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Here a degree of a polynomial knot is a minimal degree which can define a long knot. I would like to find out how this degree can be bounded below, according to the number of crossing points, for instance, or maybe some other properties. Do you know some articles or literature concerning to this problem? All materials, I found, are about particular examples of polynomial reprsentations of some knots, or just says that this is the interesting problem.

After the first answer given, I need to add some information. I know the bound given in the answer, this is natural. I would like to come up with more precise bounds (both lower and upper ones) and, ideally, to derive a formula which gives a degree according to a given knot. This problem seems to be difficult, that's why, first of all, I try to construct more narrow bounds for a degree and I would like to know what research has been done already. If you know any materials concerning to that, write it, please.

Thank you.

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1 Answer 1

Suppose one has a polynomial long knot given as the image of the function $P(t)=(p_1(t),p_2(t),p_3(t)), t\in \mathbb{R}$, where $degree(p_i)\leq d, 1\leq i\leq 3$. Then the crossing number will be bounded by the number of double points of the projection $(p_1,p_2):\mathbb{R}\to \mathbb{R}^2$, assuming this is generic. This projection is a plane curve of degree bounded by $d$ (the degree of a plane curve is the number of intersections with a generic line $ax+by=c$, and is bounded by $d$ since the number of solutions of $ap_1(t)+bp_2(t)=c$ will be bounded by $d$ by the fundamental theorem of algebra).

By Bezout's theorem, the number of double points will be bounded by $d^2$. So one obtains a bound on the crossing number $c(K)\leq d^2$, or $d\geq \sqrt{c(K)}$ since you requested a lower bound on $d$.

There is also a linear lower bound on the degree in terms of the bridge number of the knot.

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Agol, thank you for the answer. I know this bound - that is very natural. I would like to make up with more precise bounbs (both lower and upper ones). Ideally, I would like to derive a formula which gives a degree according to a given knot. As I understand, this problem is difficult, that's why, first of all, I try to construct more narrow bounds for a degree. If you know any materials concerning to that, write it, please. Thank you. –  Lukas Nov 5 '12 at 10:02
3  
Ok, from the formulation of your question, it sounded like you didn't know any bounds. I'm not an expert in this topic, so I can't help you with a literature search. –  Ian Agol Nov 5 '12 at 16:21

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