Link invariants from finite racks

Nelson, Sam

doi:10.4064/fm225-1-11

Cited by 12 publications

(38 citation statements)

References 22 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…If R is a finite rack, then the rack rank N (R) may be obtained from the rack matrix of R, as remarked in [8]: the diagonal of the rack matrix is a permutation π : R → R, given by π(x) = x x, and the rack rank of R equals the order of π ∈ S |R| . The concept of rack may be generalized by making the operator group explicit.…”

Section: Proofmentioning

confidence: 99%

“…For each framing w, the diagram (D, w) of the framed link defines its own fundamental rack R(D, w) with its own homomorphism set Hom(R(D, w), X). Luckily, since X is a finite rack, there are only finitely many framings which produce different homomorphism sets, by the following results of [8]: It follows from the Proposition 3.3 that two writhe vectors u, v ∈ Z n define the same counting rack invariant if u ≡ v mod N (meaning that u i ≡ v i mod N for i = 1, . .…”

Section: The Counting Rack Invariants Of Links In L(p 1)mentioning

confidence: 99%

“…Any link in a 3-manifold has an (augmented) fundamental rack, whose presentation may be obtained from a link diagram D, and in [4] it has been shown that the fundamental rack is a complete invariant for irreducible framed links in a 3-manifold. In [8], Sam Nelson defined computable rack invariants of classical links, which are based on counting the rack homomorphisms from the fundamental rack of a link to a fixed finite rack.…”

Section: Introductionmentioning

confidence: 99%

“…If the underlying set of a rack is finite, then the rack operation may be encoded in a square matrix with integer entries as follows [8]. Let R be a rack with n elements y 1 , .…”

mentioning

confidence: 99%

See 3 more Smart Citations

Rack invariants of links in L(p,1)

Horvat

2019

J. Knot Theory Ramifications

View full text Add to dashboard Cite

We describe a presentation for the augmented fundamental rack of a link in the lens space [Formula: see text]. Using this presentation, the (enhanced) counting rack invariants that have been defined for the classical links are applied to the links in [Formula: see text]. In this case, the counting rack invariants also include the information about the action of [Formula: see text] on the augmented fundamental rack of a link.

show abstract

Section: Proofmentioning

confidence: 99%

Section: The Counting Rack Invariants Of Links In L(p 1)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…If the underlying set of a rack is finite, then the rack operation may be encoded in a square matrix with integer entries as follows [8]. Let R be a rack with n elements y 1 , .…”

mentioning

confidence: 99%

See 2 more Smart Citations

Rack invariants of links in L(p,1)

Horvat

2019

J. Knot Theory Ramifications

View full text Add to dashboard Cite

show abstract

“…The overall conclusion is that the failure of (1.4) alone does not discard topological applications. By the way, a number of recent works consist in extending to general racks some results first established in the particular case of quandles, see for instance [78,19,79].…”

mentioning

confidence: 99%

Laver’s results and low-dimensional topology

Dehornoy

2015

Arch. Math. Logic

View full text Add to dashboard Cite

Abstract. In connection with his interest in selfdistributive algebra, Richard Laver established two deep results with potential applications in low-dimensional topology, namely the existence of what is now known as the Laver tables and the well-foundedness of the standard ordering of positive braids. Here we present these results and discuss the way they could be used in topological applications.Richard Laver established two remarkable results that might lead to significant applications in low-dimensional topology, namely the existence of a series of finite structures satisfying the left-selfdistributive law, now known as the Laver tables, and the well-foundedness of the standard ordering of Artin's positive braids. In this text, we shall explain the precise meaning of these results and discuss their (past or future) applications in topology. In one word, the current situation is that, although the depth of Laver's results is not questionable, few topological applications have been found. However, the example of braid groups orderability shows that, once initial obstructions are solved, topological applications of algebraic results involving selfdistributivity can be found; the situation with Laver tables is presumably similar, and the only reason explaining why so few applications are known is that no serious attempt has been made so far, mainly because the results themselves remain widely unknown in the topology community.Therefore this text is more a program than a report on existing results. Our aim is to provide a self-contained and accessible introduction to the subject, hopefully helping the algebraic and topological communities to better communicate. Most of the results mentioned below have already appeared in literature, a number of them even belonging to the folklore of their domain (whereas ignored outside of it). However, at least the observations about cocycles for Laver tables mentioned in Subsection 1.3 (and established in another paper) are new.Very naturally, the text comprises two sections, one devoted to Laver tables, and one devoted to the well-foundedness of the braid ordering. It should be noted that the above two topics (Laver tables, well-foundedness of the braid ordering) do not exhaust Laver's contributions to selfdistributive algebra and, from there, to potential topological applications: in particular, Laver constructed powerful tools for investigating free LD-structures, leading to applications of their own [68,69,70]. However, connections with topology are less obvious in these cases and we shall not develop them here (see the other articles in this volume).

show abstract

Twisted virtual biracks and their twisted virtual link invariants

Ceniceros

Nelson

2013

Topology and its Applications

Self Cite

View full text Add to dashboard Cite

A virtual link can be understood as a link in a trivial I-bundle over an orientable compact surface with genus. A twisted virtual link is a link in a trivial I-bundle over a not-necessarily orientable compact surface. A twisted virtual birack is an algebraic structure with axioms derived from the twisted virtual Reidemeister moves. We extend a method previously used with racks and biracks to the twisted case to define computable invariants of twisted virtual links using finite twisted virtual biracks with birack rank N ≥ 1. As an application, we classify twist structures on the virtual Hopf link.

show abstract

Link invariants from finite racks

Cited by 12 publications

References 22 publications

Rack invariants of links in L(p,1)

Rack invariants of links in L(p,1)

Laver’s results and low-dimensional topology

Twisted virtual biracks and their twisted virtual link invariants

Contact Info

Product

Resources

About