arXiv research

We give a short proof that if a non-trivial band sum of two knots results in a tight fibered knot, then the band sum is a connected sum. In particular, this means that any prime knot obtained by a non-trivial band sum is not tight fibered. Since a positive L-space knot is tight fibered, a non-trivial band sum never yie…

Connected sum affects crossing numbers of flat virtual knots.

Study connects knot polynomials with number theory sums.

Knots can be constructed and decomposed using Murasugi sums of Seifert surfaces.

Knots connected via a trivial band sum to connected sum.

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

Formula connects knot complements' invariants.

The paper studies knot Floer homology under Murasugi sum and establishes graded isomorphisms.

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

We consider the question, asked by Friedl, Livingston and Zentner, of which sums of torus knots are concordant to alternating knots. After a brief analysis of the problem in its full generality, we focus on sums of two torus knots. We describe some effective obstructions based on Heegaard Floer homology.

Proofs knot homology connected sums using grid complexes.

The warping sum $e(K)$ of a knot $K$ is the minimal value of the sum of the warping degrees of a minimal diagram of $K$ with both orientations. In this paper, knots $K$ with $e(K) \le 3$ are characterized, and some knots $K$ with $e(K)=4$ are given.

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

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…

Proves special alternating knots cannot be decomposed as non-trivial band sums.

Study Murasugi sum in 4D for knotted surfaces, defining arborescent surfaces.

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

There is a nonribbon 2-link all of whose components are trivial 2-knots and one of whose band-sums is a nonribbon 2-knot.

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.

Proved colored HOMFLY-PT polynomials for specific knots.

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

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

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

Knots from a specific band sum have similar homologies but are distinct.

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

When two boundary-parabolic representations of knot groups are given, we introduce the connected sum of these representations and show several natural properties including the unique factorization property. Furthermore, the complex volume of the connected sum is the sum of each complex volumes modulo $iπ^2$ and the twi…

We prove that the tunnel number of the sum of n knots is at least n.

The 2-twist spun trefoil is an example of a sphere that is knotted in 4-dimensional space. Here this example is shown to be distinct from the same sphere with the reversed orientation. To demonstrate this fact a state-sum invariant for classical knots and knotted surfaces is developed via a cohomology theory of racks a…

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…

New method uses quandle rings to distinguish knots and their mirrors.

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

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

We construct two knot invariants. The first knot invariant is a sum constructed using linking numbers. The second is an invariant of flat knots and is a formal sum of flat knots obtained by smoothing pairs of crossings. This invariant can be used in conjunction with other flat invariants, forming a family of invariants…

An upper bound of the superbridge index of the connected sum of two knots is given in terms of the braid index of the summands. Using this upper bound and minimal polygonal presentations, we give an upper bound in terms of the superbridge index and the bridge index of the summands when they are torus knots. In contrast…

We develop a dimer model for the Alexander polynomial of a knot. This recovers Kauffman's state sum model for the Alexander polynomial using the language of dimers. By providing some additional structure we are able to extend this model to give a state sum formula for the twisted Alexander polynomial of a knot dependin…

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

Prove integrality of genus-$g$ indices with adjoint Reidemeister torsions for twist knots and meridians.

We present new computations of tight shapes obtained using the constrained gradient descent code RIDGERUNNER for 544 composite knots with 12 and fewer crossings, expanding our dataset to 943 knots and links. We use the new data set to analyze two outstanding conjectures about tight knots, namely that the ropelengths of…

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.

We consider the recently introduced knotting-unknotting game, in which two players take turns resolving crossings in a knot diagram which initially is missing all its crossing information. Once the knot is fully resolved, the winner is decided by whether the knot is equivalent to the unknot. In this paper we determine …

Proves conjecture about integer sums of torus knot torsions.

Paper calculates L-invariant and L*-invariant for complex surface sums.

We define two new families of invariants for (3-manifold, graph) pairs which detect the unknot and are additive under connected sum of pairs and (-1/2)-additive under trivalent vertex sum of pairs. The first of these families is closely related to both bridge number and tunnel number. The second of these families is a …

Paper shows a lower bound for composite knots crossing number.

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

We establish a formula for the SL(2,C) Casson invariant of spliced sums of homology spheres along knots. Along the way, we show that the SL(2,C) Casson invariant vanishes for spliced sums along knots in the 3-sphere.

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

It is a very old conjecture that the crossing number of knots is additive under connected sum. In other words, if K#K' is the connected sum of knots K and K', then does the equality c(K#K') = c(K) + c(K') hold? We prove that c(K#K') is at most c(K) + c(K') and at least (c(K) + c(K'))/152.