**Contemporary Mathematics**

Volume: 670;
2016;
357 pp;
Softcover

MSC: Primary 57;

**Print ISBN: 978-1-4704-2257-8
Product Code: CONM/670**

List Price: $138.00

AMS Member Price: $110.40

MAA Member Price: $124.20

**Electronic ISBN: 978-1-4704-3526-4
Product Code: CONM/670.E**

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# Knot Theory and Its Applications

Share this page *Edited by *
*Krishnendu Gongopadhyay; Rama Mishra*

This volume contains the proceedings of the ICTS program Knot Theory
and Its Applications (KTH-2013), held from December 10–20, 2013,
at IISER Mohali, India.

The meeting focused on the broad area of knot theory and its
interaction with other disciplines of theoretical science. The program
was divided into two parts. The first part was a week-long advanced
school which consisted of minicourses. The second part was a
discussion meeting that was meant to connect the school to the modern
research areas.

This volume consists of lecture notes on the topics of the advanced
school, as well as surveys and research papers on current topics that
connect the lecture notes with cutting-edge research in the broad area
of knot theory.

#### Readership

Graduate students and research mathematicians interested in knot theory.

# Table of Contents

## Knot Theory and Its Applications

- Cover Cover11
- Title page iii4
- Contents ix10
- Foreword xi12
- Preface xiii14
- Lecture Notes 116
- Knot Theory 318
- Preface 318
- 1. Knot Tying And The Reidemeister Moves 419
- 2. Invariants Of Knots And Links- A First Pass 924
- 3. The Jones Polynomial 1732
- 4. The Bracket State Sum 2237
- 5. Vassiliev Invariants 3247
- 6. Vassiliev Invariants And Lie Algebras 3651
- 7. A Quick Review Of Quantum Mechanics 4257
- 8. Knot Amplitudes 4964
- 9. Topological Quantum Field Theory- First Few Steps 5570
- References 5974

- From Conway Notation to LinKnot 6378
- Surface-knots 93108
- An Introduction to Khovanov Homology 105120
- 1. Introduction 105120
- 2. Bracket Polynomial and Jones Polynomial 106121
- 3. Khovanov Homology and the Cube Category 109124
- 4. Khovanov Homology 112127
- 5. The Cube Category and the Tangle Cobordism Structure of Khovanov Homology 120135
- 6. Frobenius Algebras Implied by the Tube-Cutting Relation 125140
- 7. Other Frobenius Algebras and Rasmussenβs Theorem 129144
- 8. The Simplicial Structure of Khovanov Homology 132147
- 9. Quantum Comments 134149
- 10. Discussion 135150
- References 137152

- Knot Theory for Spatial Graphs Attached to a Surface 141156
- 1. Introduction 141156
- 2. Equivalence of a spatial graph without degree one vertices 143158
- 3. A monotone diagram, the warping degree, the complexity and the cross-index for a spatial graph without degree one vertices 148163
- 4. An unknotted graph and the induced unknotting number 151166
- 5. A π½-unknotted graph and the induced unknotting number 154169
- 6. A homological invariant of an infinite cyclic covering of a spatial graph without degree one vertices 156171
- 7. A πΎ-unknotted spatial graph and the induced unknotting number 158173
- 8. A Ξ-unknotted spatial graph and the induced unknotting number 159174
- 9. The values taken by these unknotting numbers 160175
- 10. Knotting dynamics of a spatial graph with degree one free vertices 166181
- References 168183

- Knots and Graphs: Two Centuries of Interaction 171186
- 1. Introduction 171186
- 2. Knots, graphs and their polynomials 173188
- 3. Setoids and Dichromatic Hopf Algebras 191206
- 4. Jones polynomials of alternating and adequate diagrams;Tait conjectures 202217
- 5. Application of Kauffman polynomial to alternating links 211226
- 6. Kauffman polynomial of adequate links 216231
- 7. Coefficients of Jones-Conway polynomial 231246
- 8. A π-invariant of rooted graphs 249264
- 9. Acknowledgements 250265
- References 251266

- Research Expositions 259274
- On the Welded Tube Map 261276
- On Representations of Braids as Automorphisms of Free Groups and Corresponding Linear Representations 285300
- Ribbon Graphs and Temperley-Lieb Algebra 299314
- On Twisted Knots 313328
- Tunnel Numbers of Knots 327342
- The Warping Matrix of a Knot Diagram 337352
- On Arf Invariant and Trivializing Number 345360
- Back Cover Back Cover1376