Related papers: Vassiliev invariants for pretzel knots

We compute the unknotting number of two infinite families of pretzel knots, $P(3,1,\dots,1,b)$ (with $b$ positive and odd and an odd number of 1s) and $P(3,3,3c)$ (with $c$ positive and odd). To do this, we extend a technique of Owens using…

We consider the classical pretzel knots $P(a_1, a_2, a_3)$, where $a_1, a_2, a_3$ are positive odd integers. By using continuous paths of elliptic $\mathrm{SL}_2(\mathbb R)$-representations, we show that (i) the 3-manifold obtained by…

A Chebyshev curve C(a,b,c,\phi) has a parametrization of the form x(t)=Ta(t); y(t)=T_b(t) ; z(t)= Tc(t + \phi), where a,b,c are integers, Tn(t) is the Chebyshev polynomial of degree n and \phi \in \RR. When C(a,b,c,\phi) has no double…

In this paper we construct new invariants of knotoids including the odd writhe, the parity bracket polynomial, the affine index polynomial and the arrow polynomial, and give an introduction to the theory of virtual knotoids. The invariants…

Quantum knot invariants (like colored HOMFLY-PT or Kauffman polynomials) are a distinguished class of non-perturbative topological invariants. Any known way to construct them (via Chern-Simons theory or quantum R-matrix) starts with a…

We calculate the Witte-Reshetikhi-Turaev invariant for a knot in the lens space of type L(m,1) for the N-th root of unity, and study its asymptotic behavior for large N.

Each knot invariant can be extended to singular knots according to the skein rule. A Vassiliev invariant of order at most $n$ is defined as a knot invariant that vanishes identically on knots with more than $n$ double points. A chord…

The Kontsevich integral $Z$ associates to each braid $b$ (or more generally knot $k$) invariants $Z_i(b)$ lying in finite dimensional vector spaces, for $i = 0, 1, 2, ...$. These values are not yet known, except in special cases. The…

We conjecture explicit evolution formulas for Khovanov polynomials for pretzel knots in some regions in the windings space. Our description is exhaustive for genera 1 and 2. As previously observed, evolution at T != -1 is not fully smooth:…

Oleg Viro introduced an invariant of rigid isotopy for real algebraic knots in $RP^3$ which can be viewed as a first order Vassiliev invariant. In this paper we look at real algebraic knots of degree $d$ with the maximal possible value of…

Given a virtual knot $K$, we construct a group $VG_K$ called the virtual knot group, and we use the elementary ideals of $VG_K$ to define invariants of $K$ called the virtual Alexander invariants. For instance, associated to the $k=0$ ideal…

We present a comprehensive classification of invariants of knots and links associated with irreducible representations of \uqslN{}, when the parameter of quantization $q$ is a root of unity. We demonstrate that, besides the standard…

We introduce and explore the relation between knot invariants and quiver representation theory, which follows from the identification of quiver quantum mechanics in D-brane systems representing knots. We identify various structural…

As a generalization of the classical knots, knotoids deal with the open ended knot diagrams in a surface. In recent years, many polynomial invariants for knotoids have appeared, such as the bracket polynomial, the index polynomial and the…

We exploit the Galois symmetries of the little disks operads to show that many differentials in the Goodwillie-Weiss spectral sequences approximating the homology and homotopy of knot spaces vanish at a prime $p$. Combined with recent…

The present paper produces examples of Gauss diagram formulae for virtual knot invariants which have no analogue in the classical knot case. These combinatorial formulae contain additional information about how a subdiagram is embedded in a…

We show that Vassiliev invariants separate braids on a closed oriented surface, and we exhibit an universal Vassiliev invariant for these braids in terms of chord diagrams labeled by elements of the fundamental group of the considered…

The $\Delta$-unknotting number for a knot is defined as the minimum number of $\Delta$-moves needed to deform the knot into the trivial knot. It is known that, for positive pretzel knots, the $\Delta$-unknotting number coincides with the…

We define an invariant of welded virtual knots from each finite crossed module by considering crossed module invariants of ribbon knotted surfaces which are naturally associated with them. We elucidate that the invariants obtained are non…

We show that for a large class of contact 3-manifolds the groups of Vassiliev invariants of Legendrian and of framed knots are canonically isomorphic. As a corollary, we obtain that the group of finite order Arnold's $J^+$-type invariants…