Related papers: Every Reidemeister move is needed for each knot ty…
Classical knot theory deals with {\em diagrams} and {\em invariants}. By means of horizontal {\em trisecants}, we construct a new theory of classical braids with invariants valued in {\em pictures}. These pictures are closely related to…
Motivated by the study of ribbon knots we explore symmetric unions, a beautiful construction introduced by Kinoshita and Terasaka 50 years ago. It is easy to see that every symmetric union represents a ribbon knot, but the converse is still…
We prove the first nontrivial worst-case lower bounds for two closely related problems. First, $\Omega(n^{3/2})$ degree-1 reductions, series-parallel reductions, and $\Delta$Y transformations are required in the worst case to reduce an…
We construct many examples of non-slice knots in 3-space that cannot be distinguished from slice knots by previously known invariants. Using Whitney towers in place of embedded disks, we define a geometric filtration of the 3-dimensional…
We define an invariant of tangles and framed tangles given a finite crossed module and a pair of functions, called a Reidemeister pair, satisfying natural properties. We give several examples of Reidemeister pairs derived from racks,…
We develop a theory of link projections to trivalent spines of 3-manifolds. We prove a Reidemeister Theorem providing a set of combinatorial moves sufficient to relate the projections of isotopic links. We also show that any link admits a…
We introduce \textit{dual graph diagrams} representing oriented knots and links. We use these combinatorial structures to define corresponding algebraic structures we call \textit{biquasiles} whose axioms are motivated by dual graph…
We use grid diagrams to investigate the Ozsvath-Szabo concordance invariant tau, and to prove that |tau(K_1)-tau(K_2)|<=g, whenever there is a genus g knot cobordism joining K_1 to K_2. This leads to an entirely grid diagram-based proof of…
Twisted Alexander invariants of knots are well-defined up to multiplication of units. We get rid of this multiplicative ambiguity via a combinatorial method and define normalized twisted Alexander invariants. We then show that the…
We give a first example of 2-knots with the same knot group but different knot quandles by analyzing the knot quandles of twist spins. As a byproduct of the analysis, we also give a classification of all twist spins with finite knot…
An involutive link is a link which is invariant under the standard rotation by 180 degrees in $S^3$. We establish an equivariant analogue of the work of Carter and Saito aimed at studying equivariant cobordisms between involutive links.…
An open question akin to the slice-ribbon conjecture asks whether every ribbon knot can be represented as a symmetric union. Next to this basic existence question sits the question of uniqueness of such representations. Eisermann and Lamm…
For a knot $K$ with $\Delta_K(t)\doteq t^2-3t+1$ in a homology $3$-sphere, let $M$ be the result of $2/q$-surgery on $K$. We show that an appropriate assumption on the Reidemeister torsion of the universal abelian covering of $M$ implies…
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…
The $m$-trace of a knot is the $4$-manifold obtained from $\mathbf{B}^4$ by attaching a $2$-handle along the knot with $m$-framing. In 2015, Abe, Jong, Luecke and Osoinach introduced a technique to construct infinitely many knots with the…
We show that in any symmetric monoidal category, if a weight for colimits is absolute, then the resulting colimit of any diagram of dualizable objects is again dualizable. Moreover, in this case, if an endomorphism of the colimit is induced…
It is known that every surface-link can be presented by a marked graph diagram, and such a diagram presentation is unique up to moves called Yoshikawa moves. G. Kuperberg introduced a regular isotopy invariant, called the quantum A_2…
A knot is an an embedding of a circle into three-dimensional space. We say that a knot is unknotted if there is an ambient isotopy of the embedding to a standard circle. By representing knots via planar diagrams, we discuss the problem of…
We study the following problem: Given $k$ paths that share the same vertex set, is there a simultaneous geometric embedding of these paths such that each individual drawing is monotone in some direction? We prove that for any dimension $d…
Let $K,K'$ be two-bridge knots of genus $n,k$ respectively. We show the necessary and sufficient condition of $n$ in terms of $k$ that there exists an epimorphism from the knot group of $K$ onto that of $K'$.