Link Polynomials Research Articles

THE JONES AND ALEXANDER POLYNOMIALS FOR SINGULAR LINKS

We extend the Kauffman state models of the Jones and Alexander polynomials of classical links to state models of their two-variable extensions in the case of singular links. Moreover, we extend both of them to polynomials with d + 1 variables for long singular knots with exactly d double points. These extensions can detect non-invertibility of long singular knots.

A unified framework for interpolating and approximating univariate subdivision

In this paper we show that the refinement rules of interpolating and approximating univariate subdivision schemes with odd-width masks of finite support can be derived ones from the others by simple operations on the mask coefficients. These operations are formalized as multiplication/division of the associated generating functions by a proper link polynomial. We then apply the proposed result to some families of stationary and non-stationary subdivision schemes, showing that it also provides a constructive method for the definition of novel refinement algorithms.

$SC_n$-moves and the $(n+1)$-st Coefficients of the Conway Polynomials of Links

A local move called a $C_n$-move is related to Vassiliev invariants. It is known that two knots are related by $C_n$-moves if and only if they have the same values of Vassiliev invariants of order less than $n$. In the link case, it is shown that a $C_n$-move does not change the values of any Vassiliev invariants of order less than $n$. It is also known that, if two links can be transformed into each other by a $C_n$-move, then the $n$-th coefficients of the Conway polynomials of them, which are Vassiliev invariants of order $n$, are congruent to each other modulo $2$. An $SC_n$-move is defined as a special $C_n$-move. It is shown that an $SC_n$-move does not change the values of any Vassiliev invariants of links of order less than $n+1$. In this paper, we consider the difference of the $(n+1)$-st coefficients of the Conway polynomials of two links which can be transformed into each other by an $SC_n$-move.

Link polynomials derived from magnetic graphs

We introduce a graph diagram which can be regarded as a generalized link diagram. By using it, we construct two polynomial invariants for knots and links which are distinct from both the HOMFLY and the Kauffman polynomials. We show some features of the polynomials including relationships with the HOMFLY and the Kauffman polynomials.

A VIRTUAL LINK POLYNOMIAL AND THE VIRTUAL CROSSING NUMBER

We construct a multi-variable polynomial invariant for virtual knots and links by using the concepts of a decorated virtual magnetic graph diagram and a weight map. We show that the invariant is a variety of virtual link polynomial with multiple variables introduced in [16] by the author and it gives a sharpened evaluation of the virtual crossing number of a virtual knot or link.

LOCAL MOVE IDENTITIES FOR THE ALEXANDER POLYNOMIALS OF HIGH-DIMENSIONAL KNOTS AND INERTIA GROUPS

As analogues of the well-known skein relations for the Alexander and the Jones polynomials for classical links, we present three relations that hold among invariants of high dimensional knots differing by "local moves". Two are for the Alexander polynomials and the other is for the Arf-invariants, the inertia group and the bP-subgroup.

Khovanov homology and star-like isotopies

A star-like isotopy for oriented links in 3-space is an isotopy which uses only Reidemeister moves which correspond to the following singularities of planar curves: [Display omitted] We define a link polynomial derived from the Jones polynomial which is, in general, only invariant under star-like isotopies and we categorify it.

A skein relation for the HOMFLYPT polynomials of two-cable links

We give a skein relation for the HOMFLYPT polynomials of 2-cable links. We have constructed arbitrarily many 2-bridge knots sharing the same HOMFLYPT, Kauffman, and Links-Gould polynomials, and arbitrarily many 2-bridge links sharing the same HOMFLYPT, Kauffman, Links-Gould, and 2-variable Alexander polynomials. Using the skein relation, we show their 2-cable links also share the same HOMFLYPT polynomials.

Twisted Alexander polynomials of plane algebraic curves

In the present paper, Alexander polynomials of plane algebraic curves twisted by linear representations are considered. They are shown to divide the product of the polynomials of the singularity links, for unitary representations. Moreover, their quotient is given by the determinant of the Blanchfield intersection form. Specializing to the classical case, this gives a divisibility formula in the sense of Libgober's divisibility theorem. Examples of twisted polynomials for some algebraic curves are explicitly calculated showing that they can detect Zariski pairs of equivalent Alexander polynomials and that they are sensitive to nodal degenerations.

Nontrivial Alexander polynomials of knots and links

In this paper we present a sequence of link invariants, defined from twisted Alexander polynomials, and discuss their effectiveness in distinguishing knots. In particular, we recast and extend by geometric means a recent result of Silver and Williams on the nontriviality of twisted Alexander polynomials for nontrivial knots. Furthermore we prove that these invariants decide if a genus one knot is fibered. Finally we also show that these invariants distinguish all mutants with up to 12 crossings.

Some invariants of Pretzel links

We show that nontrivial classical pretzel knots L (p, q, r) are hyperbolic with eight exceptions which are torus knots. We find Conway polynomials of n-pretzel links using a new computation tree. As applications, we compute the genera of n-pretzel links using these polynomials and find the basket number of pretzel links by showing that the genus and the canonical genus of a pretzel link are the same.

Q-DEFORMED SPIN NETWORKS, KNOT POLYNOMIALS AND ANYONIC TOPOLOGICAL QUANTUM COMPUTATION

We review the q-deformed spin network approach to Topological Quantum Field Theory and apply these methods to produce unitary representations of the braid groups that are dense in the unitary groups. Our methods are rooted in the bracket state sum model for the Jones polynomial. We give our results for a large class of representations based on values for the bracket polynomial that are roots of unity. We make a separate and self-contained study of the quantum universal Fibonacci model in this framework. We apply our results to give quantum algorithms for the computation of the colored Jones polynomials for knots and links, and the Witten–Reshetikhin–Turaev invariant of three manifolds.

Fast algorithms for computing Jones polynomials of certain links

We give a fast algorithm for computing Jones polynomials of 2-bridge links. Given the Tait graph with n edges of a 2-bridge diagram, this algorithm runs with O ( n ) additions and multiplications in polynomials of degree O ( n ) , namely in O ( n 2 log n ) time. We also propose an algorithm that, given the Tait graph with n edges of a closed 3-braid diagram, computes the Jones polynomial of the closed 3-braid link in O ( n 2 log n ) time.

Link homology theories from symplectic geometry

For each positive integer n, Khovanov and Rozansky constructed an invariant of links in the form of a doubly-graded cohomology theory whose Euler characteristic is the sl ( n ) link polynomial. We use Lagrangian Floer cohomology on some suitable affine varieties to build a similar series of link invariants, and we conjecture them to be equal to those of Khovanov and Rozansky after a collapse of the bigrading. Our work is a generalization of that of Seidel and Smith, who treated the case n = 2 .

Spin networks, quantum automata and link invariants

The spin network simulator model represents a bridge between (generalized) circuit schemes for standard quantum computation and approaches based on notions from Topological Quantum Field Theories (TQFT). More precisely, when working with purely discrete unitary gates, the simulator is naturally modelled as families of quantum automata which in turn represent discrete versions of topological quantum computation models. Such a quantum combinatorial scheme, which essentially encodes SU(2) Racah-Wigner algebra and its braided counterpart, is particularly suitable to address problems in topology and group theory and we discuss here a finite states-quantum automaton able to accept the language of braid group in view of applications to the problem of estimating link polynomials in Chern-Simons field theory.

RECURRENCE FORMULAS FOR THE ALEXANDER POLYNOMIALS OF 2-BRIDGE LINKS AND THEIR COVERING LINKS

We give a recurrence formula for calculating the Alexander polynomials of 2-bridge links by using a special type of Conway diagram. As an application we give sufficient and necessary conditions for the periodic covering link over a 2-bridge link to be a fibered link. We also give a recurrence formula to calculate the reduced Alexander polynomials of periodic covering links over 2-bridge links.

Knots and words

Knots and links are interpreted as homotopy classes of nanowords and nanophrases in an alphabet consisting of 4 letters. Similar results hold for curves on surfaces. We also discuss versions of the Jones link polynomial and the link quandles for nanophrases.

Constraints on Alexander polynomials of certain two-component links

Using Seiberg–Witten gauge theory, we give some mod 2 constraints on the Alexander polynomials of certain class of two-component links recently studied by Fintushel and Stern.

On the number of links and link polynomials

On the number of links and link polynomials

On the number of links and link polynomials

We use a recent method of Sundberg and Thistlethwaite to improve Welsh's upper bound on the rate of growth of the number of links of given crossing number and to show that the number of polynomials of alternating links drops off exponentially in the crossing number compared to the number of such links.