Fox-Milnor theorem extended to knots in thickened surfaces.
problem Extending classical knot theory to knots in thickened surfaces.
method Using Milnor torsion to prove a Fox-Milnor theorem for knots in a thickened surface.
result A Fox-Milnor theorem for concordant knots in a thickened surface.
This paper shows how to unknot 2-manifolds in 3-sphere using twistings.
problem Can all submanifolds of 3-sphere be unknotted by twistings?
method Using Fox's re-embedding theorem and twistings.
result Not all closed 2-manifolds can be unknotted by twistings.
Fox's trapezoidal conjecture for four-strand Turk's head knots is proven.
problem Proving log-concavity of the coefficient sequence of Dn(z) for four-strand Turk's head knots. method Four-block smoothing theorem for products of reciprocal quartics.
result The coefficient sequence of Dn(z) is log-concave. We introduce a norm on the real 1-cohomology of finite 2-complexes determined by the Euler characteristics of graphs on these complexes. We also introduce twisted Alexander-Fox polynomials of groups and show that they give rise to norms on the real 1-cohomology of groups. Our main theorem states that for a finite 2-com…
Understanding non-Haken 3-manifolds is central to many current endeavors in 3-manifold topology. We describe some results for closed orientable surfaces in non-Haken manifolds, and extend Fox's theorem for submanifolds of the 3-sphere to submanifolds of general non-Haken manifolds. In the case where the submanifold has…
The paper studies Fox pairings of Poincaré duality groups using group cohomology.
problem Understanding Fox pairings of Poincaré duality groups.
method Using group cohomology, the paper computes cohomology groups of Fox pairings.
result The paper suggests fundamental and higher Fox pairings.
Alternative proof of Alexander polynomial trapezoid conjecture using dimers.
problem Alexander polynomial trapezoid conjecture for special alternating links.
method Dimer model approach to Alexander polynomial.
result Shorter and more accessible proof of Azarpendar, Juhász, and Kálmán's result.
The paper connects surface embeddings to handlebody-knots and explores their properties.
problem Understanding the relationship between surface embeddings and handlebody-knots.
method Using Fox's theorem, the paper associates handlebody-knots to surface embeddings and constructs new surfaces.
result For every genus two prime bi-knotted surface, one handlebody-knot is irreducible and the other is reducible.
The paper computes groups and modules for wheel graphs using Fibonacci and Chebyshev polynomials.
problem Computing groups and modules for wheel graphs.
method Utilized Fibonacci and Chebyshev polynomials to compute the Reduced Fox Coloring Group and Alexander-Burau-Fox Module.
result Computed groups and modules for wheel graphs using Fibonacci and Chebyshev polynomials.
Log-concave coefficient sequences for two-bridge knots proved.
problem Proving log-concavity of Alexander polynomial coefficient sequences for alternating knots.
method Introducing a polynomial Δ(t) associated to Christoffel words and proving its log-concavity. result Strong Fox conjecture for two-bridge knots proved.
For a compact connected 3-submanifold with connected boundary in the 3-sphere, we relate the existence of a Seifert surface system for a surface with a Dehn surgery along a null-homologous link. As its corollary, we obtain a refinement of the Fox's re-embedding theorem.
Study on knots using 17 colors, finding specific color assignments.
problem Understanding the minimum number of colors needed for Fox colorings of knots.
method Investigated 17-colorable knots and their diagrams.
result Found that exactly 6 out of 17 colors are used in diagrams of 17-colorable knots.
Let M be an orientable closed connected 3-manifold. We introduce the notion of amalgamated Heegaard genus of M with respect to a closed separating 2-manifold F, and use it to show that the following two statements are equivalent: (i) a compact connected 3-manifold Y can be embedded in M so that the exterior of the imag…
We introduce a notion of a Fox pairing in a group algebra and use Fox pairings to define automorphisms of the Malcev completions of groups. These automorphisms generalize to the algebraic setting the action of the Dehn twists in the group algebras of the fundamental groups of surfaces. This work is inspired by the Kawa…
Study Kuperberg invariants for sutured manifolds using Fox calculus and Reidemeister torsion.
problem Computing Kuperberg invariants for sutured manifolds.
method Fox calculus and Reidemeister torsion.
result Kuperberg invariants can be computed via Fox calculus and extended to polynomial invariants.
We construct solutions to the set-theoretic Yang-Baxter equation using braid group representations in free group automorphisms and their Fox differentials. The method resembles the extensions of groups and quandles.
This paper has two-fold goal: it provides gentle introduction to Knot Theory starting from 3-coloring, the concept introduced by R. Fox to allow undergraduate students to see that the trefoil knot is non-trivial, and ending with statistical mechanics. On the way we prove various (old and new) facts about knots. We rela…
We observe that most known results of the form "v is not a finite-type invariant" follow from two basic theorems. Among those invariants which are not of finite type, we discuss examples which are "ft-independent" and examples which are not. We introduce (n,q)-finite invariants, which are generalizations of finite-type…
The paper explores knot theory from Fox colorings to Yang-Baxter homology.
problem Developing new invariants for knot theory.
method Generalizing Fox colorings to racks and quandles, then to Yang-Baxter operators and categorifying the Jones polynomial.
result Building homology of Yang-Baxter operators and speculating on co-cycle invariants.
Relations will be described between the quandle cocycle invariant and the minimal number of colors used for non-trivial Fox colorings of knots and links. In particular, a lower bound for the minimal number is given in terms of the quandle cocycle invariant.
The paper proves a generalized Kauffman-Harary conjecture for prime determinant links.
problem Proving a generalized Kauffman-Harary conjecture for prime determinant links.
method Using Fox colorings and properties of reduced alternating diagrams.
result For every pair of distinct arcs in a prime determinant link, there exists a Fox coloring that distinguishes them.
New Vassiliev invariant of order three derived from Fox-Hatcher cycles.
problem Developing a new Vassiliev invariant of order three.
method Integration of a 1-cocycle over Fox-Hatcher 1-cycles.
result Derives a Vassiliev invariant of order three.
Study on Fox's trapezoidal conjecture for specific alternating links.
problem Investigating Fox's trapezoidal conjecture for alternating links.
method Diagrammatic Murasugi sums, Alexander polynomial, and concordance analysis.
result Established inequalities and conditions for the Alexander polynomial and trapezoidal conjecture.
Goeritz and Seifert matrices derived from Dehn presentations.
problem Computing Goeritz and Seifert matrices for links.
method Using Fox's free differential calculus on modified Dehn presentations.
result Goeritz and Seifert matrices can be derived from Dehn presentations.
First geometric proof of the flyping theorem.
problem Proving Tait's flyping conjecture.
method Geometric proof using Greene's characterization, Menasco's crossing ball structures, and isotopy/re-plumbing moves.
result First entirely geometric proof of Menasco-Thistlethwaite's flyping theorem.
Study on coloring virtual tangles with integer and modular arithmetic.
problem Characterizing Fox colorings of virtual tangle diagrams.
method Analyzed classical and virtual tangle diagrams using vector representations and divisibility conditions.
result For R=Z, realizability depends on divisibility of the alternating sum. For R=Z/pZ, all vectors are realizable. We solve a century-old conjecture about Alexander polynomials of special alternating links.
problem Fox's conjecture about unimodality of Alexander polynomial coefficients.
method Proving a multivariate generalization of the Alexander polynomial is Lorentzian.
result Alexander polynomial coefficients of special alternating links form a log-concave sequence.
Formula calculates volume of two-bridge knots.
problem Calculating the volume of two-bridge knots.
method Derived from Hopf formula and Fox derivatives.
result Closed formula for the volume of two-bridge knots.
New condition for Alexander polynomial factorization in 4D knots.
problem Alexander polynomial factorization for 2-knots in S4. method Alternative notion of ribbon 2-knots to prove factorization condition.
result Topological condition for factorization of Alexander polynomial.
Mandelbrot unified diverse fields with scaling concept.
problem Understanding Mandelbrot's intellectual approach.
method Tracing Mandelbrot's contributions across math, physics, and economics.
result Scaling concept unified Mandelbrot's diverse work.
Paper uses Long-Moody construction for new braid group representations.
problem Constructing new representations of braid groups.
method Applies Fox derivation to matrix presentation of Long-Moody construction.
result Shows relation to twisted Alexander invariants.
Characterizes a specific type of alternating knot.
problem Identifying a special class of genus g alternating knots.
method Uses Ozsváth and Szabó's work on alternating knots.
result Shows that if the coefficients of Alexander polynomial satisfy a certain condition, the knot is a specific torus knot.
Study Alexander polynomials of special alternating links and generalize Fox's conjecture.
problem Distinguish special alternating links up to isotopy using polynomial invariants.
method Combinatorial and discrete geometric properties of Alexander polynomials of special alternating links.
result Generalized Alexander polynomials of special alternating links can be expressed in terms of volumes of root polytopes of unimodular matrices.
Algorithm calculates ribbon obstructions for colored knots.
problem Computing ribbon obstructions for colored knots.
method Algorithm based on dihedral branched covers and colored knot diagrams.
result Algorithm successfully computes ribbon obstructions for colored knots.
Proves a conjecture about knots and their invariants.
problem Relation between Alexander polynomial and signature invariant for two-bridge knots.
method Analyzes two-bridge knots using Fox's conjecture and signature invariant.
result Proves Hirasawa-Murasugi conjecture for two-bridge knots.
Study algebraic concordance groups for non-trivial links.
problem Invariance of signature, Fox-Milnor condition, and Blanchfield pairing under concordance.
method Defined algebraic concordance groups using generalized Seifert matrices.
result Recovery of invariance of signature, Fox-Milnor condition, and Blanchfield pairing for concordant links.
This expository paper describes how the knot invariant Fox coloring can be applied to tangles.
The paper introduces signatures for virtual knots and applies them to study concordance.
problem Investigating the concordance of virtual knots and their slice genus.
method Defined Tristram-Levine signatures for almost classical knots, used Seifert pairing, and introduced parity projection.
result Established slice obstructions for all virtual knots and determined slice status for almost classical knots.
Steenrod homotopy theory is a framework for doing algebraic topology on general spaces in terms of algebraic topology of polyhedra; from another viewpoint, it studies the topology of the lim^1 functor (for inverse sequences of groups). This paper is primarily concerned with the case of compacta, in which Steenrod homot…
Study proves Alexander polynomials of certain 4-braid knots satisfy a conjecture and gives formulas for log-concave sequences.
problem Proving the Alexander polynomials of certain 4-braid knots satisfy Fox's Trapezoidal Conjecture.
method Analyzes families of alternating 4-braids and n-braids, providing explicit formulas and verifying log-concavity. result Explicit formulas for signature and first 4 coefficients of Alexander polynomials, showing log-concavity.
New proof of trapezoidal property for Alexander polynomials of special alternating links.
problem Proving trapezoidal property of Alexander polynomials for special alternating links.
method Analyzing vector configurations from matroids and totally positive matrices.
result Alexander polynomials of special alternating links exhibit log-concavity and trapezoidal properties.
Paper introduces clock moves for plane graphs and proves Alexander polynomial properties.
problem Alexander polynomial of plane graphs and unimodality of coefficients.
method Introduces clock moves for plane graphs and develops a spanning tree model of Alexander polynomial.
result Proves unimodal property of Alexander polynomial coefficients and confirms conjectures.
The dihedral genus of a knot is related to its signature and minimal surface genus.
problem Determining the dihedral genus of knots and its relation to signatures and minimal surface genus.
method Using dihedral branched covers and properties of Fox colorings, the dihedral genus is linked to the signature of the cover and the Murasugi signature of the knot.
result An infinite family of knots with minimal genus surfaces that realize the four-genus of the knot.
Minimal coloring number found for Z-colorable links.
problem Finding the minimum number of colors needed for Z-colorings of Z-colorable links.
method Defined Z-coloring as a generalization of Fox coloring for links with zero determinants. Provided sufficient conditions for non-splittable Z-colorable links to have the least minimal coloring number.
result Sufficient conditions for non-splittable Z-colorable links to have the least minimal coloring number.
Researchers describe a spectral sequence for knots in 3D space.
problem Understanding the Sinha spectral sequence for knots in R^3.
method Explicit description using Fox Neuwirth chain complexes and multicomplex structure.
result A non-trivial third page differential found, contradicting the rational case.
Symbol calculus extended for foliations' transverse geometry.
problem Understanding index theory of transversely elliptic operators on foliations.
method Constructing Getzler rescaling calculus and Block-Fox calculus of asymptotic operators.
result Composition of AΨDOs is again an AΨDO, with a leading symbol formula. We study the number of Reidemeister type III moves using Fox n-colorings of knot diagrams.
Study links weaving knots with polynomial coefficients and lattice numbers.
problem Understanding polynomial coefficients of weaving knots and their lattice counterparts.
method Established relationships between Jones and Chebyshev polynomials, and derived explicit formulas for Alexander polynomials.
result Proved coefficients of Jones polynomial are Whitney numbers of Lucas lattices and satisfied Fox's trapezoidal conjecture.