Polynomial invariants and quandles of twisted links

Bourgoin defined the notion of a twisted link which corresponds to a stable equivalence class of links in oriented thickenings. It is a generalization of a virtual link. Some invariants of virtual links are extended for twisted links including the knot group and the Jones polynomial. In this paper, we generalize a multivariable polynomial invariant of a virtual link to a twisted link. We also introduce a quandle of a twisted link.     

Virtual knots undetected by 1- and 2-strand bracket polynomials

Kishino's knot is not detected by the fundamental group or the bracket polynomial. However, we can show that Kishino's knot is not equivalent to the unknot by applying either the 3-strand bracket polynomial or the surface bracket polynomial. In this paper, we construct two non-trivial virtual knot diagrams, KD and Km, that are not detected by the 1-strand or the 2-strand bracket polynomial. From these diagrams, we construct two infinite families of non-classical virtual knot diagrams that are not detected by the bracket polynomial. Additionally, these virtual knot diagrams are trivial as flats.     

Twisted Alexander polynomials of twist knots for nonabelian representations

In this paper, we describe the twisted Alexander polynomial of twist knots for nonabelian SL(2,C)-representations and investigate in detail the coefficient of the highest degree term as a function on the representation space of the knot group. In particular, we introduce the notion of monic representation and discuss its relation to the fiberedness of knots.     

On virtual pathwidth of virtual graphs of a virtual link

The geometric representation of a knot is not too dissimilar from a graph and this interaction has helped mathematicians to solve many problems. In this paper, we apply graph theory tools to study the classification of virtual knots and links. We define virtual planar graphs and compute virtual path width of an associated graph of a virtual link. We show that the virtual path width of an associated graph is equal to the virtual bridge number of a pseudo prime knot.     

On minimal elements for a partial order of prime knots

In this paper, applying Chebyshev polynomials we give a basic proof of the irreducibility over the complex number field of the defining polynomial of SL₂(C)-character variety of twist knots in infinitely many cases. The irreducibility, combined with a result in the paper of M. Boileau, S. Boyer, A.W. Reid and S. Wang in 2010, shows the minimality of infinitely many twist knots for a partial order on the set of prime knots defined by using surjective group homomorphisms between knot groups. In Appendix B, we also give a straightforward proof of the result of Boileau et al.     

A refinement of the Conway-Gordon theorems

In 1983, Conway-Gordon showed that for every spatial complete graph on 6 vertices, the sum of the linking numbers over all of the constituent 2-component links is congruent to 1 modulo 2, and for every spatial complete graph on 7 vertices, the sum of the Arf invariants over all of the Hamiltonian knots is also congruent to 1 modulo 2. In this article, we give integral lifts of the Conway-Gordon theorems above in terms of the square of the linking number and the second coefficient of the Conway polynomial. As applications, we give alternative topological proofs of theorems of Brown-Ramírez Alfonsín and Huh-Jeon for rectilinear spatial complete graphs which were proved by computational and combinatorial methods.     

An expression for the Homflypt polynomial and some applications

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 labeled in a braid-dependent manner and may be zero, but display a number of interesting relations. One consequence is an expression for the first three Laurent coefficient polynomials in z as a function of the other coefficient polynomials and three link invariants: the minimum v-degree and v-span of the Homflypt polynomial, and the Conway polynomial.     

The chromatic polynomial of fatgraphs and its categorification

Motivated by Khovanov homology and relations between the Jones polynomial and graph polynomials, we construct a homology theory for embedded graphs from which the chromatic polynomial can be recovered as the Euler characteristic. For plane graphs, we show that our chromatic homology can be recovered from the Khovanov homology of an associated link. We apply this connection with Khovanov homology to show that the torsion-free part of our chromatic homology is independent of the choice of planar embedding of a graph. We extend our construction and categorify the Bollobás-Riordan polynomial (a generalization of the Tutte polynomial to embedded graphs). We prove that both our chromatic homology and the Khovanov homology of an associated link can be recovered from this categorification.     

Ideal Turaev-Viro invariants

Turaev-Viro invariants are defined via state sum polynomials associated to a special spine or a triangulation of a compact 3-manifold. By evaluation of the state sum at any solution of the so-called Biedenharn-Elliott equations, one obtains a homeomorphism invariant of the manifold (“numerical Turaev-Viro invariant”). The Biedenharn-Elliott equations define a polynomial ideal. The key observation of this paper is that the coset of the state sum polynomial with respect to that ideal is a homeomorphism invariant of the manifold (“ideal Turaev-Viro invariant”), stronger than the numerical Turaev-Viro invariants. Using computer algebra, we obtain computational results on several examples of ideal Turaev-Viro invariants, for all closed orientable irreducible manifolds of complexity at most 9.     

The complete splitting number of a lassoed link

In this paper we define a lassoing on a link, a local addition of a trivial knot to a link. Let K be an s-component link with the Conway polynomial non-zero. Let L be a link which is obtained from K by r-iterated lassoings. The complete splitting number split(L) is greater than or equal to r+s−1, and less than or equal to r+split(K). In particular, we obtain from a knot by r-iterated component-lassoings an algebraically completely splittable link L with split(L)=r. Moreover, we construct a link L whose unlinking number is greater than split(L).     

Jones polynomial of knots formed by repeated tangle replacement operations

In this paper, we prove that the Jones polynomial of a link diagram obtained through repeated tangle replacement operations can be computed by a sequence of suitable variable substitutions in simpler polynomials. For the case that all the tangles involved in the construction of the link diagram have at most k crossings (where k is a constant independent of the total number n of crossings in the link diagram), we show that the computation time needed to calculate the Jones polynomial of the link diagram is bounded above by O(nk). In particular, we show that the Jones polynomial of any Conway algebraic link diagram with n crossings can be computed in O(n²) time. A consequence of this result is that the Jones polynomial of any Montesinos link and two bridge knot or link of n crossings can be computed in O(n²) time.     

The LMO-invariant of 3-manifolds of rank one and the Alexander polynomial

We prove that the LMO-invariant of a 3-manifold of rank one is determined by the Alexander polynomial of the manifold, and conversely, that the Alexander polynomial is determined by the LMO-invariant. Furthermore, we show that the Alexander polynomial of a null-homologous knot in a rational homology 3-sphere can be obtained by composing the weight system of the Alexander polynomial with the ?rhus invariant of knots. Received February 14, 2000 / Published online October 11, 2000     

Computing the Conway polynomial of several closures of oriented 3-braids

This paper deals with polynomial invariants of a class of oriented 3-string tangles and the knots (or links) obtained by applying six different closures. In Cabrera-Ibarra (2004) [1], expressions were given to compute the Conway polynomials of four different closures of the composition of two such 3-string tangles. By using the expressions and results from that reference, and using an algorithm developed on the basis of Giller?s calculations for 3-string tangles, we provide new results concerning six closures of 3-braids. Surprisingly, for 3-braids two of the closures turn out to be affine functions of the four previously defined. Among the contributions in this paper one finds computational tools to obtain the Conway polynomial of closures of 3-braids in terms of continuous fractions and their expansions. An interesting feature is that our calculations yield explicit, nonrecursive formulas in the case of 3-braids, thereby considerably lowering the time required to compute them. As a byproduct, explicit expressions are also given to obtain both numerators and denominators of continuous fractions in a nonrecursive way.     

Finite type invariants of order 3 for a spatial handcuff graph

We express a basis for the vector space of finite type invariants of order less than or equal to three for an embedded handcuff graph in a 3-sphere in terms of the linking number, the Conway polynomial, and the Jones polynomial of the sublinks of the handcuff graph.     

Finite type invariants of nanowords and nanophrases

Homotopy classes of nanowords and nanophrases are combinatorial generalizations of virtual knots and links. Goussarov, Polyak and Viro defined finite type invariants for virtual knots and links via semi-virtual crossings. We extend their definition to nanowords and nanophrases. We study finite type invariants of low degrees. In particular, we show that the linking matrix and T invariant defined by Fukunaga are finite type of degree 1 and degree 2 respectively. We also give a finite type invariant of degree 4 for open homotopy of Gauss words.     

Reidemeister torsion and lens surgeries on knots in homology 3-spheres II

We introduce the norm and the order of a polynomial and of a homology lens space. We calculate the norm of the cyclotomic polynomials, and apply it to lens surgery problem for a knot whose Alexander polynomial is the same as an iterated torus knot.     

An endomorphism of the Khovanov invariant

We construct an endomorphism of the Khovanov invariant to prove H-thinness and pairing phenomena of the invariants for alternating links. As a consequence, it follows that the Khovanov invariant of an oriented nonsplit alternating link is determined by its Jones polynomial, signature, and the linking numbers of its components.     

New perspectives on self-linking

We initiate the study of classical knots through the homotopy class of the nth evaluation map of the knot, which is the induced map on the compactified n-point configuration space. Sending a knot to its nth evaluation map realizes the space of knots as a subspace of what we call the nth mapping space model for knots. We compute the homotopy types of the first three mapping space models, showing that the third model gives rise to an integer-valued invariant. We realize this invariant in two ways, in terms of collinearities of three or four points on the knot, and give some explicit computations. We show this invariant coincides with the second coefficient of the Conway polynomial, thus giving a new geometric definition of the simplest finite-type invariant. Finally, using this geometric definition, we give some new applications of this invariant relating to quadrisecants in the knot and to complexity of polygonal and polynomial realizations of a knot.     

A geometric construction of the Conway potential function

We give a geometric construction of the multivariable Conway potential function for colored links. In the case of a single color, it is Kauffmans definition of the Conway polynomial in terms of a Seifert matrix.