This is indeed an interesting test case. Dropped or unexpressed complements such as this are context dependent, and in some contexts may actually be ambiguous between several options. As this is in poetry, however, we tag it in the syntax database as a case of ellipsis, and identify the null complement as linking back to 'me' in the previous clause. Preference for this analysis also emerges from the relationship between vv. 1 and 2, in which v. 2 appears to specify God's knowledge of the psalmist, stated more generally in v. 1.
Yhwh, you have examined me and you know (me):
You know my sitting and standing; you perceive my thoughts at a distance . . .
The more troublesome question is why drop it at all? It is not clear that poetic structure (line length constraints) or style (alliteration, etc.) account for it. It may be that its absence leads so well into the following line. That is, even though I maintain there is ellipsis here and 'me' is restored based on the previous clause, its non-overt character leads into the specificity of what God knows about 'me' in v. 2: 'Yhwh, you have examined me and you know . . . (dramatic pause).
Professor of Old Testament
and Director of Hebrew Language Instruction
Asbury Theological Seminary, Wilmore, KY