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[1601.07153] Alexander and writhe polynomials for virtual knots
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This paper is an introduction to virtual knot theory and an exposition of new ideas and constructions, including the parity bracket polynomial, the arrow polynomial, the parity arrow polynomial.
The virtual domain, and it is an open problem whether these counterexamples are equivalent (by addition and subtraction of empty handles) to classical knots and links. Virtual knot theory is a significant domain to be investigated for its own sake and for a deeper understanding of classical knot theory.
Virtual knot theory is an extension of classical knot theory to stabilized embeddings of circles into thickened orientable surfaces o f genus possibly greater than zero.
When virtual knot theory arose, it and invites the reader to generate their.
Aug 28, 2019 for virtual knots which are generalizations of affine index polynomial.
An invitation to abstract mathematics the only undergraduate textbook to teach both classical and virtual knot theory an invitation to knot theory: virtual.
Knot polynomials the bracket polynomial the normalized kauffman bracket polynomial the state sum the image of the f-polynomial surfaces surfaces constructions of virtual links genus of a virtual link bracket polynomial ii states and the boundary property proper states diagrams with one virtual crossing the checkerboard framing.
Nov 8, 2019 video from 19w5118: unifying 4-dimensional knot theory. Robin gaudreau, university of concordance for framed and twisted virtual knots.
Sep 18, 2020 knots are fundamental objects of study in low dimensional topology and appear in diverse areas of sciences.
By virtual knot theory we mean the study knots and and links in thickened surfaces $\sigma\times i$ modulo stabilization, where $\sigma$ is compact orientable surface (not necessarily closed), and i is the closed unit interval. The goal of the present paper is to study classical knots using the methods of virtual knot theory.
The only undergraduate textbook to teach both classical and virtual knot theory an invitation to knot theory: virtual and classical gives advanced undergraduate students a gentle introduction to the field of virtual knot theory and mathematical research.
We give a new interpretation of the alexander polynomial $δ_0$ for virtual knots due to sawollek and silver and williams, and use it to show that, for any virtual knot, $δ_0$ determines the writhe polynomial of cheng and gao (equivalently, kauffman's affine index polynomial). We also use it to define a second-order writhe polynomial, and give some applications.
This paper is an introduction to virtual knot theory and an exposition of new ideas and constructions, including the parity bracket polynomial, the arrow polynomial, the parity arrow polynomial and categorifications of the arrow polynomial. The paper is relatively self-contained and it describes virtual knot theory both combinatorially and in terms of the knot theory in thickened surfaces.
In this sense, virtual knot theory is an extension of the classical theory. Let k be a nontrivial classical knot, and assume that s is a subset of the crossings of k such that the knot obtained by switching these crossings is trivial. Let kv denote virtual knot that is obtained from k by switching the crossings of s but also placing a virtual.
In virtual knot theory, given a virtual knot diagram d there are three kinds of crossingnumbers: thenumberofrealcrossingsc r(d),thenumberofvirtualcross-ingsc v(d),andthenumberoftotalcrossingsc(d). Amongallthevirtualdiagrams ofavirtualknotk, the minimal valuesof themare invariants ofk,saythereal crossingnumberc r(k),thevirtualcrossingnumberc.
Virtual knot theory is a generalization of classical knot theory that utilizes a crossing that is not there. This virtual crossing can be interpreted as a detour through a handle that is attached to the plane of projection, and the theory of virtual knots can be understood as a theory of (stabilized) knots and links in thickend surfaces.
Dec 26, 2018 lecture at quantum knot invariants and supersymmetric gauge theories held at kitp, santa barbara, nov5-dec14, 2018.
The subject of virtual knot theory is relatively new, having been introduced by kauffman and by goussarov, polyak and viro around 1996. Virtual knot theory can be learned right along with classical knot theory, as this book demonstrates, and it is a current research topic as well.
I am a topologist working in knot theory and its relationships with statistical mechanics, contact roger at [roger dot fenn at gmail dot com] and he will add your address to the invitations list.
Reviews for an invitation to knot theory: virtual and classical this book will be greatly helpful and perfect for undergraduate and graduate students to study knot theory and see how ideas and techniques of mathematics learned at colleges or universities are used in research.
A skein theory for the virtual jones polynomial can be obtained from its original version with the addition of a virtual crossing.
Dec 22, 2016 from that perspective, virtual knot theory generalizes classical knot theory by considering knots as embeddings of circles into thickened.
Many of the usual theorems from classical knot theory carry over to virtual knot theory with little or no modification; the jones polynomial, the quandle, the fundamental group and others extend quite nicely.
Virtual knot theory is a generalization (discovered by the author in 1996) of knot theory to the study of all oriented gauss codes. ) graph theory studies non-planar graphs via graphical diagrams with virtual crossings.
The only undergraduate textbook to teach both classical and virtual knot theoryan invitation to knot theory: virtual and classical gives advanced undergraduate students a gentle introduction to the field of virtual knot theory and mathematical research.
A particular surface embedding, but it does apply to such embeddings.
To each virtual link diagram one can associate an abstract surface: ¡! ¡! problem: what is the relation with knot theory on the correspond-ing thickened surface? † how can one compare diagrams when they each live on their own surface? † what is the meaning of classical / virtual reidemeister moves in this context? carter, kamada, saito.
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