Narrow Results

We consider the classical pretzel knots $P(a_1,a_2,a_3)$, where $a_1,a_2,a_3$ are positive odd integers. By using continuous paths of elliptic ${\mathrm{\text{SL}}}_2(ℝ)$-representations, we show that (i) the 3-manifold obtained by $\frac{m}{l}$-surgery on $P(a_1,a_2,a_3)$ has left orderable fundamental group if $\frac{m}{l}<1$ and (ii) the $n\mathrm{\text{th}}$-cyclic branched cover of $P(a_1,a_2,a_3)$ has left orderable fundamental group if $n>2\pi /arccos(1-2/(1+a_1{a}_2+a_2{a}_3+a_3{a}_1))$.

We classify Dehn surgeries on (p,q,r) pretzel knots resulting in a manifold M(α) having cyclic fundamental group and analyze those leading to a finite fundamental group. The proof uses the theory of cyclic and finite surgeries developed by Culler, Shalen, Boyer, and Zhang. In particular, Culler-Shalen seminorms play a central role.

We show that the -character variety of the (-2, 3, n) pretzel knot consists of two (respectively three) algebraic curves when 3 ∤ n (respectively 3 | n) and given an explicit calculation of the Culler-Shalem seminorms of these curves. Using this calculation, we describe the fundamental polygon and Newton polygon for these knots and give a list of Dehn surgerise yielding a manifold with finite or cyclic fundamental group. This constitutes a new proof of property P for these knots.

The Kauffman–Harary conjecture states that for any reduced alternating diagram K of a knot with a prime determinant p, every non-trivial Fox p-coloring of K assigns different colors to its arcs. We generalize this conjecture by stating it in terms of homology of the double cover of S³. In this way we extend the scope of the conjecture to all prime alternating links of arbitrary determinants. We first prove the Kauffman–Harary conjecture for pretzel knots and then we generalize our argument to show the generalized Kauffman–Harary conjecture for all Montesinos links. Finally, we discuss on the relation between the conjecture and Menasco's work on incompressible surfaces in exteriors of alternating links.

We calculate the dihedral quandle cocycle invariants of twist-spins of alternating odd pretzel knots. The calculation leads us to the conclusion that there exist non-ribbon 2-knots which admit a non-trivial coloring by the dihedral quandle R_p and all of whose cocycle invariants derived from ℤ_p-valued 3-cocycles on R_p take value in ℤ ⊂ ℤ[ℤ_p] for any odd prime integer p.

Computing unlinking number is usually very difficult and complex problem, therefore we define BJ-unlinking number and recall Bernhard–Jablan conjecture stating that the classical unknotting/unlinking number is equal to the BJ-unlinking number. We compute BJ-unlinking number for various families of knots and links for which the unlinking number is unknown. Furthermore, we define BJ-unlinking gap and construct examples of links with arbitrarily large BJ-unlinking gap. Experimental results for BJ-unlinking gap of rational links up to 16 crossings, and all alternating links up to 12 crossings are obtained using programs LinKnot and K2K. Moreover, we propose families of rational links with arbitrarily large BJ-unlinking gap and polyhedral links with constant non-trivial BJ-unlinking gap. Computational results suggest existence of families of non-alternating links with arbitrarily large BJ-unlinking gap.

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.

In this paper, we give infinitely many non-Haken hyperbolic genus three 3-manifolds each of which has a finite cover whose induced Heegaard surface from some genus three Heegaard surface of the base manifold is reducible but can be compressed into an incompressible surface. This result supplements [A. Casson and C. Gordon, Reducing Heegaard splittings, Topology Appl. 27 (1987) 275–283] and extends [J. Masters, W. Menasco and X. Zhang, Heegaard splittings and virtually Haken Dehn filling, New York J. Math. 10 (2004) 133–150].

Let K_s be a (-2, 3, 2s + 1)-type pretzel knot (s ≧ 3) and E_Ks(p/q) be a closed manifold obtained by Dehn surgery along K_s with a slope p/q. We prove that if q > 0, p/q ≧ 4s + 7 and p is odd, then E_Ks(p/q) cannot contain an ℝ-covered foliation. This result is an extended theorem of a part of works of Jun for (-2, 3, 7)-pretzel knot.

We compute integer valued knot concordance invariants of a family of general pretzel knots if the invariants are equal to the negative values of signatures for alternating knots. Examples of such invariants are Rasmussen s-invariants and twice Ozsváth–Szabó knot Floer homology τ-invariants. We use the crossing change inequalities of Livingston and the fact that pretzel knots are almost alternating. As a consequence, for the family of pretzel knots given in this paper, s-invariants are twice τ-invariants.

We prove that many four-strand pretzel knots of the form $K=P(2n,m,-2n\pm 1,-m)$ are not topologically slice, even though their positive mutants $P(2n,-2n\pm 1,m,-m)$ are ribbon. We use the sliceness obstruction of Kirk and Livingston [Twisted Alexander invariants, Reidemeister torsion, and Casson–Gordon invariants, Topology38 (1999) 635–661], related to the twisted Alexander polynomials associated to prime power cyclic covers of knots.

We study 2-adjacency between a classical (3-strand) pretzel knot and the trefoil knot or the figure-eight knot by using the early results about classical pretzel knots and their polynomials and elementary number theory. We show that except for the trefoil knot or the figure-eight knot, a nontrivial classical pretzel knot is not 2-adjacent to either of them, and vice versa.

Pseudo-alternating knots and links are defined constructively via their Seifert surfaces. By performing Murasugi sums of primitive flat surfaces, such a knot or link is obtained as the boundary of the resulting surface. Conversely, it is hard to determine whether a given knot or link is pseudo-alternating or not. A major difficulty is the lack of criteria to recognize whether a given Seifert surface is decomposable as a Murasugi sum.

In this paper, we propose a new idea to identify non-pseudo-alternating knots. Combining with the uniqueness of minimal genus Seifert surface obtained through sutured manifold theory, we demonstrate that two infinite classes of pretzel knots are not pseudo-alternating.

A pair of surgeries on a knot is chirally cosmetic if they result in homeomorphic manifolds with opposite orientations. Using recent methods of Ichihara, Ito, and Saito, we show that, except for the (2, 5) and (2, 7)-torus knots, the genus 2 and 3 alternating odd pretzel knots do not admit any chirally cosmetic surgeries. Further, we show that for a fixed genus, at most finitely many alternating odd pretzel knots admit chirally cosmetic surgeries.

The Δ-unknotting number for a knot is defined to be the minimum number of Δ-unknotting operations which deform the knot into the trivial knot. We determine the Δ-unknotting numbers for torus knots, positive pretzel knots, and positive closed 3-braids.

We prove that genus one, three-bridge knots are pretzel knots.