An Orientation-Sensitive Vassiliev Invariant for Virtual Knots
نویسنده
چکیده
It is an open question whether there are Vassiliev invariants that can distinguish an oriented knot from its inverse, i.e., the knot with the opposite orientation. In this article, an example is given for a first order Vassiliev invariant that takes different values on a virtual knot and its inverse. The Vassiliev invariant is derived from the Conway polynomial for virtual knots. Furthermore, it is shown that the zeroth order Vassiliev invariant coming from the Conway polynomial cannot distinguish a virtual link from its inverse and that it vanishes for virtual knots.
منابع مشابه
A Sequence of Degree One Vassiliev Invariants for Virtual Knots
For ordinary knots in 3-space, there are no degree one Vassiliev invariants. For virtual knots, however, the space of degree one Vassiliev invariants is infinite dimensional. We introduce a sequence of three degree one Vassiliev invariants of virtual knots of increasing strength. We demonstrate that the strongest invariant is a universal Vassiliev invariant of degree one for virtual knots in th...
متن کاملOn Finiteness of Vassiliev Invariants and a Proof of the Lin-wang Conjecture via Braiding Polynomials
Using the new approach of braiding sequences we give a proof of the Lin-Wang conjecture, stating that a Vassiliev invariant v of degree k has a value Ov(c(K)k) on a knot K, where c(K) is the crossing number of K and Ov depends on v only. We extend our method to give a quadratic upper bound in k for the crossing number of alternating/positive knots, the values on which suffice to determine uniqu...
متن کاملThe Polynomial Behaviour of Some Knot Invariants
Using the new approach of braiding sequences we reprove the Lin-Wang conjecture, giving a quadratic upper bound for the crossing number of alternating/positive knots, determining uniquely a Vassiliev invariant, and thus making orientation and mutation sensitivity of Vassiliev invariants decidable on alternating/positive knots/mutants only. We give an exponential upper bound for the number of Va...
متن کاملDetecting knot invertibility
We discuss the consequences of the possibility that Vassiliev invariants do not detect knot invertibility as well as the fact that quantum Lie group invariants are known not to do so. On the other hand, finite group invariants, such as the set of homomorphisms from the knot group to M11, can detect knot invertibility. For many natural classes of knot invariants, including Vassiliev invariants a...
متن کاملVirtual Knot Theory
This paper is an introduction to the subject of virtual knot theory, a generalization of classical knot theory that I discovered in 1996 [2]. This paper gives the basic definitions, some fundamental properties and a collection of examples. Subsequent papers will treat specific topics such as classical and quantum link invariants and Vassiliev invariants for virtual knots and links in more detai...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2002