Etymology
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*kes- 
Proto-Indo-European root meaning "to cut."

It forms all or part of: caret; cashier (v.) "dismiss;" cassation; caste; castellan; castellated; Castile; castle; castigate; castrate; castration; chaste; chastity; chateau; chatelaine; Chester; forecastle; incest; quash (v.) "make void, annul."

It is the hypothetical source of/evidence for its existence is provided by: Sanskrit sastra- "knife, dagger;" Greek keazein "to split;" Latin carere "to be cut off from," cassus "empty, void;" Old Church Slavonic kosa "scythe."
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*gleubh- 

Proto-Indo-European root meaning "to tear apart, cleave." 

It forms all or part of: cleave (v.1) "to split, part or divide by force;" cleft; clever; clevis; clove (n.2)  "slice of garlic;" glyptodon; hieroglyphic; petroglyph.

It is the hypothetical source of/evidence for its existence is provided by: Greek glyphe "a carving," glyphein "to hollow out, cut out with a knife, engrave, carve;" Latin glubere "to peel, shell, strip;" Old High German klioban, Old English cleofan, Old Norse kljufa "to cleave," Old Norse klofi, Middle Dutch clove "a cleft."  

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*skel- (1)
also *kel-, Proto-Indo-European root meaning "to cut."

It forms all or part of: coulter; cutlass; half; halve; scale (n.1) "skin plates on fish or snakes;" scale (n.2) "weighing instrument;" scalene; scallop; scalp; scalpel; school (n.2) "group of fish;" sculpture; shale; sheldrake; shelf; shell; shield; shoal (n.2) "large number;" skoal; skill.

It is the hypothetical source of/evidence for its existence is provided by: Latin culter "knife," scalpere "to cut, scrape;" Old Church Slavonic skolika "mussel, shell," Russian skala "rind, bark," Lithuanian skelti "split," Old English scell "shell," scalu "drinking cup, bowl, scale of a balance."
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*sek- 

Proto-Indo-European root meaning "to cut." It forms all or part of: bisect; dissect; hacksaw; insect; intersect; resect; saw (n.1) "cutting tool;" Saxon; scythe; secant; secateurs; sect; section; sector; sedge; segment; skin; skinflint; skinny; transect.

It is the hypothetical source of/evidence for its existence is provided by: Hittite shakk- "to know, pay attention to;" Latin secare "to cut," sectio "a cutting, cutting off, division;" Old Church Slavonic seko, sešti "to cut," sečivo "ax, hatchet," Russian seč' "to cut to pieces;" Lithuanian įsėkti "to engrave, carve;" Albanian šate "mattock;" Old Saxon segasna, Old English sigðe "scythe;" Old English secg "sword," seax "knife, short sword;" Old Irish doescim "I cut."

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