A link invariant with values in the Witt ring

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).

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Witt ring

Bilinear Algebra ◽

10.1201/9781315138237-15 ◽

2017 ◽

pp. 191-206

Author(s):

Kazimierz Szymiczek

Keyword(s):

Witt Ring

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Link invariant and $G_2$ web space

Hiroshima Mathematical Journal ◽

10.32917/hmj/1492048846 ◽

2017 ◽

Vol 47 (1) ◽

pp. 19-41

Author(s):

Takuro Sakamoto ◽

Yasuyoshi Yonezawa

Keyword(s):

Link Invariant ◽

Web Space

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The graded Witt ring and Galois cohomology. II

Transactions of the American Mathematical Society ◽

10.1090/s0002-9947-1989-0964897-9 ◽

1989 ◽

Vol 314 (2) ◽

pp. 745-745 ◽

Cited By ~ 5

Author(s):

J{ón Kr. Arason ◽

Richard Elman ◽

Bill Jacob

Keyword(s):

Galois Cohomology ◽

Witt Ring

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Graded Witt ring and unramified cohomology

K-Theory ◽

10.1007/bf00961333 ◽

1992 ◽

Vol 6 (1) ◽

pp. 29-44 ◽

Cited By ~ 4

Author(s):

R. Parimala ◽

R. Sridharan

Keyword(s):

Witt Ring ◽

Unramified Cohomology

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ALL LINK INVARIANTS FOR TWO DIMENSIONAL SOLUTIONS OF YANG–BAXTER EQUATION AND DRESSINGS

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216506005147 ◽

2006 ◽

Vol 15 (10) ◽

pp. 1279-1301

Author(s):

N. AIZAWA ◽

M. HARADA ◽

M. KAWAGUCHI ◽

E. OTSUKI

Keyword(s):

Jones Polynomial ◽

Alexander Polynomial ◽

Baxter Equation ◽

Two Dimensional ◽

Link Invariant ◽

Three Solutions ◽

Polynomial Invariants ◽

Link Invariants ◽

Higher Dimensional

All polynomial invariants of links for two dimensional solutions of Yang–Baxter equation is constructed by employing Turaev's method. As a consequence, it is proved that the best invariant so constructed is the Jones polynomial and there exist three solutions connecting to the Alexander polynomial. Invariants for higher dimensional solutions, obtained by the so-called dressings, are also investigated. It is observed that the dressings do not improve link invariant unless some restrictions are put on dressed solutions.

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Basic facts about symmetric bilinear forms, and definition of the Witt ring

Wittrings ◽

10.1007/978-3-322-84382-1_1 ◽

1982 ◽

pp. 1-18

Author(s):

Manfred Knebusch ◽

Manfred Kolster

Keyword(s):

Bilinear Forms ◽

Witt Ring ◽

Basic Facts ◽

Definition Of

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Making Links Between Solutions to an Unstructured Problem

Advances in Educational Technologies and Instructional Design - Handbook of Research on Driving STEM Learning With Educational Technologies ◽

10.4018/978-1-5225-2026-9.ch008 ◽

2017 ◽

pp. 152-175

Author(s):

Sheila Evans

Keyword(s):

Link Invariant ◽

Teaching Resources ◽

Student Responses ◽

Solution Strategies ◽

Invariant Properties

In the study described here, teaching resources have been developed to provide students with explicit opportunities to link invariant properties across a range of different solution strategies, and make comparative judgments about the same solutions. After tackling an unstructured problem, students complete, compare and critique pre-designed student responses to the same problem. The framework used to analyze the data focuses on the types of links students may make between responses. The findings indicate students made varied links when completing them. The outcome of these links appeared to be influenced by how students perceived the representation being completed. Students made further assorted links that focused on invariant properties and the comparative validity of the completed responses.

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