We show that every proper, smooth 2-knot is ambient isotopic to a polynomial embedding from $\mathbb{R}^2$ to $\mathbb{R}^4$. This representation is unique up to a polynomial isotopy. Using polynomial representation of classical long knots we show that all twist spun knots posses polynomial parametrization. We construct such parametrizations for few spun and twist spun knots and provide their $3$ dimensional projections using Mathematica.

Polynomially knotted 2-spheres

Chapter 1: The Real and Complex Number Systems Introduction Ordered Sets Fields The Real Field The Extended Real Number System The Complex Field Euclidean Spaces Appendix Exercises Chapter 2: Basic Topology Finite, Countable, and Uncountable Sets Metric Spaces Compact Sets Perfect Sets Connected Sets Exercises Chapter 3: Numerical Sequences and Series Convergent Sequences Subsequences Cauchy Sequences Upper and Lower Limits Some Special Sequences Series Series of Nonnegative Terms The Number e The Root and Ratio Tests Power Series Summation by Parts Absolute Convergence Addition and Multiplication of Series Rearrangements Exercises Chapter 4: Continuity Limits of Functions Continuous Functions Continuity and Compactness Continuity and Connectedness Discontinuities Monotonic Functions Infinite Limits and Limits at Infinity Exercises Chapter 5: Differentiation The Derivative of a Real Function Mean Value Theorems The Continuity of Derivatives L'Hospital's Rule Derivatives of Higher-Order Taylor's Theorem Differentiation of Vector-valued Functions Exercises Chapter 6: The Riemann-Stieltjes Integral Definition and Existence of the Integral Properties of the Integral Integration and Differentiation Integration of Vector-valued Functions Rectifiable Curves Exercises Chapter 7: Sequences and Series of Functions Discussion of Main Problem Uniform Convergence Uniform Convergence and Continuity Uniform Convergence and Integration Uniform Convergence and Differentiation Equicontinuous Families of Functions The Stone-Weierstrass Theorem Exercises Chapter 8: Some Special Functions Power Series The Exponential and Logarithmic Functions The Trigonometric Functions The Algebraic Completeness of the Complex Field Fourier Series The Gamma Function Exercises Chapter 9: Functions of Several Variables Linear Transformations Differentiation The Contraction Principle The Inverse Function Theorem The Implicit Function Theorem The Rank Theorem Determinants Derivatives of Higher Order Differentiation of Integrals Exercises Chapter 10: Integration of Differential Forms Integration Primitive Mappings Partitions of Unity Change of Variables Differential Forms Simplexes and Chains Stokes' Theorem Closed Forms and Exact Forms Vector Analysis Exercises Chapter 11: The Lebesgue Theory Set Functions Construction of the Lebesgue Measure Measure Spaces Measurable Functions Simple Functions Integration Comparison with the Riemann Integral Integration of Complex Functions Functions of Class L2 Exercises Bibliography List of Special Symbols Index

Principles of mathematical analysis

Diagrams of knotted surfaces Moving knotted surfaces Braid theory in dimension four Combinatorics of knotted surface diagrams The fundamental group and the Seifert algorithm Algebraic structures related to knotted surface diagrams Bibliography Index.

/pdf/knotted-surfaces-and-their-diagrams-5ar1scuebp.pdf

Knotted Surfaces and Their Diagrams

Basic notions and notation Classical braids and links: Braids Braid automorphisms Classical links Braid presentation of links Deformation chain and Markov's theorem Surface knots and links: Surface links Surface link diagrams Motion pictures Normal forms of surface links Examples (Spinning) Ribbon surface links Presentations of surface link groups Surface braids: Branched coverings Surface braids Products of surface braids Braided surfaces Braid monodromy Chart descriptions Non-simple surface braids 1-handle surgery on surface braids Braid presentation of surface links: The normal braid presentation Braiding ribbon surface links Alexander's theorem in dimension four Split union and connected sum Markov's theorem in dimension four Proof of Markov's theorem in dimension four Surface braids and surface links: Knot groups Unknotted surface braids and surface links Ribbon surface braids and surface links 3-braid 2-knots Unknotting surface braids and surface links Seifert algorithm for surface braids Basic symmetries in chart descriptions Singular surface braids and surface links Bibliography Index.

Braid and knot theory in dimension four

Zur Isotopie zweidimensionaler Flächen imR 4

(1963). A Method for Obtaining Polynomials of Bernstein Type of Two Variables. The American Mathematical Monthly: Vol. 70, No. 3, pp. 260-264.

A Method for Obtaining Polynomials of Bernstein Type of Two Variables

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Polynomially knotted 2-spheres

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References

Principles of mathematical analysis

Knotted Surfaces and Their Diagrams

Braid and knot theory in dimension four

Zur Isotopie zweidimensionaler Flächen imR 4

A Method for Obtaining Polynomials of Bernstein Type of Two Variables