arXiv research

Connected sum affects crossing numbers of flat virtual knots.

Formula connects knot complements' invariants.

New tiles allow efficient knot mosaics for small knots.

Knots' Morse-Novikov number behaves additively under connected sum and unchanged by cabling.

If a knot is a nontrivial connected sum of positive torus knots, then it is not concordant to an L-space knot.

Paper proves corner connection tiles can represent knots with fewer tiles.

Knots connected via a trivial band sum to connected sum.

Every shadow can be a knot or a connected sum of trefoils.

Study connects knot polynomials with number theory sums.

Proofs knot homology connected sums using grid complexes.

We study the behavior of Legendrian and transverse knots under the operation of connected sums. As a consequence we show that there exist Legendrian knots that are not distinguished by any known invariant. Moreover, we classify Legendrian knots in some non-Legendrian simple knot types.

We prove a formula for the conjugation action on the knot Floer complex of the connected sum of two knots. Using the formula we construct a homomorphism from the smooth concordance group to an abelian group consisting of chain complexes with homotopy automorphisms, modulo an equivalence relation. Using our connected su…

The Strong Slope Conjecture is proven for specific knot types.

New knot invariant defined using sheaf theory and SL(2,C) connections.

Study on knot unknotting numbers and their behavior under connected sums.

Study concordance of alternating torus knots to L-space knots.

The paper characterizes and contrasts knots with high 4D clasp numbers.

A knot is an a-small knot if its exterior does not contain closed incompressible surfaces disjoint from some incompressible Seifert surface for the knot. Using circular thin position for knots we prove that the handle number is additive under the connected sum of two a-small knots. As a consequence the Morse-Novikov nu…

Deep learning uncovers patterns between knot types.

A short proof for a theorem about composite knots.

Study knotted surfaces in simply-connected 4-manifolds with trivial boundary.

The purpose of this paper is to discuss how topology and geometry provide, in many instances, the connective tissue that enables logical comprehension. We illustrate this theme with many examples including Venn diagrams, knot diagrams, knot-logical diagrams and an arrow of reference that elucidates self-reference and G…

Proves properties of instanton knot Floer homology and connected sum formula.

Knots 4_1 and 5_1 can be paired to show unknotting number is not additive.

Paper generalizes Satoh's result on knots and projective planes.

Research confirms non-trivial knots in S^3 do not admit purely cosmetic surgeries.

The paper defines and analyzes the adjoint Reidemeister torsion for connected sums of knots.

Introduces knot theory via surface perspectives.

We study the set of Crowell states for alternating knot projections and show that for prime alternating knots the space of states for a reduced projection is connected, a result similar to that for Kauffman states. As an application we give a new proof of a result of Ozsvath and Szabo characterizing (2,2n+1) torus knot…

Study of singular knots connects knot theory with quantum algebra.

We present two different representations of (1,1)-knots and study some connections between them. The first representation is algebraic: every (1,1)-knot is represented by an element of the pure mapping class group of the twice punctured torus. The second representation is parametric: every (1,1)-knot can be represented…

New connections found between knot invariants and Rozansky-Witten theory.

Study on factorizations of knot polynomials for up to 12 crossings.

New hyperbolic knots with convex Upsilon invariants constructed.

In this paper, we describe the relation between the study of closed connected surfaces embedded in $S^3$ and the theory of handlebody-knots. By Fox's theorem, a pair of handlebody-knots is associated to a closed connected surface embedded in $S^3$ in the sense that their exterior components are pairwise homeomorphic. W…

Paper connects knot invariants and Morse flow loops.

We study 2-string free tangle decompositions of knots with tunnel number two. As an application, we construct infinitely many counter-examples to a conjecture in the literature stating that the tunnel number of the connected sum of prime knots doesn't degenerate by more than one.

The AJ conjecture is verified for certain connected sums of torus knots.

New subgroup found in knot homology concordance group.

The Gluck twist preserves the diffeomorphism type of certain satellite 2-knots.

KnotMosaics package simplifies knot theory computations in SageMath.

In this article we prove that, if $X$ is a smooth $4$-manifold containing an embedded double node neighborhood, all knot surgery $4$-manifolds $X_K$ are mutually diffeomorphic to each other after a connected sum with $\mathbb{CP}^2$. Hence, by applying to the simply connected elliptic surface $E(n)$, we also show that …

The paper connects knot volume to $A$-polynomial structure.

New invariant for knotted tori, similar to classical invariant.

Knot contact homology is an invariant of knots derived from Legendrian contact homology which has numerous connections to the knot group. We use basic properties of knot groups to prove that knot contact homology detects every torus knot. Further, if the knot contact homology of a knot is isomorphic to that of a cable …

Agent finds unknotting sequences for complex knots.

By proving a connected sum formula for the Legendrian invariant $λ_+$ in knot Floer homology we exhibit infinitely many transversely non simple knots.

Jones polynomial bounds and crossing numbers of knots.