Alexander polynomials relate to KP hierarchy via 1-hook property.
problem Relating knot polynomials to the KP hierarchy.
method Kontsevich construction and linear equations reformulation.
result Solutions of reformulated system induce KP equations in Hirota form.
This paper studies the Riley polynomial of 2-bridge knots using Chebyshev polynomials.
problem Understanding the Riley polynomial of 2-bridge knots and its splitting property.
method Introducing ε-Chebyshev polynomials to express and split the Riley polynomial.
result Explicit formula for the splitting polynomial as ε-Chebyshev polynomials.
New method proves Jones Polynomial's connect sum property.
problem Jones Polynomial's behavior under connect sums.
method Trip matrix method for calculating Jones Polynomial.
result Jones Polynomial is multiplicative under connect sums.
Direct proof of Alexander polynomial scaling for L-shaped representations.
problem Proving scaling property of Alexander polynomials for specific representations.
method Direct use of Reshetikhin-Turaev formalism to compute R-matrices.
result Normalized Alexander polynomial for one-hook representations scales with q∣R∣. Formula for interior polynomial of bipartite graphs derived from knot theory.
problem Deriving a formula for the interior polynomial of bipartite graphs.
method Applied knot theory, Ehrhart reciprocity, flyping and mutation.
result Proved a mirroring formula for the interior polynomial of bipartite graphs.
A 20-crossing tangle forms knots with unique polynomial properties.
problem Constructing knots with specific polynomial properties.
method Using a 20-crossing tangle T20 undetectable by Kauffman bracket modulo 2.
result Jones polynomial of knots formed equals 1 modulo 2r.
Formula for 2-head of colored Jones polynomial for pretzel knots proved.
problem Calculating the 2-head of colored Jones polynomial for pretzel knots.
method Skein-theoretic techniques and stability properties of coefficients.
result Formula for 2-head of colored Jones polynomial proved for pretzel knots.
This paper gives a polynomial invariant for flat virtual links. In the case of one component, the polynomial specializes to Turaev's virtual string polynomial. We show that Turaev's polynomial has the property that it is non-zero precisely when there is no filamentation of the knot, as described by Hrencecin and Kauffm…
Researchers compute A-polynomials of manifolds using symplectic properties and cluster algebras.
problem Computing A-polynomials of infinite families of knots and related manifolds is difficult.
method Starting with a triangulation, they use symplectic properties of the Neumann-Zagier matrix to simplify the computation.
result The defining equations of A-polynomials of manifolds obtained by Dehn filling are Ptolemy equations.
Generalizes index polynomial to virtual tangles.
problem No specific problem stated; generalization of polynomial invariant.
method Generalized index polynomial to virtual tangles.
result Three polynomial invariants result from the generalization.
The Links-Gould invariant of alternating links has log-concave coefficients.
problem Log-concavity of Links-Gould coefficients for alternating links.
method Experimental and computational evidence.
result The Links-Gould coefficients of alternating links are log-concave.
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.
The paper studies SDP feasibility and sos ranks for specific polynomials.
problem Characterizing sos representations of nonnegative polynomials.
method Explicit SDP formulation based on Clifford systems.
result Quantitative rank bounds for sos representations, with rigidity.
New polynomial invariants defined for long virtual knots.
problem Defining and studying polynomial invariants for long virtual knots.
method Intersection numbers of cycles on a closed surface, considering crossing order.
result Intersection polynomials are finite-type invariants of degree two under crossing changes, but not under virtualizations.
New polynomials detect non-rotatable knotoid shapes.
problem Detecting non-rotatable knotoid shapes.
method Defined homotopy index polynomials for knotoids.
result Homotopy polynomials detect non-rotatable spherical knotoids.
Groups with hyperbolic properties don't have strong Property (T).
problem Proving groups with hyperbolic properties don't have strong Property (T).
method Constructing an unbounded affine representation with polynomial growth.
result Groups with hyperbolic properties do not have strong Property (T).
We study relationships between the colored Jones polynomial and the A-polynomial of a knot. We establish for a large class of 2-bridge knots the AJ conjecture (of Garoufalidis) that relates the colored Jones polynomial and the A-polynomial. Along the way we also calculate the Kauffman bracket skein module of all 2-brid…
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.
Liouville property proven for certain harmonic functions on metric measure spaces.
problem Proving Liouville property for f-harmonic functions with polynomial growth. method Using Bakry-Émery Ricci curvature nonnegativity and sublinear diameter growth of geodesic spheres.
result Liouville property established for f-harmonic functions with polynomial growth. In this paper, we study the properties of the colored HOMFLY polynomials via HOMFLY skein theory. We prove some limit behaviors and symmetries of the colored HOMFLY polynomial predicted in some physicists' recent works.
Study shows dense roots of Yamada polynomial for certain graphs.
problem Understanding the roots of Yamada polynomials for spatial graphs.
method Construction and analysis of Yamada polynomial for spatial graphs.
result Found an infinite family of graphs with dense roots of Yamada polynomials.
New F-polynomial distinguishes knotoid diagrams not previously possible.
problem Polynomial invariants for knotoids.
method Introduced F-polynomial and constructed distinguishing examples. result New invariant F distinguishes knotoids not previously possible. Polynomial growth elements found in all subgroups of Out(F_n).
problem Understanding polynomial growth in subgroups of Out(F_n).
method Analyzing conjugacy classes and elements of Out(F_n).
result Polynomial growth elements exist in all subgroups of Out(F_n).
New knots show colored Jones polynomials grow linearly.
problem Stability and tail in colored Jones polynomials for alternating knots.
method Infinite family of knots exhibiting linear growth in first coefficient of n-colored Jones polynomials.
result Stability and tail concept in colored Jones polynomials does not generalize to all knots.
Extended symmetric union with multiple tangle regions and Alexander polynomial properties.
problem Characterizing knots with multiple tangle regions.
method Generalizing the symmetric union construction to include multiple tangle regions and analyzing the Alexander polynomial.
result The Alexander polynomial of the constructed knot is the product of the Alexander polynomials of the tangles and the square of the partial knot's Alexander polynomial.
The Jones polynomial of a knot in 3-space is a Laurent polynomial in q, with integer coefficients. Many people have pondered why is this so, and what is a proper generalization of the Jones polynomial for knots in other closed 3-manifolds. Our paper centers around this question. After reviewing several existing defin…
Smoothly slice a knot with specific properties.
problem Existence of certain types of knots.
method Proving the existence of a specific knot with given properties.
result Existence of a smoothly doubly slice, amphicheiral knot with Alexander polynomial 1 and unknotting number 5.
New knot models analyze local entanglement for robust curve analysis.
problem Lack of local structural information in classical knot theory.
method Proposed multiscale and persistent Jones polynomials.
result Models are stable to small perturbations, robust for real-world applications.
Proves non-properness set of 3D polynomial homeos can't be a line.
problem Non-properness set of 3D polynomial homeomorphisms.
method Proof by contradiction, using topological properties.
result Non-properness set cannot be homeomorphic to the real line.
Extends interior polynomial to signed bipartite graphs and connects to HOMFLY polynomial.
problem Invariants of signed bipartite graphs and their relation to HOMFLY polynomial.
method Extending interior polynomial to signed bipartite graphs and showing equality to HOMFLY polynomial part.
result Interior polynomial of signed bipartite graphs equals part of HOMFLY polynomial for planar case.
New polynomials defined for virtual knots, calculated up to crossing 4.
problem Defining and calculating invariants for virtual knots.
method Intersection number of curves on a closed surface.
result Intersection polynomials calculated up to crossing 4.
Novel knot polynomials from Gaussian calculus show half vanish and determine Jones polynomials.
problem Understanding and characterizing knot polynomials from Gaussian calculus.
method Gaussian calculus of generating series for noncommutative algebras, connected sum of knots.
result Half of the polynomials vanish and three polynomials are explicitly given.
The eigenvalue conjecture is supported for colored Alexander polynomials.
problem Supporting the eigenvalue conjecture for colored Alexander polynomials.
method Connecting Alexander polynomials and eigenvalues of braid group generators.
result Support for the eigenvalue conjecture for i>2, where direct evaluation is difficult.
The paper defines and classifies Cappell-Shaneson polynomials.
problem Characterizing Cappell-Shaneson polynomials.
method Algebraic conditions on polynomials, reduction modulo primes, and construction of infinite series.
result Complete lists of Cappell-Shaneson polynomials of degrees 4 and 5, and several infinite series of degree 6.
This paper studies HOMFLY polynomials of specific and infinite classes of knots.
problem Computing HOMFLY polynomials in general is difficult; this paper examines specific cases.
method Examined two specific knots and a general infinite class of knots.
result Observed apparent patterns in the polynomials of specific knots and conjectured properties of the general class.
Study polynomial growth harmonic functions on infinite penny graphs.
problem Finite-dimensional property of polynomial growth harmonic functions on infinite penny graphs.
method Asymptotically sharp dimensional estimate for ancient solutions of the heat equation.
result Proved the asymptotically sharp dimensional estimate.
Quantum polynomials are derived from a specific tribracket structure.
problem Quantum enhancement polynomials for oriented links.
method Defined using a canonical two-element tribracket, proving polynomials can be derived from five specific ones.
result Universal quantum enhancement polynomials are strictly stronger than the Jones polynomial.
The article extends a knot polynomial algorithm for singular knots.
problem Computing the colored Jones polynomial for singular knots.
method Extended Masbaum and Vogel's algorithm to singular knots.
result Introduced the tail of the colored Jones polynomial and proved a Ramanujan identity.
Generalized Wriggle polynomial for virtual tangles.
problem Defining a polynomial invariant for virtual tangles.
method Generalization of the Wriggle polynomial to virtual tangles, proving invariance.
result The generalized Wriggle polynomial is a Vassiliev invariant of order one for virtual knots.
Study Markov cubature rules for polynomial processes.
problem Tractability of path-dependent tasks in polynomial process models.
method Discretizations using finite state Markov processes with moment matching conditions.
result Markov cubature rules aid American option pricing.
Study of Alexander modules for hyperplane arrangements, distinguishing complements.
problem Distinguishing homotopy equivalent but non-homeomorphic hyperplane arrangement complements.
method Analysis of twisted Alexander modules and polynomials.
result Distinguish non-homeomorphic homotopy equivalent arrangement complements.
Defines a new knot invariant and studies its properties.
problem Understanding the structure of knot invariants.
method Defining and analyzing the twisted Alexander vanishing order.
result Characterizes knots with a zero-twisted Alexander polynomial.
Associated with each oriented link is the two variable Homflypt polynomial. The Morton-Franks-Williams (MFW) inequality gives rise to an expression for the Homflypt polynomial with MFW coefficient polynomials. These MFW coefficient polynomials are labelled in a braid-dependent manner and may be zero, but display a numb…
The notion of chckerboard colorability for virtual links and abstract links is introduced. We study the Jones polynomials of virtual links and abstruct links. It is proved that a certain property of the Jones polynomials of classical links is valid for virtual links which admit checkerboard colorings.
New knots with specific properties have identical polynomial values.
problem Identifying knots with matching polynomial values after braiding.
method Constructing infinitely many hyperbolic knots and analyzing their braided satellites.
result Mutually distinct hyperbolic knots have identical HOMFLY polynomial values up to given z-degrees. Study revisits Alexander-Conway and Kauffman bracket polynomials for pretzel links.
problem Understanding polynomial invariants of pretzel links.
method Revisits Alexander-Conway and Kauffman bracket polynomials for pretzel links P(1,1,n). result Reveals properties of Alexander-Conway and Kauffman bracket polynomials for P(1,1,n). The paper reveals a property of chromatic homology for complete graphs.
problem Understanding the chromatic homology of complete graphs.
method Introduced a combinatorial description of enhanced states and used it to analyze the homology.
result Showed a splitting property of the chromatic homology for certain graphs.
Study of panhandle polynomials of torus links with geometric applications.
problem Characterizing the HOMFLY-PT polynomial of torus knots and links.
method Utilizing quantum group representations and the Rosso-Jones formula.
result Established panhandle-like structure of HOMFLY-PT polynomials for torus knots and links.