scholarly journals Nielsen number is a knot invariant

2007 ◽  
Author(s):  
Alexander Fel'shtyn
Keyword(s):  
Nielsen Number ◽  
Knot Invariant

Pseudo-cycles of surface-knots

2016 ◽  
Vol 25 (13) ◽  
pp. 1650068 ◽  
Author(s):  
Tsukasa Yashiro
Keyword(s):  
Cell Complex ◽  
Two Dimensional ◽  
Rectangular Cell ◽  
Knot Invariant

In this paper, we describe a two-dimensional rectangular-cell-complex derived from a surface-knot diagram of a surface-knot. We define a pseudo-cycle for a quandle colored surface-knot diagram. We show that the maximal number of pseudo-cycles is a surface-knot invariant.

scholarly journals Kontsevich’s integral for the Kauffman polynomial

1996 ◽  
Vol 142 ◽  
pp. 39-65 ◽  
Author(s):  
Thang Tu Quoc Le ◽  
Jun Murakami
Keyword(s):  
Finite Type ◽  
Knot Invariants ◽  
Chord Diagrams ◽  
Knot Invariant ◽  
Kauffman Polynomial ◽  
Zamolodchikov Equation

Kontsevich’s integral is a knot invariant which contains in itself all knot invariants of finite type, or Vassiliev’s invariants. The value of this integral lies in an algebra A0, spanned by chord diagrams, subject to relations corresponding to the flatness of the Knizhnik-Zamolodchikov equation, or the so called infinitesimal pure braid relations [11].

On Weight Systems Derived from Heisenberg Lie Algebras

2003 ◽  
Vol 12 (05) ◽  
pp. 589-604
Author(s):  
Hideaki Nishihara
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Weight System ◽  
Infinite Dimensional ◽  
Conway Polynomial ◽  
Knot Invariant ◽  
Polynomial Representations ◽  
Solvable Lie Algebras

Weight systems are constructed with solvable Lie algebras and their infinite dimensional representations. With a Heisenberg Lie algebra and its polynomial representations, the derived weight system vanishes on Jacobi diagrams with positive loop-degree on a circle, and it is proved that the derived knot invariant is the inverse of the Alexander-Conway polynomial.

An index and a Nielsen number for n-valued multifunctions

1984 ◽  
Vol 124 (3) ◽  
pp. 207-219 ◽  
Author(s):  
Helga Schirmer
Keyword(s):  
Nielsen Number

scholarly journals The SL(2, C) Casson Invariant for Knots and the Â-polynomial

2016 ◽  
Vol 68 (1) ◽  
pp. 3-23 ◽  
Author(s):  
Hans U. Boden ◽  
Cynthia L. Curtis
Keyword(s):  
Large Class ◽  
Product Formula ◽  
Integral Homology ◽  
Connected Sum ◽  
Connected Sums ◽  
Knot Invariant ◽  
Definition Of ◽  
Arbitrary Knots

AbstractIn this paper, we extend the definition of the SL(2,ℂ) Casson invariant to arbitrary knots K in integral homology 3-spheres and relate it to the m-degree of the Â-polynomial of K. We prove a product formula for the Â-polynomial of the connected sum K1#K2 of two knots in S3 and deduce additivity of the SL(2,ℂ) Casson knot invariant under connected sums for a large class of knots in S3. We also present an example of a nontrivial knot K in S3 with trivial Â-polynomial and trivial SL(2,ℂ) Casson knot invariant, showing that neither of these invariants detect the unknot.

scholarly journals A complete knot invariant from contact homology

2017 ◽  
Vol 211 (3) ◽  
pp. 1149-1200 ◽  
Author(s):  
Tobias Ekholm ◽  
Lenhard Ng ◽  
Vivek Shende
Keyword(s):  
Contact Homology ◽  
Knot Invariant

scholarly journals THE QUANDARY OF QUANDLES: A BOREL COMPLETE KNOT INVARIANT

2019 ◽  
Vol 108 (2) ◽  
pp. 262-277 ◽  
Author(s):  
ANDREW D. BROOKE-TAYLOR ◽  
SHEILA K. MILLER
Keyword(s):  
Isomorphism Problem ◽  
Algebraic Structures ◽  
Knot Invariant ◽  
Borel Reducibility ◽  
Special Case

We show that the isomorphism problems for left distributive algebras, racks, quandles and kei are as complex as possible in the sense of Borel reducibility. These algebraic structures are important for their connections with the theory of knots, links and braids. In particular, Joyce showed that a quandle can be associated with any knot, and this serves as a complete invariant for tame knots. However, such a classification of tame knots heuristically seemed to be unsatisfactory, due to the apparent difficulty of the quandle isomorphism problem. Our result confirms this view, showing that, from a set-theoretic perspective, classifying tame knots by quandles replaces one problem with (a special case of) a much harder problem.

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