These crocheted bags have the shape of Klein bottles:

4 Klein bottles
4 Klein bottles

Actually, they are immersions of the Klein bottle in three dimensional space where a disk has been left out at the location of self intersection, so that one can access the interior of the bag through the hole in the bottom.

Instructions

  1. Take some yarn and make a slipped overhand knot:
    Crochet instructions
    Tighten it, but not the loop:
    Crochet instructions
  2. Use the crochet hook to pull the thread through the loop:
    Crochet instructions
    Now pull the first loop closed, but not too tight. This is the first chain stitch. The crochet hook always stays in the loop made last.
  3. Crochet a whole chain of chain stitches: Crochet instructions
  4. As soon as the chain is somewhat longer than the intended circumference of the bag's neck, it is closed to form a circle. For this, insert the hook through the first chain stitch...
    Crochet instructions
  5. ... and pull the thread through both loops:
    Crochet instructions
  6. To make the next stitch, insert the hook into the next loop of the previous row and again pull the the thread through both loops:
    Crochet instructions
    Crochet instructions
  7. Repeat this so that a cylinder develops. You should skip a loop of the previous row from time to time, especially at the beginning, so that the circumference of the cylinder decreases until the desired neck diameter is reached. The skipped loops should be all on the same side of the cylinder, for example below the starting position, so that a protrusion forms. This protrusion will later belong to the lower part of the curve of the neck:
    Crochet instructions
  8. continue crocheting the cylinder with constant circumference until you reach a position that will later not be outwardly visible. There you should switch the crocheting direction by turning the cylinder inside out and crocheting back on top of the loops you just made:
    Crochet instructions
    If you do not do that, there will later be a visible discontinuity of the crochet pattern because the Klein bottle is nonorientable, so at some place the left side and the right side have to lie next to each other. In the following picture of a different Klein bag, where I did not change the crocheting direction, one can see how two different structures clash at the location of the self intersection:
    negative example
  9. Continue the cylinder until the desired length is reached:
    Crochet instructions
  10. Then, start inserting about 6.5 chain stitches per turn, so that the diameter grows evenly and the bottom of the bag forms:
    Crochet instructions
  11. When the bottom has reached the desired diameter, stop inserting chain stitches. Thus the outer wall of the bag develops:
    Crochet instructions
  12. When the wall is high enough, stitch it to the inner cylinder, opposite to the location where the protrusion is. To do that, insert the hook through a loop in the cylinder wall before putting it through the loop of the previous row, so that you have three loops on the hook:
    Crochet instructions
    Then pull the thread through all three loops. Overall, make a few of these stitches:
    Crochet instructions
  13. After that, continue normally until you get again to the place where the cylinder was joined to the wall. There, crochet along the exposed side of the cylinder by inserting the hook through a loop in the cylinder wall and then pulling the thread through both loops on the hook:
    Crochet instructions
    Thus a row of loops forms, on top of which you can crochet along in the next turn.
  14. Crochet further in circles, but when you get to the corners, skip loops from the previous row, lest the fabric pucker and so that the diameter of the hole decreases further until the desired width of the neck of the bag is reached:
    Crochet instructions
  15. Then you stop skipping loops and continue at constant circumference until you get to the height where the protrusion of the first cylinder begins:
    Crochet instructions
  16. From there on, insert chain stitches on the side facing the protrusion, so that a second protrusion forms:
    Crochet instructions
  17. Both protrusions are joined by changing over to the other rim at the point of contact of the two protrusions, crocheting one turn on the other rim and then changing back:
    Crochet instructions
    Thus the two rims become a single one.
  18. Crochet along on the united rim. While doing that, from time to time skip a loop of the previous row when near the point of contact of the two rims, so that a transition from the 8-shaped rim to a 0-shaped rim develops:
    Crochet instructions
    Crochet instructions
  19. At first only at the locations lying furthest out on the rim, then everywhere, skip loops in order to contract the rim into an ever smaller circle:
    Crochet instructions
  20. Finally, skip every second loop, so that the circular rim contracts into a point:
    Crochet instructions
  21. Cut the thread and you're done!
    Crochet instructions
    Crochet instructions
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