Latin <digitus> 'finger, toe' and Germanic *taik- (represented by English <token> and <teach>) are commonly referred to PIE *deik^- 'to show', with one ad-hoc mechanism or another invoked to explain the unexpected reflexes of */k^/. In my view no such mechanisms are required.
One problem with referring <digitus> to *deik^- (or its ad-hoc variant *deig^-) is that its supposed sense 'pointer, indicator' is active, whereas the morphology of the presumed derivation is generally passive, as in <libitum>, <alitum>; one does not find /o/-stem nouns *libitus 'lover, pleaser', *alitus 'feeder, nourisher'. Thus I prefer to regard <digitus> as a back-formation from a frequentative verb. The starting point is an Old Latin verb *end-ago: 'I drive (something) in', leading regularly to Early Classical Latin *indigo: 'id.', with frequentative *indigito: 'I repeatedly drive (something) in'. (We may compare Class. Lat. <indigeo:> from OL *end-egeo: 'I am lacking in', and the freq. <agito:> from <ago:>.) One likely application of *indigito: is to sexual foreplay, commonly performed with the middle finger. After the loss of productivity of the prefix *end(o)-, *ind(i)-, the etymology of *indigito: would be opaque, its frequentative sense would have faded with use, and the extraction of <digitus> 'middle finger' would be straightforward, making *indigito: look like an ordinary denominative, 'I insert my middle finger'. Finally, the sense of <digitus> would be generalized to '(any) finger', and even 'toe'. (The usual explanation of <digitus> as 'pointer' is no better in this regard, since it requires 'index finger' to be similarly generalized.) I do not insist that <digitus> necessarily entered mainstream Latin from the realm of the bedroom; it might well have come from the potter's or baker's craft, but foreplay is easier to illustrate.
PIE *déik^- is regularly reflected as Gmc. *teih- (Gothic <gateihan>, Old English <te:on>, Old High German <zi:han> 'to announce'), with the /o/-grade derivative *dóik^wa:- becoming Gmc. *taihwo:- 'toe' (OE <ta:he>, <ta:>, OHG <ze:ha>, Old Norse <ta:>). But Gmc. *taik- also agrees so well in sense with PIE *deik^- that the connection can hardly be abandoned. The difficulty with the ordinary application of Kluge's Law is that PIE noun-formation does not typically involve a root in the /o/-grade followed by an oxytone /n/-suffix. Kluge himself (Die germanische Consonantendehnung, PBB 9:149-186 [1884]) derived Gmc. *taikno-, *taikni- from PIE *dóigno-, *dóigni- [sic velars] in which the */g/ is supposedly explained by an old IE soundlaw proposed by Osthoff which lumps together Sanskrit sandhi with Latin <dignus> and <vi:ginti:> (pp. 180-1). In fact these Latin words are unconnected with Skt. sandhi and involve independent soundlaws (*dek-nos > *degnos > dignus with [n,] as allophone of /g/; *veikm.ti: > *veign.ti: > vi:ginti:). Now, apart from /n/-suffixation, we also have oxytone /o/-grade deverbatives from PIE roots with final /n/, like Greek <phoné:> 'homicide, slaughter' from PIE *gWHoná:- (or if you prefer *gWHonéh2-), root *gWHen- 'to strike, slay'. Here we also find Greek <poiné:> 'payment, price, penalty' (beside paroxytone <Poíne:> 'Vengeance') from PIE *kWei-n- 'to pay, atone, compensate' which appears with and without /n/-extension in Grk. <tío:> 'I pay honor, esteem, value' and <tíno:> 'I pay (a penalty or debt)' (hence <poinè:n tîsai> 'to pay a penalty' is an old cognate-accusative phrase obscured by sound-shifting). It appears here that <poiné:> represents a PIE *kWoiná:- (or if you prefer *kWoinéh2-) formed from the adjective *kWóino- after the fixation of ablaut; otherwise we should expect *kWiná:- directly from *kWei-n- leading to Grk. *tiné:. Regardless, <poiné:> provides direct evidence for /o/-grade oxytone formation from a root extended by /n/.
This mechanism when applied to *deik^-n- yields a PIE adjective *dóik^no- and a feminine noun *doik^ná:- 'show, sign, illustration' vel sim. which would yield a strong feminine *taikko:- in Germanic after applying the applicable soundlaws up to Kluge's. Kluge himself demanded the Common Germanic shortening of geminates created by this process after long vowels and diphthongs (p. 183), so *taiko:- would result. On the other hand Kauffmann (Zur Geschichte des germanischen Consonantismus, PBB 12:504-547 [1887]) found it hard to explain West Gmc. (especially OHG) data this way, so he proposed instead that geminates followed immediately by other consonants were shortened (pp. 512-5). In this view Bavarian <zagng> 'sign' continues the gemination of *taikko:-, but the forms reflecting single */k/ require further steps. To me the simplest solution is to propose that an adjective *taikka- 'shown, manifest' vel sim. was extracted from *taikko:-, and then used to form new nouns *taikkni- f., *taikkna- n. 'sign', which were reduced to *taikni-, *taikna- by Kauffmann's shortening (whence Goth. <taikns> and <taikn>, OHG <zeihhan>, OE <ta:c(e)n>, Eng. <token>). From *taikna- came the verb *taikjan 'to make shown, demonstrate, instruct' (whence OE <tæ:c(e)an>, Eng. <teach>).
PIE *peik^- 'to be hostile toward, betray, deceive' is similarly given a variant *peig^- by some scholars on the basis of Lat. <piget> 'it annoys, disgusts, shames', <piger> 'disinclined, unwilling, lazy', and some Gmc. forms. The Latin words do not fit well semantically and are better assigned to a separate root *peig(^)- 'to tire, fatigue, disincline' vel sim. The /o/-grade *póik^o- is regularly reflected in Gmc. *(ga)faiha- 'hostile (with)' (OE <(ge)fa:h> 'enemy', Eng. <foe>), with *póik^a:- in Goth. <faiho:> 'deceit, fraud' and *poik^yó- in Gmc. *faigja- (OE <fæge> 'fated to die', Eng. <fey>). As with *deik^-n-, from PIE *peik^-n- I derive *poik^ná:- 'betrayal, deception, fraud' vel sim., Gmc. *faikko:-, from which *faikka- 'deceptive, fraudulent' was extracted and a new noun formed, *faik(k)na- n. 'deceit, evil, crime' (OE <fa:c(e)n>, Old Saxon <fe:can>, OHG <feihhan>; ON <feikn> 'ruin'). Since I reject the variant *peig^-, I must explain OE <ficol> 'treacherous, fickle', <gefic> 'deceit, fraud', and <befician> 'to betray' by recourse to a lost zero-grade weak noun, PIE *pik^nón- 'deceiver, betrayer', Gmc. *fikkan- (OE *ficca), and analogy with OE reflexes of *steig- 'to prick, goad, stick'. Here we find a weak noun *stignón- 'pricker, goad' (Gmc. *stikkan-, OE <sticca> 'stick'), an adjective *stígolo- 'prickly, sticky, difficult' (Gmc. *stikala-, OE <sticol> 'steep'), and a denominative verb *stíga:- (WGmc *stiko:jan, OE <stician> 'to stab, stick'). I presume that the WGmc analogical forms *fikala- 'treacherous' and *bi-fiko:jan 'to betray' stood to *fikkan- as the inherited *stikala- and *stiko:jan stood to *stikkan-, while <gefic> could have been formed within Old English itself.
DGK