced by Hass, Rubinstein and Thompson [Knots and k-width, Geom. Dedicata143 (2009) 7–18] as a natural generalization of the bridge number introduced by Schubert [Über eine numerisch knoteninvariante, Math. Z.61 (1954) 245–288]. We show that the 2-bridge number of a torus knot of type (p, q) is (p - 1)q + 2 if 1 < p < q.

In our works with Stoimenow, Vdovina and with Byberi, we introduced the virtual canonical genus g_vc(K) and the virtual bridge number vb(K) invariants of virtual knots. One can see from the definitions that for a classical knot K the values of these invariants are less than or equal to the classical canonical genus g_c(K) and the bridge number b(K) respectively. We use Manturov's projection from the category of virtual knot diagrams to the category of classical knot diagrams, to show that for every classical knot type K we have g_vc(K) = g_c(K) and vb(K) = b(K).

We construct various functorial maps (projections) from virtual knots to classical knots. These maps are defined on diagrams of virtual knots; in terms of Gauss diagram each of them can be represented as a deletion of some chords. The construction relies upon the notion of parity. As corollaries, we prove that the minimal classical crossing number for classical knots. Such projections can be useful for lifting invariants from classical knots to virtual knots. Different maps satisfy different properties.

A Gauss code for a virtual knot diagram is a sequence of crossing labels, each repeated twice and assigned a + or - symbol to identify over and undercrossings. Eliminate these symbols and what remains is a Gauss code for the shadow of the diagram, one type of virtual pseudodiagram. While it is now impossible to determine which particular virtual diagram the shadow resulted from, we can consider the collection of all diagrams, called resolutions of the shadow, that would yield such a code. We compute the average virtual bridge number over all these diagrams and show that for a shadow with n classical precrossings, the average virtual bridge number is .

Using Gauss diagrams, one can define the virtual bridge number vb(K) and the welded bridge number wb(K), invariants of virtual and welded knots satisfying wb(K) ≤ vb(K). If K is a classical knot, Chernov and Manturov showed that vb(K) = br(K), the bridge number as a classical knot, and we ask whether the same thing is true for welded knots. The welded bridge number is bounded below by the meridional rank of the knot group G_K, and we use this to relate this question to a conjecture of Cappell and Shaneson. We show how to use other virtual and welded invariants to further investigate bridge numbers. Among them are Manturov's parity and the reduced virtual knot group Ḡ_K, and we apply these methods to address Questions 6.1–6.3 and 6.5 raised by Hirasawa, Kamada and Kamada in their paper [Bridge presentation of virtual knots, J. Knot Theory Ramifications 20(6) (2011) 881–893].

Meier and Zupan proved that an orientable surface ${𝒦}$ in $S^4$ admits a tri-plane diagram with zero crossings if and only if ${𝒦}$ is unknotted, so that the crossing number of ${𝒦}$ is zero. We determine the minimal crossing numbers of nonorientable unknotted surfaces in $S^4$, proving that $c({{𝒫}}^{n,m})=max\{1,|n-m|\}$, where ${{𝒫}}^{n,m}$ denotes the connected sum of $n$ unknotted projective planes with normal Euler number $+2$ and $m$ unknotted projective planes with normal Euler number $-2$. In addition, we convert Yoshikawa’s table of knotted surface ch-diagrams to tri-plane diagrams, finding the minimal bridge number for each surface in the table and providing upper bounds for the crossing numbers.

Let $\alpha$ be a map from the set of all knot types ${𝒦}$ to a set $X$. Let $\beta$ be a map from ${𝒦}$ to a set $Y$. We define the relation between $\alpha$ and $\beta$ to be the image of a map $(\alpha ,\beta )$ from ${𝒦}$ to $X\times Y$ sending an element $K$ of ${𝒦}$ to $(\alpha (K),\beta (K))$. We determine the relations between $\alpha$ and $\beta$ for certain $\alpha$ and $\beta$ such as crossing number, unknotting number, bridge number, braid index, genus and canonical genus. This is a study of geography problem in knot theory.

We use the new approach of braiding sequences to prove exponential upper bounds for the number of Vassiliev invariants on knots with bounded braid index, bounded bridge number and arborescent knots.

We prove, that any Vassiliev invariant of degree k is determined by its values on knots with braid index at most k + 1.

We define the virtual bridge number vb(K) and the virtual unknotting number vu(K) invariants for virtual knots. For ordinary knots K they are closely related to the bridge number b(K) and the unknotting number u(K) and we have vu(K) ≤ u(K), vb(K) ≤ b(K).

There are no ordinary knots K with b(K) = 1. We show there are infinitely many homotopy classes of virtual knots each of which contains infinitely many isotopy classes of K with vb(K) = 1. In fact for each i ∈ ℕ there exists K virtually homotopic (but not virtually isotopic) to the unknot with vb(K) = 1 and vu(K) = i.