3 Local Moves on Knots and Products of Knots (original) (raw)
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Local moves on knots and products of knots
Banach Center Publications, 2014
We show a relation between products of knots, which are generalized from the theory of isolated singularities of complex hypersurfaces, and local moves on knots in all dimensions. We discuss the following problem. Let K be a 1-knot which is obtained from another 1-knot J by a crossing change (resp. pass-move). For a given knot A, what kind of relation do the products of knots, K ⊗ A and J ⊗ A, have? We characterize these kinds of relation between K ⊗ A and J ⊗ A by using local moves on high dimensional knots. We also discuss a connection between local moves and knot invariants. We show a new form of identities for knot polynomials associated with a local move.
Local-moves on knots and products of knots II
Journal of Knot Theory and Its Ramifications, 2021
We use the terms, knot product and local-move, as defined in the text of this paper. Let [Formula: see text] be an integer [Formula: see text]. Let [Formula: see text] be the set of simple spherical [Formula: see text]-knots in [Formula: see text]. Let [Formula: see text] be an integer [Formula: see text]. We prove that the map [Formula: see text] is bijective, where [Formula: see text]Hopf, and Hopf denotes the Hopf link. Let [Formula: see text] and [Formula: see text] be 1-links in [Formula: see text]. Suppose that [Formula: see text] is obtained from [Formula: see text] by a single pass-move, which is a local-move on 1-links. Let [Formula: see text] be a positive integer. Let [Formula: see text] denote the knot product [Formula: see text]. We prove the following: The [Formula: see text]-dimensional submanifold [Formula: see text] [Formula: see text] is obtained from [Formula: see text] by a single [Formula: see text]-pass-move, which is a local-move on [Formula: see text]-submani...
An Invariant for Singular Knots
Journal of Knot Theory and Its Ramifications, 2009
In this paper we introduce a Jones-type invariant for singular knots, using a Markov trace on the Yokonuma–Hecke algebras Y d,n(u) and the theory of singular braids. The Yokonuma–Hecke algebras have a natural topological interpretation in the context of framed knots. Yet, we show that there is a homomorphism of the singular braid monoid SBn into the algebra Y d,n(u). Surprisingly, the trace does not normalize directly to yield a singular link invariant, so a condition must be imposed on the trace variables. Assuming this condition, the invariant satisfies a skein relation involving singular crossings, which arises from a quadratic relation in the algebra Y d,n(u).
2018
We prove that given a Conway algebraic link diagram D with n crossings then D can be embedded on the cubic lattice with a length bounded above by en, where cis a positive constant independent of D and n. This implies that the ropelength of alternating Conway algebraic knots growths at most linear with their crossing number.
Polynomial invariants of singular knots and links
Journal of Knot Theory and Its Ramifications, 2021
We generalize the notion of the quandle polynomial to the case of singquandles. We show that the singquandle polynomial is an invariant of finite singquandles. We also construct a singular link invariant from the singquandle polynomial and show that this new singular link invariant generalizes the singquandle counting invariant. In particular, using the new polynomial invariant, we can distinguish singular links with the same singquandle counting invariant.
A New Polynomial Invariant of Knots and LINKS1
1985
The purpose of this note is to announce a new isotopy invariant of oriented links of tamely embedded circles in 3-space. We represent links by plane projections, using the customary conventions that the image of the link is a union of transversely intersecting immersed curves, each provided with an orientation, and undercrossings are indicated by broken lines. Following Conway [6], we use the symbols L+, Lo, L_ to denote links having plane projections which agree except in a small disk, and inside that disk are represented by the pictures of Figure 1. Conway showed that the one-variable Alexander polynomials of L+, Lo, L_ (when suitably normalized) satisfy the relation
A new polynomial invariant of knots and links
Bulletin of the American Mathematical Society, 1985
The purpose of this note is to announce a new isotopy invariant of oriented links of tamely embedded circles in 3-space. We represent links by plane projections, using the customary conventions that the image of the link is a union of transversely intersecting immersed curves, each provided with an orientation, and undercrossings are indicated by broken lines. Following Conway [6], we use the symbols L+, Lo, L_ to denote links having plane projections which agree except in a small disk, and inside that disk are represented by the pictures of Figure 1. Conway showed that the one-variable Alexander polynomials of L+, Lo, L_ (when suitably normalized) satisfy the relation