SEEING FOUR-DIMENSIONAL SPACE AND BEYOND: USING KNOTS!

According to string theory, our universe exists in a 10- or 11-dimensional space. However, the idea the space beyond 3 dimensions seems hard to grasp for beginners. This book presents a way to understand four-dimensional space and beyond: with knots! Beginners can see high dimensional space although they have not seen it.

With visual illustrations, we present the manipulation of figures in high dimensional space, examples of which are high dimensional knots and n-spheres embedded in the (n+2)-sphere, and generalize results on relations between local moves and knot invariants into high dimensional space.

Local moves on knots, circles embedded in the 3-space, are very important to research in knot theory. It is well known that crossing changes are connected with the Alexander polynomial, the Jones polynomial, HOMFLYPT polynomial, Khovanov homology, Floer homology, Khovanov homotopy type, etc. We show several results on relations between local moves on high dimensional knots and their invariants.

The following related topics are also introduced: projections of knots, knot products, slice knots and slice links, an open question: can the Jones polynomial be defined for links in all 3-manifolds? and Khovanov-Lipshitz-Sarkar stable homotopy type. Slice knots exist in the 3-space but are much related to the 4-dimensional space. The slice problem is connected with many exciting topics: Khovanov homology, Khovanv–Lipshits–Sarkar stable homotopy type, gauge theory, Floer homology, etc. Among them, the Khovanov–Lipshitz–Sarkar stable homotopy type is one of the exciting new areas; it is defined for links in the 3-sphere, but it is a high dimensional CW complex in general.

Much of the book will be accessible to freshmen and sophomores with some basic knowledge of topology.

Contents:

• About the Author
• Acknowledgment
• Introduction
• Local Moves On Knots: For Beginners
• Four-Dimensional Space ℝ⁴
• Local Moves in Higher-Dimensional Space: For Beginners
• Knotted-Objects in 4-Space and Beyond
• Local Moves on High-Dimensional Knots and Related Invariants
• The Alexander and Jones Polynomials of One-Dimensional Links in ℝ³
• Local Moves and Knot Polynomials in Higher Dimensions
• Important Topics in Knot Theory
• References
• Index

Readership: Freshmen and sophomores of science and people with similar mathematical background; general public interested in science, sci-fiction, and high-dimensional space; novice researchers in knot theory.

Key Features:

• High dimensional space is of interest not only to mathematicians and physicists, but also the lay public
• The book is introductory and suitable for students and novice researchers in mathematics and physics
• The first half will be easily digestible by freshmen and sophomores and provide the foundations for the second half
• Graduate students can follow the book without much difficulty
• A novice working mathematician will gain a new viewpoint for understanding knot invariants from this book