## Knit fabrics are constructed by interloping one or more sets of yarns

Knit fabrics are constructed by interloping one or more sets of yarns
Common examples of apparel utilizing weft knitted fabric are socks. Knitting is a more versatile manufacturing process, as entire garments can be manufactured on a single knitting machine, and it is much faster than weaving. However, due to the looping, more yarn is required to manufacture a knitted garment than a comparable woven garment. Thus any cost savings gained in manufacturing speed are offset by the higher materials cost.

Knits are comfortable fabrics, as they adapt to body movement. The loop structure contributes to elasticity beyond what is capable of the yarns or fibers alone. A knit fabric is prone to snagging, and has a higher potential shrinkage than a woven fabric. The loop structure also provides many cells to trap air, and thus provides good insulation in still air. Knits are not typically very wind- or water-repellent.
Knit fabrics are composed of intermeshing loops of yarns. There are two major types of knits: weft knits and warp knits, as illustrated in Fig. 4.7. In weft knits, each weft yarn lies more or less at right angles to the direction in which the fabric is produced, and the intermeshing yarn traverses the fabric crosswise. In warp knits, each warp yarn is more or less in line with the direction in which the fabric is produced, and the intermeshing yarn traverses the fabric lengthwise. Similar to the way that woven fabrics have warps and wefts, knit fabrics have courses and wales, which lie in the crosswise and lengthwise direction, respectively. However, unlike woven fabrics, courses and wales are not composed of different sets of yarns; rather are formed by a single yarn.
Weft blend knitted fabrics are produced predominantly on circular knitting machines. The simplest of the two major weft knitting machines is a jersey machine. Generally, the terms circular knit and plain knit refer to jersey goods. The loops are formed by knitting needles and the jersey machine has one set of needles. Typical fabrics are hosiery, T-shirts, and sweaters.

Rib knitting machines have a second set of needles at approximately right angles to the set found in a jersey machine. They are used for the production of double-knit fabrics. In weft knits, design effects can be produced by altering needle movements to form tuck and miss stitches for texture and color patterns, respectively. Instead of a single yarn, several yarns can be used in the production of these structures. This increases the design possibilities.

‘Loop’ is the basic unit of knit fabric. As illustrated in Fig, 4.7a, in weft knits, a loop, called a needle loop, consists of a head and two legs, and the section of yarn connecting two adjacent needle loops is called the sinker. In warp knits, the needle loop is divided into overlap and underlap, as illustrated in Fig. 4.7b. Each loop in a printed fabric is a stitch. Alternative to fabric count for woven fabrics, cut (or gauge) and stitch density are used to represent the closeness of the intermeshing loops. Cut or gauge indicates the number of knitting needles per unit length along the crosswise or lengthwise direction. The greater the number, the closer together the loops are to each other. Stitch density is the number of stitches per unit area, obtained by multiplying the number of courses per inch (25 mm) by the number of wales per inch (25 mm). Like woven fabrics, a knit fabric also has a technical face and a technical back and can differ in appearance on each side. The technical face is the side where the loops are pulled toward the viewer. Knit fabric also has an effect side, which is intended to be used outermost on a garment or other textile product. In some cases, the technical face and the effect side are the same; but in others, they are opposite.
Gel Knit® fabric is a small diameter weft knitted tube. This is knitted on a small diameter circular knitting machine with the provision for the positive feeding of two separate yarns. Positive feeding is used to ensure the good quality assurance required for a medical product.

The main yarn that is knitted is a staple (spun) yarn of cellulose. This may be any normal cellulosic textile material such as cotton or a number of different reconstituted cellulose materials such as lyocel or viscose. As will be described later in this paper, the cellulose is the precursor material since it will be chemically converted after knitting into the Gel forming material.

The second yarn is a very thin continuous filament nylon which acts as reinforcement and holds the fabric together after the Gel has been formed and the Gel forming yarn has lost all form and stability. The total nylon content of the fabric is about 10%.

Warp knit fabric is similar to that of a woven fabric in that yarns are supplied from warp beams. The fabric is produced, however, by intermeshing loops in the knitting elements rather than interlacing warps and wefts as in a weaving machine. Warp knitted fabric is knitted at a constant continuous width. This is achieved by supplying each needle with a yarn (or yarns) and all needles knit at the same time, producing a complete course (row) at once. It is also possible to knit a large number of narrow width fabrics within a needle bed width to be separated after finishing. In comparison with weft-knit structures, warp knits are typically run-resistant and are closer, flatter and less elastic.

The two common warp-knit fabrics are tricot and raschel (Fig. 10.9). Tricot, solely composed of knit stitches, represents the largest quantity of warp knit. It is characterized by fine, vertical wales on the surface and crosswise ribs on the back. Tricot fabrics may be plain, loop-raised or corded, ribbed, cropped velour or patterned designs. It is commonly used for lingerie owing to its good drapability. It is used for underwear, night-wear, dresses, blouses and outerwear.4 Tricot fabric is used in household products such as sheets and pillowcases. It is also be used for upholstery fabrics for car interiors.Most warp cotton stripe jersey knit fabrics tend to curl, including the most important type known as Jersey stitch (in the USA) or Locknit stitch (in the UK). If they receive appropriate heat treatment, synthetic warp knit fabrics do not curl. In dyeing, finishing, cutting and sewing garments, it helps to know the face and back of the fabric and its curling propensity. When a greige nylon Jersey stitch fabric is put on a table technically upright (having the loop side up), the top and bottom edges of the fabric will curl upwards or towards the loop side or technical face. However, the side edges will curl under the fabric towards the float or technical backside of the fabric.

If nylon Jersey stitch fabric is heat set it will not curl, but if that fabric is laid on the table technically upright and it is pulled sideways on the top edge of the fabric, the fabric will curl towards the loop side. There are some warp knit structures that will not curl in the greige state.Plain warp-and weft-knitted structures are not commonly used for composite applications due to their inherent anisotropy in the wale and course directions. This causes the fabric preform to roll up on itself making handling and manufacturing more difficult. This problem is solved by using weft-knit structures such as the 1 × 1 rib and milano rib, which exhibit balanced properties because of their through-thickness symmetry. However, the highly curved fibre architecture, or crimp, present in these and any knitted structure, means that composites produced using these structures exhibit relatively poor mechanical performance. Characteristics of high conformability and low strength make them ideally suited to producing semi-structural complexly shaped components.

To help increase mechanical performance, insert yarns can be placed between the planes of loops in either the warp or weft direction. The technique can be used for both warp-and weft-knitted fabrics which allow the insert yarns to remain perfectly straight, giving a greater yarn to fabric translational strength. This results in an increase in the composite stiffness and strength along the insert direction. Warp-and weft-knitted fabrics with inlay yarns are termed unidirectional knitted fabrics and the incorporation of insert yarns in two directions creates biaxial knitted fabrics.

8.4.1 Multiaxial warp knits
Multiaxial Warp Knit (MWK) fabric is a further development of this idea by utilising layers of insertion yarns for the in-plane reinforcement and warp stitch yarns for the through-thickness reinforcement. They consist of one or more parallel layers of yarns held together by a warp knit loop system. Theoretically, as many layers as preferred can be used but typical commercially available machines only allow four layers (Du and Ko, 1996). The purpose of the knit loops is to hold the layers of unidirectional yarns together, but it has also been proven to be the key to increasing the damage tolerance of the material (Zhou et al., 2005).

These types of knitted structure are termed non-crimp structures and can be produced in a single knitting process (Du and Ko, 1996). They are particularly suitable for thin to medium thickness parts. The combination of the warp-knitted structure and non-crimp yarns means they have the ability to conform to complex shapes as well as the potential to meet the demands of primary load bearing applications.

MWKs have evolved through structural modifications of warp-knitted fabrics and are predominantly fabrics with inlay yarns in the warp (90°), wale (0°) and bias (± θ°) directions. Warp, weft and bias yarns are held together by a chain or tricot stitch through the thickness of the fabric (Du and Ko, 1996). Layers of 0° need to be placed somewhere other than the top or bottom layer to ensure structural integrity. The amount of fibre and the orientation of the inlay yarns can be controlled, which is advantageous for preform engineering. As a result, the insert yarns are made from a much higher linear density yarn than the stitch yarns, since they form the load-bearing component of the fleece fabric structure (Du and Ko, 1996). Figure 8.4 shows the configuration of the chain and tricot MWK structures.Yarns in a simple weft-knitted structure, as shown in Figure 11.13a, lack the long continuous paths found in woven fabrics and there would be stress concentrations where yarns cross one another. This limits their mechanical performance, but as shown in Chapter 3, they do have applications as composites. In the free state, the knit fabric shows a low resistance to extension and shear, with accompanying area change, until the yarns jam together. This means that they are easily draped into complex shapes.

## Knit fabrics are constructed by interloping one or more sets of yarns

Knit fabrics are constructed by interloping one or more sets of yarns
Common examples of apparel utilizing weft knitted fabric are socks. Knitting is a more versatile manufacturing process, as entire garments can be manufactured on a single knitting machine, and it is much faster than weaving. However, due to the looping, more yarn is required to manufacture a knitted garment than a comparable woven garment. Thus any cost savings gained in manufacturing speed are offset by the higher materials cost.

Knits are comfortable fabrics, as they adapt to body movement. The loop structure contributes to elasticity beyond what is capable of the yarns or fibers alone. A knit fabric is prone to snagging, and has a higher potential shrinkage than a woven fabric. The loop structure also provides many cells to trap air, and thus provides good insulation in still air. Knits are not typically very wind- or water-repellent.
Knit fabrics are composed of intermeshing loops of yarns. There are two major types of knits: weft knits and warp knits, as illustrated in Fig. 4.7. In weft knits, each weft yarn lies more or less at right angles to the direction in which the fabric is produced, and the intermeshing yarn traverses the fabric crosswise. In warp knits, each warp yarn is more or less in line with the direction in which the fabric is produced, and the intermeshing yarn traverses the fabric lengthwise. Similar to the way that woven fabrics have warps and wefts, knit fabrics have courses and wales, which lie in the crosswise and lengthwise direction, respectively. However, unlike woven fabrics, courses and wales are not composed of different sets of yarns; rather are formed by a single yarn.
Weft blend knitted fabrics are produced predominantly on circular knitting machines. The simplest of the two major weft knitting machines is a jersey machine. Generally, the terms circular knit and plain knit refer to jersey goods. The loops are formed by knitting needles and the jersey machine has one set of needles. Typical fabrics are hosiery, T-shirts, and sweaters.

Rib knitting machines have a second set of needles at approximately right angles to the set found in a jersey machine. They are used for the production of double-knit fabrics. In weft knits, design effects can be produced by altering needle movements to form tuck and miss stitches for texture and color patterns, respectively. Instead of a single yarn, several yarns can be used in the production of these structures. This increases the design possibilities.

‘Loop’ is the basic unit of knit fabric. As illustrated in Fig, 4.7a, in weft knits, a loop, called a needle loop, consists of a head and two legs, and the section of yarn connecting two adjacent needle loops is called the sinker. In warp knits, the needle loop is divided into overlap and underlap, as illustrated in Fig. 4.7b. Each loop in a printed fabric is a stitch. Alternative to fabric count for woven fabrics, cut (or gauge) and stitch density are used to represent the closeness of the intermeshing loops. Cut or gauge indicates the number of knitting needles per unit length along the crosswise or lengthwise direction. The greater the number, the closer together the loops are to each other. Stitch density is the number of stitches per unit area, obtained by multiplying the number of courses per inch (25 mm) by the number of wales per inch (25 mm). Like woven fabrics, a knit fabric also has a technical face and a technical back and can differ in appearance on each side. The technical face is the side where the loops are pulled toward the viewer. Knit fabric also has an effect side, which is intended to be used outermost on a garment or other textile product. In some cases, the technical face and the effect side are the same; but in others, they are opposite.
Gel Knit® fabric is a small diameter weft knitted tube. This is knitted on a small diameter circular knitting machine with the provision for the positive feeding of two separate yarns. Positive feeding is used to ensure the good quality assurance required for a medical product.

The main yarn that is knitted is a staple (spun) yarn of cellulose. This may be any normal cellulosic textile material such as cotton or a number of different reconstituted cellulose materials such as lyocel or viscose. As will be described later in this paper, the cellulose is the precursor material since it will be chemically converted after knitting into the Gel forming material.

The second yarn is a very thin continuous filament nylon which acts as reinforcement and holds the fabric together after the Gel has been formed and the Gel forming yarn has lost all form and stability. The total nylon content of the fabric is about 10%.

Warp knit fabric is similar to that of a woven fabric in that yarns are supplied from warp beams. The fabric is produced, however, by intermeshing loops in the knitting elements rather than interlacing warps and wefts as in a weaving machine. Warp knitted fabric is knitted at a constant continuous width. This is achieved by supplying each needle with a yarn (or yarns) and all needles knit at the same time, producing a complete course (row) at once. It is also possible to knit a large number of narrow width fabrics within a needle bed width to be separated after finishing. In comparison with weft-knit structures, warp knits are typically run-resistant and are closer, flatter and less elastic.

The two common warp-knit fabrics are tricot and raschel (Fig. 10.9). Tricot, solely composed of knit stitches, represents the largest quantity of warp knit. It is characterized by fine, vertical wales on the surface and crosswise ribs on the back. Tricot fabrics may be plain, loop-raised or corded, ribbed, cropped velour or patterned designs. It is commonly used for lingerie owing to its good drapability. It is used for underwear, night-wear, dresses, blouses and outerwear.4 Tricot fabric is used in household products such as sheets and pillowcases. It is also be used for upholstery fabrics for car interiors.Most warp cotton stripe jersey knit fabrics tend to curl, including the most important type known as Jersey stitch (in the USA) or Locknit stitch (in the UK). If they receive appropriate heat treatment, synthetic warp knit fabrics do not curl. In dyeing, finishing, cutting and sewing garments, it helps to know the face and back of the fabric and its curling propensity. When a greige nylon Jersey stitch fabric is put on a table technically upright (having the loop side up), the top and bottom edges of the fabric will curl upwards or towards the loop side or technical face. However, the side edges will curl under the fabric towards the float or technical backside of the fabric.

If nylon Jersey stitch fabric is heat set it will not curl, but if that fabric is laid on the table technically upright and it is pulled sideways on the top edge of the fabric, the fabric will curl towards the loop side. There are some warp knit structures that will not curl in the greige state.Plain warp-and weft-knitted structures are not commonly used for composite applications due to their inherent anisotropy in the wale and course directions. This causes the fabric preform to roll up on itself making handling and manufacturing more difficult. This problem is solved by using weft-knit structures such as the 1 × 1 rib and milano rib, which exhibit balanced properties because of their through-thickness symmetry. However, the highly curved fibre architecture, or crimp, present in these and any knitted structure, means that composites produced using these structures exhibit relatively poor mechanical performance. Characteristics of high conformability and low strength make them ideally suited to producing semi-structural complexly shaped components.

To help increase mechanical performance, insert yarns can be placed between the planes of loops in either the warp or weft direction. The technique can be used for both warp-and weft-knitted fabrics which allow the insert yarns to remain perfectly straight, giving a greater yarn to fabric translational strength. This results in an increase in the composite stiffness and strength along the insert direction. Warp-and weft-knitted fabrics with inlay yarns are termed unidirectional knitted fabrics and the incorporation of insert yarns in two directions creates biaxial knitted fabrics.

8.4.1 Multiaxial warp knits
Multiaxial Warp Knit (MWK) fabric is a further development of this idea by utilising layers of insertion yarns for the in-plane reinforcement and warp stitch yarns for the through-thickness reinforcement. They consist of one or more parallel layers of yarns held together by a warp knit loop system. Theoretically, as many layers as preferred can be used but typical commercially available machines only allow four layers (Du and Ko, 1996). The purpose of the knit loops is to hold the layers of unidirectional yarns together, but it has also been proven to be the key to increasing the damage tolerance of the material (Zhou et al., 2005).

These types of knitted structure are termed non-crimp structures and can be produced in a single knitting process (Du and Ko, 1996). They are particularly suitable for thin to medium thickness parts. The combination of the warp-knitted structure and non-crimp yarns means they have the ability to conform to complex shapes as well as the potential to meet the demands of primary load bearing applications.

MWKs have evolved through structural modifications of warp-knitted fabrics and are predominantly fabrics with inlay yarns in the warp (90°), wale (0°) and bias (± θ°) directions. Warp, weft and bias yarns are held together by a chain or tricot stitch through the thickness of the fabric (Du and Ko, 1996). Layers of 0° need to be placed somewhere other than the top or bottom layer to ensure structural integrity. The amount of fibre and the orientation of the inlay yarns can be controlled, which is advantageous for preform engineering. As a result, the insert yarns are made from a much higher linear density yarn than the stitch yarns, since they form the load-bearing component of the fleece fabric structure (Du and Ko, 1996). Figure 8.4 shows the configuration of the chain and tricot MWK structures.Yarns in a simple weft-knitted structure, as shown in Figure 11.13a, lack the long continuous paths found in woven fabrics and there would be stress concentrations where yarns cross one another. This limits their mechanical performance, but as shown in Chapter 3, they do have applications as composites. In the free state, the knit fabric shows a low resistance to extension and shear, with accompanying area change, until the yarns jam together. This means that they are easily draped into complex shapes.

## Does the symmetric group $S_{10}$ factor as a knit product of symmetric subgroups $S_6$ and $S_7$?

By knit product (alias: Zappa-Szép product), I mean a product $$AB$$ of subgroups for which $$A\cap B=1$$. In particular, note that neither subgroup is required to be normal, thus making this a generalization of the semidirect product.

Synopsis of questions (in order):

(1) Can someone provide subgroups $$A,B$$ of $$S_{10}$$ for which $$S_{10}=AB$$, $$A\cong S_6$$, and $$B\cong S_7$$? (Note that by cardinality considerations, necessarily $$A\cap B=1$$ if this happens, in which case $$S_{10}$$ really is the knit product of the two.)

(2) Can it be proven, without a computer exhaust, that $$S_{10}$$ does not have such a decomposition?

(3) How would one go about, with a computer exhaust, showing $$S_{10}$$ does not have such a decomposition? This has as a subquestion: how would we know we captured all the weird ways each $$S_k$$ with $$k=6,7$$ embeds as a subgroup of $$S_{10}$$?

For reference, this is similar to the question here, but even there it was pointed out there were additional ways the embeddings could occur.

History: Once upon a time (i.e., a number of years ago), I was contemplating ways one could factor a symmetric group $$S_n$$ as a knit product of two symmetric subgroups $$A\cong S_a$$ and $$B\cong S_b$$ with positive integers $$a,b$$. Obviously, a necessary condition for this to happen is that $$n! = a!b!$$, so a natural question to ask is the corresponding number theory problem: when is it possible to write $$c!$$ as a product $$a!b!$$ ? Via computer runs, I quickly discovered two infinite families (breaking the symmetry between $$a$$ and $$b$$, I’ll only write triples with $$a\leq b$$), which are $$(a,b,c) = (1,n,n)$$ over integers $$n\geq 1$$ and $$(a,b,c)=(n,n!-1,n!)$$ over integers $$n\geq 3$$, and an outlier example $$(a,b,c)=(6,7,10)$$.

Returning these examples to the motivating group theory question, the first family obviously corresponds to the (extremely trivial) product of $$1=S_1$$ and $$S_n$$. Meanwhile, a Frattini argument applied to the right regular action of $$S_n$$ on itself can be used to show $$\mathrm{Sym}(S_n)$$ is the knit product of $$\mathrm{Sym}(S_n\smallsetminus \langle1\rangle)$$ with the group $$H$$ which is the image of the Cayley embedding $$S_n\hookrightarrow\textrm{Sym}(S_n)$$. This then yields the second family of factorizations.

All of this leads to the question: is there a factorization of $$S_{10}$$ as a product of $$S_6$$ and $$S_7$$, thus providing group theoretic reason for the triple $$(6,7,10)$$? I seem to recall, but cannot find the e-mail, that a friend of mine did a computer run to verify there is no copy-of-$$S_6$$, copy-of-$$S_7$$ pair for which the product is $$S_{10}$$ and which intersect trivially.

If I’m wrong in my recollection, and there does exist a decomposition of $$S_{10}$$ as a knit product of a copy-of-$$S_6$$ times a copy-of-$$S_7$$, I would appreciate enough details to be convinced it is true, including knowledge about which copy of each $$S_k$$ is being considered (e.g., generating set of the $$S_k$$-copy, or a monomorphism $$S_k\rightarrow S_{10}$$).

If I do recall correctly that there is no such factorization, then can someone provide a proof of that fact (directly or via reference)?

Barring the first being true and the second being fulfilled, my fallback position is that I would like to reproduce that computation for myself, except I don’t have a solid feel for how many different ways each $$S_k$$, $$k=6,7$$ can embed into $$S_{10}$$. Therefore, a necessary step in an algorithmic process is coming up with a full list of copies-of-$$S_k$$.

Likely the best way to gather that information would be to provide a representative for each conjugacy class. (If there is a better way to perform the computation, I am all ears.)

As to the conjugacy classes of which I am aware:

$$\bullet$$ The symmetric groups that move exactly $$k$$ letters among the $$10$$ letters are the conjugates of the usual subgroup interpretation of $$S_k$$.

$$\bullet$$ There is, generally speaking, an embedding $$S_k$$ into $$A_{k+2}$$ given by mapping members of $$A_k$$ to themselves and mapping $$\sigma(1\;2)$$ in the coset $$A_k(1\;2)$$ to $$\sigma(1\;2)(k+1\;k+2)$$. This yields the conjugacy class representative $$A_k\cup \bigl(A_k(1\;2)(k+1\;k+2)\bigr)$$.

As an aside for anyone who might be interested, while I have been given reason to believe $$10! = 6!7!$$ does not come up as a symmetric group factorization (via the aforementioned, now lost e-mail), it does come up as a permutation group factorization. Via a Frattini argument applied to the sharply $$3$$-transitive action of the Mathieu group $$M_{10}$$ on $$10$$ letters, the symmetric group $$S_{10}$$ is the knit product of $$S_7$$ and $$M_{10}$$, and $$|M_{10}|=720=6!$$. This makes me think that the sporadic example really is sporadic, in that it (likely) arises through similar “happy accidents” of small numbers that allows $$A_6$$ to have nontrivial outer automorphisms. I am very curious if the two families and this sporadic example really do represent the only solutions $$(a,b,c)$$ to $$c!=a!b!$$, but even if true a proof of that fact is not likely to materialize any time soon.