Related papers: Homotopy ribbon concordance and Alexander polynomi…
In this article, we identify the existence of a divisibility relationship between the number of ring homomorphisms and surjective group homomorphisms. We demonstrate that for finite cyclic structures, the number of ring homomorphisms from…
Ribbon concordances between knots generalize the notion of ribbon knots. Agol, building on work of Gordon, proved ribbon concordance gives a partial order on knots in $S^3$. In previous work, the author and Greene conjectured that positive…
Link homotopy has been an active area of research for knot theorists since its introduction by Milnor in the 1950s. We introduce a new equivalence relation on spatial graphs called component homotopy, which reduces to link homotopy in the…
Given a link diagram L we construct spectra X^j(L) so that the Khovanov homology Kh^{i,j}(L) is isomorphic to the (reduced) singular cohomology H^i(X^j(L)). The construction of X^j(L) is combinatorial and explicit. We prove that the…
In this paper we look at the knot complement problem for L-space $\mathbb{Z}$-homology spheres. We show that an L-space $\mathbb{Z}$-homology sphere $Y$ cannot be obtained as a non-trivial surgery along a knot $K\subset Y$. As a…
It is known that the minimal degree of the Jones polynomial of a positive knot is equal to its genus, and the minimal coefficient is 1. We extend this result to almost positive links and partly identify the 3 following coefficients for…
The energy spectrum of the three-particle Hamiltonian obtained by replacing the two-body trigonometric potential of the Sutherland problem by a three-body one of a similar form is derived. When expressed in appropriate variables, the…
We show that if $V^3$ is a handlebody in $\R^3$, with curves $J_1, ..., J_g \subset \partial V$ which are the attaching curves for a Heegaard splitting of a homology sphere, then there exists a homeomorphism $h\colon V \to V$ so that each…
Three-component links in the 3-dimensional sphere were classified up to link homotopy by John Milnor in his senior thesis, published in 1954. A complete set of invariants is given by the pairwise linking numbers p, q and r of the…
We give simple homological conditions for a rational homology 3-sphere Y to have infinite order in the rational homology cobordism group, and for a collection of rational homology spheres to be linearly independent. These translate…
Inspired by the combinatorial constructions in earlier work of the authors that generalized the classical Alexander polynomial to a large class of spatial graphs with a balanced weight on edges, we show that the value of the Alexander…
In this survey we summarize results regarding the Kauffman bracket, HOMFLYPT, Kauffman 2-variable and Dubrovnik skein modules, and the Alexander polynomial of links in lens spaces, which we represent as mixed link diagrams. These invariants…
We determine lens surgeries (i.e.\ Dehn surgery yielding a lens space) along the $n$-twisted Whitehead link. To do so, we first give necessary conditions to yield a lens space from the Alexander polynomial of the link as: (1) $n=1$ (i.e.…
Free ribbon lemma that every free sphere-link in the 4-sphere is a ribbon sphere-link is shown in an earlier paper by the author. In this paper, another proof of this lemma is given.
The trace of $n$-framed surgery on a knot in $S^3$ is a 4-manifold homotopy equivalent to the 2-sphere. We characterise when a generator of the second homotopy group of such a manifold can be realised by a locally flat embedded 2-sphere…
We introduce the notion of adjacency in three-manifolds. A three-manifold $Y$ is $n$-adjacent to another three-manifold $Z$ if there exists an $n$-component link in $Y$ and surgery slopes for that link such that performing Dehn surgery…
Fox conjectured the Alexander polynomial of an alternating knot is trapezoidal, i.e. the coefficients first increase, then stabilize and finally decrease in a symmetric way. Recently, Hirasawa and Murasugi further conjectured a relation…
We extend the notion of link colorings with values in an Alexander quandle to link colorings with values in a module $M$ over the Laurent polynomial ring $\Lambda_{\mu}=\mathbb{Z}[t_1^{\pm1},\dots,t_{\mu}^{\pm1}]$. If $D$ is a diagram of a…
When does the double cover of the three-sphere branched along an alternating link bound a rational homology ball? Heegaard Floer homology generates a necessary condition for it to bound: the link's chessboard lattice must be cubiquitous,…
A braid-like isotopy for links in 3-space is an isotopy which uses only those Reidemeister moves which occur in isotopies of braids. We define a refined Jones polynomial and its corresponding Khovanov homology which are, in general, only…