Topology for Knitters

Spoilers:  Have you ever wondered why sleeves come in so many shapes?  Why it is possible to take one design and size it from new born to extra-large adult?  Why color charts can be worked flat and circularly?  Why, even though there are sixteen different ways to make knit and purl stitches, there are only six different stockinette fabrics?  The answer is a single word: Topology.  It pervades knitting from the twist of the yarn to the shape of the finished object, and a better understanding of this encompassing subject will help answer all of the above questions, as well as provide us with some common vocabulary for a few of the very cool upcoming blog posts. End Spoilers.

What is Topology?  And what does it have to do with Knitting?

Topology appears to be many things to many people.  What it is not is the study of how rough the roads are – that belongs to topography.  The basic definition of topology I want you to keep in mind while reading this post is: topology is a branch of mathematics which lets a person (preferably a mathematician, but occasionally a knitter) take a shape and study how that shape can be changed into another shape without tearing the original shape.

Fortunately, just because topology is part of math, doesn’t mean you have to be able to do the math to understand what topology is all about; just like you can love to eat french pastry and have no idea how to make that delectable flaky dough yourself.  So:

No Math Skills are Required To Read This Post

No adding, subtracting, dividing, multiplying, fractions, decimals, or anything else involving the math problems we spent hours solving in school.

What does topology have to do with knitting?  To start answering that, I’m going to define knitting as an exploration of the number and types of shapes a knitter can make using yarn, knitting needles, and knit stitches.  Knitting is an incredibly flexible medium; not only does it have great floppiness, but it can also be used to form shapes of almost any kind.  Often, I have found knitting’s only limits are the limits I impose on it.

Thinking about what shapes it is possible to knit leads to a much bigger question: What shapes are possible, period?  Is it possible to think up shapes which can’t exist in real life?  What would those shapes look like if they can’t really exist?

Fortunately, if you read the above sentences without too much mind boggling, you will make it through the rest of this post with ease, because topology is much, much easier than imagining shapes which can’t exist.  Topology, in this post, is about taking a shape you already have, and changing it into another shape without scissors or a glue stick.  When tapering the knitting from the shoulder of a sleeve to the wrist, that taper is topology.

So.  Because we are talking about changing one shape to another in a knitting and in a topological context, it is better for this post if you imagine that all of the shapes mentioned are made out of rubber sheets, play dough, silly putty, or something equally malleable which can be squished or stretched into shapes, rather than out of paper which must be cut and glued into new shapes.  I’d hate to see what would happen if you cut and glued your sweater to get it to fit properly.

A Definition of Topology:

According to Wikipedia:

Topology, as a branch of mathematics, can be formally defined as ‘the study of qualitative properties of certain objects (called topological spaces) that are invariant under a certain kind of transformation (called a continuous map), especially those properties that are invariant under a certain kind of equivalence (called homeomorphism).’  To put it more simply, topology is the study of continuity and connectivity.

Can anyone say circular knitting?  But I get ahead of myself.

Wikipedia continues:

The term topology is also used to refer to a structure imposed upon a set X, a structure that essentially ‘characterizes’ the set X as a topological space by taking proper care of properties such as convergence, connectedness and continuity, upon transformation.

So, say we have a game board–set X in the above paragraph–and divide it into squares for chess or checkers.  The squares would be a structure imposed upon set X, transforming it from any old game board to a chess board.  This allows the board to be used (or understood) in a new manner.

 The motivating insight behind topology is that some geometric problems depend not on the exact shape of the objects involved, but rather on the way they are put together.  For example, the square and the circle have many properties in common: they are both one-dimensional objects (from a topological point of view) and both separate the plane into two parts, the part inside and the part outside.

[…] To deal with these problems that do not rely on the exact shape of the objects, one must be clear about just what properties these problems do rely on.  From this need arises the notion of homeomorphism.  […]  Intuitively, two spaces are homeomorphic if one can be deformed into the other without cutting or gluing.  – Wikipedia

So.  We have two topological objects: a square and a circle.  These shapes are stretchy, and without pulling out scissors or glue-stick, we can squish the circle’s edges until they are straight like a square, and we can squish the square’s corners until they are rounded like a circle.  Because we didn’t cut or glue, and because the two can be squished to look the same, they are topologically equivalent and homeomorphic.

A second kind of topological equivalence is homotopy.  Homeomorphism relates two shapes to each other.  Homotopy relates those same two shapes to a third shape.

The Mathematical Definition (feel free to skip)

Let X be a set and let T be a family of subsets of X.  Then T is called a topology on X if:

    • Both the empty set and X are elements of T
    • Any union of elements of T is an element of T
    • Any intersection of finitely many elements of T is an element of T

If T is a topology on X, then the pair (X, T) is called a topological space. […]  If two spaces are homemorphic, they have identical topological properties, and are considered topologically the same.  The cube and the sphere are homeomorphic, as are the coffee cup and the doughnut.  But the circle is not homeomorphic to the doughnut.

And my interpretation (using the chess board from earlier):

Let X be a chess board and let T be all the squares on the chess board.  Then T is called a topology of X if:

  • The squares and the chess board we are discussing are part of the same thing.  It’s one chess board, not a cutting board and squares of a checkers board.  Even the board without the squares is still a chess board.
  • A group of chess squares looks like chess squares, and not like a soccer ball.
  • You count the four corners where the chess squares meet as part of the chess squares, instead of calling it no man’s land.

If all the above things are true, and the chess squares are on the chess board (not thrown on the floor), then the chess board and the chess squares together may be called a topological space.

To Review:

Topology is the part of math where we think about how shapes are similar to each other.  Their similarity is determined by rules, including homeomorphism and homotopy.

The Good Stuff:  Where the Math and the Knitting Meet.

At this point, it should be fairly obvious that knitting has quite a bit to do with topology, if only because we are talking about shapes.

Let’s ditch the play dough and chess board and pull out some mental yarn and needles.

Imagine the most basic type of pullover sweater:  A square for the front, a square for the back, and two rectangles for the sleeves.  According to topology, these shapes are all homeomorphically equivalent.  Why?  Because they can all be knit in pretty much the same manner.  To knit a square, one casts on a number of stitches, then knits until the length of the sides is equal to the length of the cast on.  To knit a rectangle, one casts on a number of stitches and knits until the length of the sides is longer than the length of the cast on.  Pretty much the same procedure; the only difference is we have stretched the square into a rectangle by knitting a little longer.

So let’s get more complicated.

Imagine a women’s fitted pullover.  It narrows at the waist, increases at the bust, and gradually widens from the wrist to the top of the arm.  To a knitter, this is a very different sweater from the one above, but to a topologist the two sweaters are identical.  The rectangles and squares have merely been squished and stretched using increases and decreases.  They are still homeomorphically equivalent.

Now we get to knitting in the round.  To a knitter, knitting in the round is very similar to knitting flat–we just knit all the stitches and skip the purl rows.  Color charts, textured patterns, lace work, and cables can all be knit circularly without to much difficulty.  However to a topologist, this is radically different.  Why?  because a cylinder (the basic shape resulting from circular knitting) is made by taking a flat square and gluing  the opposite sides.  As soon as gluing or cutting is involved, the two shapes are no longer similar.

However, a topologist will have trouble discerning between knit hats, socks, mittens, gloves, bags and tea cozies; because these are all homeomorphic shapes even if their knit construction is radically different.

Topology allows us to delve deeper still though, beyond the shapes we can create with knitting and into the actual knit fabric itself.

Each knit stitch is something of a topological wonder.  Knots are a very important part of topology, and any knitted object is basically an unknot.  An unknot is a closed loop without any sort of knot.  While knit fabric does have a knot at the beginning, and maybe at the end (or where you join a new color), it still qualifies, because the knot is not what is holding the fabric together, it is just what keeps it from unraveling.

There are plenty of knitters who already know what I am struggling to explain.  They knit exotic things: möbius strips, klein bottles, toruses, and nonorientable surfaces.  It’s a far cry from socks I was going to give out as Christmas gifts. :)

So.  Topology is a huge, encompassing subject embedded in my sweaters I was utterly ignorant of.  And now it’s here, and I can see it everywhere.

Happy Knitting!

 

 

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