Stable-ribbon surface-links have unique handle-irreducible summands.
problem Characterize handle-irreducible summands of stable-ribbon surface-links.
method Combining old results on stably trivial surface-links and surface-knots with infinite cyclic fundamental groups.
result Every handle-irreducible summand of a stably trivial surface-link is a trivial 2-link.
Generalizes ribbonness result for surface-links.
problem Characterizing ribbon surface-links.
method Analyzes handle-irreducible summands and uses equivalences.
result Every stable-ribbon surface-link is a ribbon surface-link.
We introduce the notion of a ribbon-clasp surface-link, which is a generalization of a ribbon surface-link. We generalize the notion of a normal form on embedded surface-links to the case of immersed surface-links and prove that any (immersed) surface-link can be described in a normal form. It is known that an embedded…
Paper extends method of presenting surface-links to immersed surface-links.
problem Presenting immersed surface-links in 4-space.
method Use marked graph diagrams with moves preserving isotopy classes.
result Extended method for presenting immersed surface-links.
A new method for describing surface-links in 4-space.
problem Describing surface-links in 4-dimensional space.
method Introducing a plat form method using braided surfaces.
result Every surface-link can be described in a plat form.
Surface-links with specific groups are always trivial.
problem Understanding when surface-links are trivial.
method Analyzing surface-links with meridian-based free fundamental groups.
result Disconnected surface-links with meridian-based free fundamental groups are trivial.
New concept of quasi-ribbon surface-links simplifies complex surface-links.
problem Complexity in surface-links of trivial components.
method Introducing quasi-ribbon surface-links as a generalization of ribbon surface-links.
result Every F-link of trivial components on a surface F with at most one aspheric component is a quasi-ribbon surface-link.
Invariants from biquasile colorings distinguish surface-links.
problem Counting and distinguishing surface-links.
method Coloring oriented surface-links using biquasiles and marked graph diagrams.
result Invariants can distinguish closed surface-links and cobordisms.
For an oriented surface link S, we can take a satellite construction called a 2-dimensional braid over S, which is a surface link in the form of a covering over S. We demonstrate that 2-dimensional braids over surface links are useful for showing the distinctness of surface links. We investigate non-trivial examp…
Revised proof shows ribbonness of surface-links in 4-sphere.
problem Determining ribbonness of surface-links in 4-sphere.
method Surgery along fusion 1-handle systems to prove ribbonness.
result Surface-links in 4-sphere are ribbon if certain conditions are met.
Enhanced bikei modules distinguish unoriented and non-orientable surface-links.
problem Distinguishing unoriented and non-orientable surface-links.
method Extending biquandle module invariants to unoriented surface-links using bikei modules.
result Enhanced bikei modules are more effective at distinguishing non-orientable surface-links than bikei homset cardinality alone.
This paper shows how to create surface-links with many triple points.
problem Creating surface-links with a large number of triple points.
method Analogous to knot diagrams, the paper uses broken sheet diagrams to project surface-links and analyze their triple points.
result There are non-split surface-links with arbitrarily many triple points.
Free surface-links are shown to be ribbon links.
problem Characterizing surface-links as ribbon links.
method Four proofs are provided, including a stabilization approach.
result Every free surface-link is a ribbon surface-link.
Paper classifies surface-links using charts with specific properties.
problem Classifying surface-links with specific charts.
method Using quandle colorings to differentiate charts representing different surface-links.
result Charts in the second class represent different surface-links.
New method finds infinitely many surface knots with specific bridge numbers.
problem Finding numerical invariants for surface links.
method Colorings of surface links by keis to prove bridge number existence.
result Existence of infinitely many surface knots with bridge number n for n ≥ 4.
Enhances knot counting using mosaic diagrams.
problem Counting and classifying surface-links and knots.
method Marked graph diagrams and mosaic numbers.
result Established bounds on mosaic numbers for surface-links.
Upper bounds for surface-links in the Yoshikawa table are estimated.
problem Estimating Kirby-Thompson invariants of surface-links.
method Using tri-plane diagrams and L-, L*-invariants.
result Upper bounds for surface-links in the Yoshikawa table are obtained.
In this paper, we formulate a construction of ideal coset invariants for surface-links in 4-space using invariants for knots and links in 3-space. We apply the construction to the Kauffman bracket polynomial invariant and obtain an invariant for surface-links called the Kauffman bracket ideal coset invariant of sur…
Paper discusses biquandle cohomology and invariants for surface-links.
problem Computing invariants for surface-links using biquandle theory.
method Developed (co)homology theory of biquandles, introduced new invariants using broken surface diagrams and marked graph diagrams.
result Constructs new invariants for surface-links using biquandle theory.
Characterizes knot groups and symmetric quandles of surface-links.
problem Characterize knot groups and symmetric quandles of surface-links.
method Used plat presentations for surface-links and closed 2-dimensional braids.
result Generalized results to include non-orientable surface-links and showed that dihedral quandles can be realized as symmetric quandles of surface-links.
We study surface links whose link groups are free abelian, and construct various stimulating and highly non-trivial examples of such surface links.
A new mosaic system for immersed surface-links is introduced.
problem Constructing a mosaic system for immersed surface-links.
method Using singular marked graph diagrams, the mosaic number for immersed surface-links is defined and discussed.
result A mosaic system for immersed surface-links is established.
Paper computes knot symmetric quandle for surface-links and finds infinitely many distinct surface-knots.
problem Computing knot symmetric quandle for surface-links.
method Using plat form presentations, the paper computes the knot symmetric quandle for surface-links.
result Infinitely many distinct surface-knots of genus g with plat indices m.
Bounding twist number of surface links using polynomial coefficients.
problem Bounding the twist number of alternating surface links.
method Introducing a generalized homological Kauffman bracket and applying it to surface link diagrams.
result A bound for the twist number of alternating surface links in terms of polynomial coefficients.
Extends Milnor invariants to surface-links using cut-diagrams.
problem Tackles invariants for surface-links in 4-space.
method Introduces cut-diagrams and groups associated to them.
result Yields concordance and link-homotopy invariants for surface-links.
New invariants for surface-links using biquandle modules.
problem Defining invariants for surface-links that are not determined by Alexander ideals.
method Enhancing biquandle counting invariant with biquandle modules.
result New invariants not determined by Alexander ideals.
We characterize the first Alexander Z[Z]-modules of ribbon surface-links in the 4-sphere fixing the number of components and the total genus, and then the first Alexander Z[Z]-modules of surface-links in the 4-sphere fixing the number of components. Using the result of ribbon torus-links, we also characterize the first…
In-degree quiver polynomials for surface-links computed.
problem Computing in-degree quiver polynomials for surface-links.
method Defined using a quandle and set of endomorphisms, computed for surface-links with ch-index up to 10.
result Example computations for surface-links with ch-index up to 10.
This paper compiles and calculates triple point numbers for surface-links in Yoshikawa's table.
problem Determining the triple point number of surface-links in Yoshikawa's table.
method Using broken sheet diagrams, the paper compiles known triple point numbers and calculates or bounds the remaining ones.
result Compilation and calculation of triple point numbers for surface-links in Yoshikawa's table.
A marked graph diagram is a link diagram possibly with marked 4-valent vertices. S. J. Lomonaco, Jr. and K. Yoshikawa introduced a method of representing surface-links by marked graph diagrams. Specially, K. Yoshikawa gave local moves on marked graph diagrams, nowadays called Yoshikawa moves. It is now known that two…
We introduce the notion of a quandle with a good involution and its homology groups. Carter et al. defined quandle cocycle invariants for oriented links and oriented surface-links. By use of good involutions, quandle cocyle invariants can be defined for links and surface-links which are not necessarily oriented or orie…
A. S. Lipson constructed two state models yielding the same classical link invariant obtained from the Kauffman polynomial F(a,z). In this paper, we apply Lipson's state models to marked graph diagrams of surface-links, and observe when they induce surface-link invariants.
This paper establishes a correspondence between biquandle and quandle colorings for classical and surface links.
problem Finding refined invariants for classical and surface links using biquandles.
method Explicit one-to-one correspondence between biquandle colorings and quandle colorings.
result Biquandle homotopy invariants and quandle homotopy invariants are equivalent.
The paper studies knitted surfaces and surface-links, showing their isotopy and closure properties.
problem Understanding the isotopy and closure properties of knitted surfaces and surface-links.
method Analyzing the structure and closure of knitted surfaces and surface-links in R4. result Any surface-link is ambient isotopic to the closure of a 2-dimensional knit.
Paper introduces tensor product of quandles for knot classification.
problem Classifying knot invariants of surface-links with 1-handles.
method Introduces tensor product of quandles and applies to surface-links.
result Tensor product of knot quandles/classical quandles can classify surface-link invariants.
There are many studies about twisted Alexander invariants for knots and links, but calculations of twisted Alexander invariants for spatial graphs, handlebody-knots, and surface-links have not been demonstrated well. In this paper, we give some remarks to calculate the twisted Alexander ideals for spatial graphs, handl…
Unified invariant for immersed surface-links using biquandle cocycles.
problem Invariants for immersed surface-links in 4-space.
method Biquandle cohomology, Roseman moves, singular biquandle 3-cocycles.
result Invariance of state-sum invariant under all generating moves, including singular move (h).
This note proves properties of surface-links with trivial components.
problem Properties of surface-links with trivial components.
method Analyzes closed oriented disconnected surfaces and their links.
result For certain surface-links, trivial components result in ribbon links.
By using the cohomology theory of quandles, quandle cocycle invariants and shadow quandle cocycle invariants are defined for oriented links and surface-links via broken surface diagrams. By using symmetric quandles, symmetric quandle cocycle invariants are also defined for unoriented links and surface-links via broken …
We submit a new way to detect pairs of non-cobordant surface-links. We find a new example of a pair of non-cobordant surface-links with the following properties: Orr invariant, Cochran sequence, Sato-Levine invariant, the alinking number and one of Stallings's theorems cannot distinguish them. However our new way can d…
Proves a theorem for surface links, simplifying diagrams of links.
problem Understanding minimal diagrams of surface links.
method Two-variable generalization of Jones polynomial for surface links.
result Reduced alternating diagrams of surface links have minimal crossing number.
Roseman moves are seven types of local modification for surface-link diagrams in 3-space which generate ambient isotopies of surface-links in 4-space. In this paper, we focus on Roseman moves involving triple points, one of which is the famous tetrahedral move, and discuss their independence. For each diagram of an…
The paper constructs exotic surface links in 4-ball, proving their Brunnian nature.
problem Investigating exotic surface links in 4-ball.
method Two constructions of Brunnian exotic surface links, using satellite operations and covering links.
result Provided constructions of exotic Brunnian surface links with varied properties.
A quandle coloring obstruction prevents a specific link from being ribbon concordant.
problem Obstructing a specific link from being ribbon concordant.
method Symmetric dihedral quandle coloring analysis.
result A symmetric dihedral quandle of order 4 cannot color a specific surface-link, obstructing ribbon concordance.
For an oriented surface link F in R4, we consider a satellite construction of a surface link, called a 2-dimensional braid over F, which is in the form of a covering over F. We introduce the notion of an m-chart on a surface diagram π(F)⊂R3 of F, which is a finite graph on $π(F)…
Study uses graph Laplacians to analyze surface links.
problem Analyzing virtual genus of surface links.
method Laplacian matrices of weighted graphs in surfaces are used to define invariants.
result Obtained information about virtual genus.
We consider surface links in the 4-space which are presented by the form of simple branched coverings over the standard torus, which we call torus-covering links. In this paper, we study unknotting numbers of torus-covering links. In some cases, we can determine the unknotting numbers.
The paper extends surface link coloring theory to triplane diagrams and knots.
problem Understanding the topological properties of knots and surfaces in 4-space.
method Translated Niebrzydowski's theory of region colorings to triplane diagrams and movies of knots, providing inequalities and applications.
result Yoshikawa's 2-knots 91 and 102 are non-invertible.