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2011•Journal of Knot Theory and Its RamificationsOpen access

REAL ALGEBRAIC KNOTS OF LOW DEGREE

Johan Björklund

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Abstract

In this paper, we study rational real algebraic knots in ℝP 3 . We show that two real rational algebraic knots of degree ≤ 5 are rigidly isotopic if and only if their degrees and encomplexed writhes are equal. We also show that any smooth irreducible knot which admits a plane projection with less than or equal to four crossings has a rational parametrization of degree ≤6. Furthermore an explicit construction of rational knots of a given degree with arbitrary encomplexed writhe (subject to natural restrictions) is presented.

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In this paper, we study rational real algebraic knots in ℝP 3 . We show that two real rational algebraic knots of degree ≤ 5 are rigidly isotopic if and only if their degrees and encomplexed writhes are equal. We also show that any smooth irreducible knot which admits a plane projection with less than or equal to four crossings has a rational parametrization of degree ≤6. Furthermore an explicit construction of rational knots of a given degree with arbitrary encomplexed writhe (subject to natural restrictions) is presented.

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Available abstract

In this paper, we study rational real algebraic knots in ℝP 3 . We show that two real rational algebraic knots of degree ≤ 5 are rigidly isotopic if and only if their degrees and encomplexed writhes are equal. We also show that any smooth irreducible knot which admits a plane projection with less than or equal to four crossings has a rational parametrization of degree ≤6. Furthermore an explicit construction of rational knots of a given degree with arbitrary encomplexed writhe (subject to natural restrictions) is presented.

Key concepts: Mathematics, Knot (papermaking), Degree (music), Writhe, Algebraic number, Knot theory, Pure mathematics, Birational geometry

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