Arf invariant of a knot
Okay kiddo, let's talk about a knot and something called the arf invariant. So imagine you have a piece of string and you tie it up into a knot, like the ones you make with your shoe laces.
The arf invariant is a way to help us understand different kinds of knots. It's like a secret code that tells us if two knots are the same or different. It's called an "invariant" because it doesn't change no matter how you move or twist the knot.
The arf invariant works like this: we take our knot and use some math to describe it as a special kind of shape called a "link." Then, we count how many tiny loops the link has in it. This number can be either even or odd. That's the arf invariant! It's either an even number or an odd number.
Now, we can take another knot and do the same thing to it, turning it into a link and counting the number of loops. If this knot has an even number of loops and the first knot also had an even number of loops, then the two knots have the same arf invariant. But if the second knot has an odd number of loops and the first knot had an even number of loops, then the two knots have different arf invariants.
So you see, the arf invariant is a special way of describing knots that helps us tell if two knots are the same or not. It's like a fingerprint for knots!
The arf invariant is a way to help us understand different kinds of knots. It's like a secret code that tells us if two knots are the same or different. It's called an "invariant" because it doesn't change no matter how you move or twist the knot.
The arf invariant works like this: we take our knot and use some math to describe it as a special kind of shape called a "link." Then, we count how many tiny loops the link has in it. This number can be either even or odd. That's the arf invariant! It's either an even number or an odd number.
Now, we can take another knot and do the same thing to it, turning it into a link and counting the number of loops. If this knot has an even number of loops and the first knot also had an even number of loops, then the two knots have the same arf invariant. But if the second knot has an odd number of loops and the first knot had an even number of loops, then the two knots have different arf invariants.
So you see, the arf invariant is a special way of describing knots that helps us tell if two knots are the same or not. It's like a fingerprint for knots!