Properties of Glued Knots

Abstract

Abstract Properties of Glued Knots The study of knots and links in the real projective space has been motivated by one of the questions from the famous list of problems given by Hilbert in 1900. A real rational knot in RP 3 is an embedding of RP 1 → RP 3 given by homogeneous polynomials. All knots in RP 3 are isotopic to some real rational knot and therefore have a real rational representative. The knots which do not intersect the plane at infinity are also called affine knots and correspond to the classical knots in R 3 . However, there are very less known results which tell the relationships between classical knot theoretic properties and real rational knots. Björklund introduced a method of "gluing" two real rational knots of degree d 1 and d 2 to get a real rational knot of degree d 1 +d 2 . The method is very geometric and he was able to construct all real rational knots up to degree 5 by joining lines. This method was used to show that each classical knot has a real rational representative glued out of ellipses. In this thesis we establish some results relating the degree of the glued knots and the values of classical knot invariants that these knots possess. We construct a family of real rational knots whose members have the desired number of 3-colour invariant with a bound on the degree of the knot. We then calculate the maximum writhe number that an affine glued knot of certain degree could achieve and give the examples of knots that have that much writhe. The diagrams that glued knots can have are of special types due to the rigidity of ellipses that are used to make them. We show that the figure eight knot has a rigid diagram with least number of crossings and with least number of degree whereas trefoil can not have such a diagram. The maximum number of double points a knot diagram of degree d could have is (d−1)(d−2) but we prove that there could be no alternating glued knot with this much crossing number. Then we show that with degree 2n, a knot can be constructed with crossing number 2n − 2 for n≥3. There is a strong relation between affine glued knots and links made of rigid ellipses. We exploit this connection along with the Skein Relations of polynomial invariants of the knots and links to get results concerned with maximum degrees2 of polynomial invariants of these two. We then classify all the affine glued real rational knots up to degree 6 and consequently get all the prime 3-component links which have all 3 components as ellipses. We show that classification up to rigid isotopy in the preglued sense is strictly stronger than rigid isotopy by showing two degree 6 curves which are rigidly isotopic but one can not be isotoped to the other just by using glued knots obtained from preglued curves at each step. We also produce representative of all knots having crossing number≤ 6 and show they are all realizable as curves with degree≤8. We also classify projective knots glued out of lines for low degree up to smooth isotopy.