Tensile properties of cellulose fibres
Lignin and cellulose in cotton, work together like the epoxy resin and  the glass fibers in  fiberglass composite materials (a fiberglass boat, for example). So, the cellulose is the fibrous component and is the primary load-bearing element.  While lignin is the polymeric matrix that provides stiffness and rigidity. 

The more cellulose is in cotton, the higher is the tensile strength.