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Cotton is 99% pure cellulose. Cellulose molecules form long strands that repeat, forming the fibre that makes up cotton. 
Cotton is actually stronger wet than dry. This occurs because the hydrogen atoms in the water create extra bonds with those of the cellulose. Cotton cellulose has a high degree of polymerization and a high degree of crystallinity. Crystallinity means that the fibre molecules are closely packed and parallel to one another. 
The higher the crystallinity and polymerization in polymers, the stronger they tend to be. The cellulose chains within the cotton fibres are held in place by hydrogen bonding. These hydrogen bonds occur between the hydroxyl groups of adjacent molecules and are more prevalent between the parallel, closely packed molecules in the crystalline areas of the fibre. When the cotton is wet, the water present forms additional hydrogen bonds increasing the strength of the cotton.
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