Symmetrical Single Increase

 When you want to increase one knit stitch to two, there are not many ways to do it that are symmetrical. If you know of any others, let me know!

This way is probably not ideal, but I've seen none others like it.

If you're familiar with the "knit-one-yarn-over-knit-one" increase, that's the first step in this new increase. K1-yo-k1 is what I'll call it from now on.

K1-yo-k1 is a symmetric increase, but you've added two stitches when you work it. The way to turn this into a single decrease is to decrease away (symmetrically) the yarnover on the return row, leaving just two knit stitches symmetrically arranged.

Here is the increase:

Row 1, right side: Work to stitch to be increased. K1-yo-k1 in next stitch. work to end of row.

Row 2, wrong side: Work to increased stitch. Purl 2 together, the knit stitch and the yarnover. Slip the next stitch to the right needle knitwise, and replace on the left needle. Next work a "purl-2-together through back loops" of the stitch just slipped, using that loop and also using the left needle to pick up the left leg of the yarnover  (that was just worked together in the first p2tog).

This is brand new, so I don't think the instructions are that great. A video would probably help.

I just wanted to stake my claim on this technique!

Knitting axioms

Writing about knitting is not like writing about math. 

 Terms are ambiguous, sometimes in a good way, sometimes not.

A "stitch" is sometimes a loop. Sometimes it is an interaction between loops.

A "course" is a length of yarn pulled through previous loops, creating new loops on the right needle. I'm talking about conventional, right-to-left hand-knitting.

With circular knitting, it's all the same fucking course, man. No, but it is really a spiral with no back-tracking like flat knitting. Or at least, not exclusively back-tracking. Short rounds are a thing, like short rows, I guess.

 A great "aha" moment came for me when I realized that a knit stitch is not a loop. Stitches on the needles are loops.

 Pulling a loop through a loop is knitting a stitch.

 The stitch participates in (usually) at least two courses. Yarnovers are granted an exception.

Suppose you have knitted back-and-forth in one color, in stockinette stitch. Suppose the color is red.

Suppose you knit one course (row) using black yarn. 

You break off the red yarn and recommence knitting with red.

 The fabric, when you're done, has one "row" of black knit stitches.

 Wrong!

The fabric has two rows of knit stitches that are half-black, half red.

 You could legitimately say that there is one course knitted with black, but each course of stockinette stitch participates in two rows of knitting. 

 With few exceptions, each worked stitch participates in at least two courses. Each course is (always) attached to two other courses. Except cast-on or bind-off.

Now when casting on, you're actually creating two courses. With the long-tail cast-on, you are creating a course of e-wraps and a course of loops. The e-wraps are the exception, they are sui generis, not attached to another course at the base. They are their own base.

Every course you knit after that is pulled through a previous course and serves as the base for another course.