Green’s Dictionary of Slang

conk n.2

also konk
[Congolene, a fiery liquid, combining lye, eggs, potatoes and other ingredients, used in the artificial straightening of naturally kinky black (racially) hair]
(US black)

1. pomade; hair grease.

[US]Pittsburgh Courier (PA) 9 Sept. 11/1: Brother Jones has cropped his hair short to save the [cash] he once spent on ‘Konk’ and now his moss isn’t glossy any more .
[US] ‘Jiver’s Bible’ in D. Burley Orig. Hbk of Harlem Jive.

2. hair that has been straightened through the application of a special mixture.

[US]D. Burley N.Y. Amsterdam News 19 Oct, 20: This is my conk, and I don’t give a honk.
[US]C. Himes ‘Let Me at the Enemy’ in Coll. Stories (1990) 36: This George Brown was strictly an icky, drape-shaped in a fine brown zoot with a pancho conk slicker’n mine.
[US]N.Y. Age 19 Aug. 9: [headline] me’n my konk.
[US]H. Simmons Corner Boy 14: The little man with the conk tossed Jake a dollar.
[US]H. Rap Brown Die, Nigger Die! (con. late 1950s) 24: He’d drop by the school and be vined down. He was clean, jim. Had him a conk then and he knew he was ready.
[US]E. Torres Carlito’s Way 9: You had a pimp name of Red Conk on account of he conked his hair red.
[US](con. 1986) G. Pelecanos Sweet Forever 44: French cuffs . . . man had a conk and shit.
[US]‘Touré’ Portable Promised Land (ms.) 156: We Words (My Favorite Things) [...] Chump. Crunk. Conk.
[US]T. Swerdlow Straight Dope [ebook] [He] rocks a “conk” he’s probably had since doo-wop night at the Shrine Auditorium.

In compounds

conkhead (n.)

1. (US) one who uses hair oil on artificially straightened hair, thus adj. conk-headed.

[US]letter in N.Y. Age 29 Dec. 9/4: Nothing nauseates me more than the sight of these ‘conkheads.’ I often wonder how much barrel of grease they must ladle onto their skulls.
[US]R.D. Pharr S.R.O. (1998) 39: He was short, dark, conk-headed.

2. (US black) a black person.

[US]Hal Ellson ‘Decoy’ in Tell Them Nothing (1956) 40: A conkhead who don’t know he’s living says [etc.]. [Ibid.] ‘I Didn’t See a Thing’ 141: Two men. Bug-Eye and a conkhead.
conk job (n.)

(US black) a black hairstyle that has been straightened.

[US]G. Tate ‘Atomic Dog’ in Flyboy in the Buttermilk (1992) 29: My man was chemically processing hair – mixing down wave, curl, and conk jobs for all the slick splib hustlers.