For example, consider these red/gray/black fishes on the left. Many paintings are not just a study in wallpaper groups but also in group-subgroup pairs. However, there is another layer to his tessellations. Plus, he has so many beautiful drawings to choose from. Escher is the best painter among mathematicians and the best mathematician among painters. I use Escher’s tessellations to teach wallpaper groups. I crocheted a lot of links, and now my students and I have no problem calculating the linking numbers. In this case, the linking number is the difference between the two. It is possible for a loop to wind clockwise and then counterclockwise. The only thing to remember is that while counting the number of windings, I need to consider the direction. Now, it is easy to see that the yellow loop winds around the blue one 3 times, making the linking number 3. I simply slid the yellow loop along the blue one until I could clearly see a piece of the blue loop as a straight segment and the yellow loop circling around it. For example, the second picture shows the same link as the first but slightly rearranged. It only became easy after I started crocheting. When I studied the linking number, I would look at a picture of a link trying to calculate this number. For example, if it is possible to pull the two curves apart, the linking number is zero. Intuitively, it represents the number of times that each curve winds around the other. The linking number is a simple numerical invariant of a link. The first picture shows an example of a link with one yellow curve and one blue. What would happen with the borders if we increase the number of degrees in a twist? Can you figure it out? Are you willing to take up crocheting to solve this puzzle?Ī link is defined as two closed curves in three-dimensional space. The last object has one border, and the color helps you notice that its border is a trefoil knot! The border of the piece in the middle consists of two loops, and the different colors make it obvious that the two borders are linked. For example, it is easy to see that the Möbius strip’s border is a circle. The point of using extra colors for the borders is to make them more prominent. In other cases, the resulting surface is not orientable and has only one border, so I only used one color for the border. I used two different colors to emphasize this fact. When the twist in the strip is a multiple of 360 degrees, the resulting surface is orientable and has two borders. I used green yarn for the internal part of the strips. For the other two objects, I made 360 and 540 degree twists, respectively. Then, instead of connecting the short sides to form a cylinder, I twisted one side 180 degrees before stitching them together. I made it by crocheting a long rectangle. Why would I complicate my life by crocheting colored borders onto different strips?Īnswer: I wanted to emphasize their borders.ĭo you recognize the objects in the picture? The leftmost one is a Möbius strip. In fact, the Whitehead link is the simplest linked link with the linking number zero, but I already wrote about it in my post Whitehead Links for Ukraine. But its linking number is zero, which is unusual for a linked link. It has the crossing number of 5, so it is slightly more complex than Solomon’s link. Here is where my favorite link, the Whitehead link, comes in. So far, it looks like the linking number grows with the crossing number. In the second picture, one can see that the red loop winds around the green loop twice but in the same direction, making the linking number two. What’s important is to keep the direction in mind. Then, I can count the number of times the red loop goes around the green segment. The second image shows the same link, but I slid the red loop so it twists around a small portion of the green loop, which now looks like a segment. The first image shows the crocheted standard representation of Solomon’s link. This is why I started to crochet links: finding their linking numbers by fiddling with them is easy. I often have difficulty calculating the linking number by looking at a picture of a link. It equals zero for the unlink, one for a Hopf link, and two for Solmon’s knot. The linking number is an invariant of a link, which describes how many times one loop goes around the other. However, the linking number created the confusion, so I will explain it. The unlink has a crossing number of 0, the Hopf link has a crossing number of 2, and the Solomon’s knot has a crossing number of 4. Usually, the simplicity of a link corresponds to its crossing number, which is the smallest number of crossings of a projection of the link onto a plane. The Solomon’s knot is the next after that. The simplest one is the unlink, the link where two loops are not linked. It is famous for being one of the simplest linked links. It is a link because it consists of two loops. Solomon’s knot is actually not a knot in a mathematical sense.
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