Related papers: Plumbers' knots
For a genus-1 1-bridge knot in the 3-sphere, that is, a (1,1)-knot, a middle tunnel is a tunnel that is not an upper or lower tunnel for some (1,1)-position. Most torus knots have a middle tunnel, and non-torus-knot examples were obtained…
We develop a skein exact sequence for knot Floer homology, involving singular knots. This leads to an explicit, algebraic description of knot Floer homology in terms of a braid projection of the knot.
We define homotopy-theoretic invariants of knots in prime 3-manifolds. Fix a knot J in a prime 3-manifold M. Call a knot K in M concordant to J if it cobounds a properly embedded annulus with J in MxI, and call K J-characteristic if there…
We study petal diagrams of knots, which provide a method of describing knots in terms of permutations in a symmetric group $S_{2n+1}$. We define two classes of moves on such permutations, called trivial petal additions and crossing…
Ng constructed an invariant of knots in ${\mathbb{R}}^3$, a combinatorial knot contact homology. Extending his study, we construct an invariant of surface-knots in ${\mathbb{R}}^4$ using diagrams in ${\mathbb{R}}^3$.
It is known that for every knotted curve in space, there is a line intersecting it in four places, a quadrisecant. Comparing the order of the four points along the line and knot we can distinguish three types of quadrisecants; the…
It is known that each of the successive quotient groups of the grope and solvable filtrations of the knot concordance group has an infinite rank subgroup. The generating knots of these subgroups are constructed using iterated doubling…
We study spectral gaps of cellular differentials for finite cyclic coverings of knot complements. Their asymptotics can be expressed in terms of irrationality exponents associated with ratios of logarithms of algebraic numbers determined by…
We give a combinatorial treatment of transverse homology, a new invariant of transverse knots that is an extension of knot contact homology. The theory comes in several flavors, including one that is an invariant of topological knots and…
In the loop representation the quantum constraints of gravity can be solved. This fact allowed significant progress in the understanding of the space of states of the theory. The analysis of the constraints over loop dependent wavefunctions…
We study inequalities between integer-valued knot invariants arising from classical knot theory, four-dimensional topology, knot homologies, and knot polynomials. We present a directed graph consisting of 48 inequalities between 33 knot…
The knot Floer complex and the concordance invariant $\varepsilon$ can be used to define a filtration on the smooth concordance group. We exhibit an ordered subset of this filtration that is isomorphic to $\mathbb{N} \times \mathbb{N}$ and…
We study knots whose $\mathrm{SL}_2(\mathbb{C})$-character varieties have a component of dimension greater than one. We call such knots $\mathcal{X}$-large and introduce two diagrammatic constructions that produce $\mathcal{X}$-large knots.…
We study knots in $\mathbb{S}^3$ obtained by the intersection of a minimal surface in $\mathbb{R}^4$ with a small 3-sphere centered at a branch point. We construct examples of new minimal knots. In particular we show the existence of…
We study collections of planar curves that yield diagrams for all knots. In particular, we show that a very special class called potholder curves carries all knots. This has implications for realizing all knots and links as special types of…
The Alexander polynomial of a knot has been generalized in three different ways to give twisted invariants. The resulting invariants are usually referred to as twisted Alexander polynomials, higher-order Alexander polynomials and…
The list of knots with up to 10 crossings is commonly referred to as the Rolfsen Table. This paper presents a way to generate the Rolfsen table in a simple, clear, and reproducible manner. The methods we use are similar to those used by J.…
We describe an error in the proof of a key proposition, which was necessary for the proof of the main result. Alternate proofs of the main result are given by Ozsvath-Stipsicz-Szabo and Dai-Hom-Stoffregen-Truong.
The motivation of this work is to define cohomology classes in the space of knots that are both easy to find and to evaluate, by reducing the problem to simple linear algebra. We achieve this goal by defining a combinatorial graded cochain…
Knots are deeply entangled with every branch of science. One of the biggest open challenges in knot theory is to formalise a knot invariant that can unambiguously and efficiently distinguish any two knotted curves. Additionally, the…