Related papers: Achirality of knots and links

Families of alternating knots (links) and tangles are studied using as building block the conway defined as the twisting of two strands. The regular representation of knots assumes the projection has the minimal number of overpassings, and…

We show that if the branched double cover of an alternating link arises as $p/q \in \mathbb{Q} \setminus \mathbb{Z}$ surgery on a knot in $S^3$, then this is exhibited by a rational tangle replacement in an alternating diagram.

We use Heegaard Floer homology to give obstructions to unknotting a knot with a single crossing change. These restrictions are particularly useful in the case where the knot in question is alternating. As an example, we use them to classify…

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…

Using the techniques on annulus twists, we observe that $6_3$ has infinitely many non-characterizing slopes, which affirmatively answers a question by Baker and Motegi. Furthermore, we prove that the knots $6_2$, $6_3$, $7_6$, $7_7$, $8_1$,…

The contents of this 6-page paper have been subsumed into the 13-page paper, "A note on closed 3-braids", arXiv:0802.1072 [math.GT]. This paper is correct, but contains less information than the new one. The topological classification of…

We use technology from sutured manifold theory and the theory of Heegaard splittings to relate genus reducing crossing changes on knots in S^3 to twists on surfaces arising in circular Heegaard splittings for knot complements. In a separate…

We present a complete classification of spherical knotoids with up to six crossings and conjecture that our classification up to seven crossings is complete. Our work extends the tradition of knot tabulation to the setting of knotoids…

The A-polynomial of a knot in S^3 defines a complex plane curve associated to the set of representations of the fundamental group of the knot exterior into SL(2,C). Here, we show that a non-trivial knot in S^3 has a non-trivial…

We study some comparison between a bilinear cohomology pairing in local coefficients and the Blanchfield pairing of a knot. We show that the former pairing is an $S$-equivalent invariant, and give a criterion to a relation between the two…

We present two families of knots which have straight number higher than crossing number. In the case of the second family, we have computed the straight number explicitly. We also give a general theorem about alternating knots that states…

Binary representations of the trefoil and other knots of up to ten crossings in the simple cubic lattice were created. The BiEntropy of each knot was computed using a variety of binary encodings and compared against controls. This showed…

We introduce an invariant of alternating knots and links (called here WRP), namely a pair of integer polynomials associated with their two checkerboard planar graphs from their minimal diagram. We prove that the invariant is well-defined…

Let $K\subset S^3$ be a knot, $X:= S^3\setminus K$ its complement, and $\mathbb{T}$ the circle group identified with $\mathbb{R}/\mathbb{Z}$. To any oriented long knot diagram of $K$, we associate a quadratic polynomial in variables…

Traditionally, knot theorists have considered projections of knots where there are two strands meeting at every crossing. A triple crossing is a crossing where three strands meet at a single point, such that each strand bisects the…

Let K' be a knot that admits no cosmetic crossing changes and let C be a non-trivial, prime, non-cable knot. Then any knot that is a satellite of C with winding number zero and pattern K' admits no cosmetic crossing changes. As a…

Bankwitz characterized an alternating diagram representing the trivial knot. A non-alternating diagram is called almost alternating if one crossing change makes the diagram alternating. We characterize an almost alternaing diagram…

We show that the triple-crossing number of any knot is greater or equal to twice its (canonical) genus and we show an even stronger bound in the case of links. As an application we show that this bound is strong enough to obtain the…

A conjecture proposed by J. Tripp in 2002 states that the crossing number of any knot coincides with the canonical genus of its Whitehead double. In the meantime, it has been established that this conjecture is true for a large class of…

Final revision. To appear in the Journal of Differential Geometry. This paper studies knots that are transversal to the standard contact structure in $\reals^3$, bringing techniques from topological knot theory to bear on their transversal…