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You can find photos of these stitches in  Encyclopedia of Tunisian Crochet!
Back Crossed Tunisian Crochet Stitches

Back Crossed Double Tss/Purl Combo- BXDcTss/Tps
Any number of chs + 2 extra for the edges.
Row 1 Forward & Return: Work as you normally would for Double Tks or Tss.
Row 2: Ch 1. Sk 1 st.  *Work 1 Double Tps, Cross in front of the current st.  Work 1 Double Tss.  Sk 1 st. Repeat from * to end of row.
Row 2 Return: Same as Row 1 Return.
I have had the best results when I use the Simple stitch foundation for the front stitch as opposed to the Knit stitch foundation.

Back Crossed Double Knit and Purl Combo-BXDcTks/Tps
Any number of chs + 2 extra for the edges.
Row 1 Forward and Return: Work as you normally would for Double Tks.
Row 2: Ch 1. Sk next st. *Work 1 Double Tps. Cross in front of the current st. Work 1 Tks in skipped st. Sk 1 st. Repeat from * to end of row.
Row 2 Return: Same as Row 1 Return.
I have had the best results when I use the Simple stitch foundation for the front stitch as opposed to the Knit stitch foundation.

Eyelet Stitch
Eyelets are small openings in a ‘crochet fabric’ used to create a visual pattern.

Tunisian Eyelets are formed by:
On the Forward pass, wrap yarn over hook, skip 1 stitch, and continue picking up loops as you normally would for Tunisian Crochet.  
On the Return, drop the loops off as you normally would, AND treat the Yo thread as a loop, working it off just as you do the loops.
The Tunisian Knit or Simple Stitches are the best choices for the background stitch when using Eyelets.

Detailed Instructions:
Row 1 Forward: Pull loops onto hook as you normally would for Tks or Tss.  When you come to the Eyelet placement, Yo hk and skip one st.  Continue alternating Loops and Yo/Sk St to create your desired visual pattern.
Row 1 Return: Drop loops off as you normally would for Tunisian Crochet.

Treat each Yo sp as a loop and drop it off as you do the other loops.  This will create a small *hole* in the fabric of your piece.  Placement of eyelets can create many different visual patterns such as diagonal stripes, argyle criss cross, or use a Cross Stitch pattern to create a simple picture.

3 Stitch Checkerboard Patterns

Tks/Tss Combo
Make a chain of desired length based on a factor of 6 (36, 42, 48, etc) + 2
Row 1 Forward & Return: Work each st as you would for Tunisian Knit St.
Row 2 Forward: Sk 1st st. *Work 3 Tunisian Knit sts. Work 3 Tunisian Simple Sts. Repeat from * across to the end. Work a Tks into last st of every row.
Row 2 Return: Return as you normally would.
Rows 3: repeat Row 2 Forward and Return.
Row 4 Forward: Sk 1st St. * Work 3 Simple Sts. Work 3 Knit sts. Repeat from * across to the end. Work a Tks into last st of every row.
Row 4 Return: Return as you normally would.
Rows 5 & 6: repeat Row 4 Forward and Return.
To create the pattern, Work 3 rows of Row 2, and 3 rows of Row 4, alternating back and forth between the pattern groupings.

For all Checker Board Patterns follow the previous instructions, but substitute other stitches.
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For a diagram of all the various Threads of a Tunisian Crochet Stitch,
Click Here!
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Abbreviations:  
Beg: Beginning
 Ch: Chain stitch   
Lp: Loop  
Hk: Hook
DO: drop off
PU: pick up
Sk: skip stitches
Sl St: slip stitch
Yo: Yarn Over hook
Tks-Knit st  
Tss-Simple/Afghan St Tps-Purl St
 Trs-Reverse St
 Tfs-Full/Plain St

Trps-Reverse Purl st