Did Yato cut Hiyori's ties to the Far-Shore in episode 6 of Aragoto?
I found the perfect answer to this question on Reddit:

It's supposed to look ambiguous, but if he had, Hiyori probably wouldn't have even seen him in the first place, let alone remember who he is. Plus, she runs away from him in her spirit form and even has a conversation with him afterwards when she looks for him to tell him to back off. Tenjin also confirms that he didn't go through with it.
I think that part is better played in the manga - we really had to wait for the answer, since they changed some of the order of what happens in the anime and the reason Hiyori gets annoyed with Yato as well. If you'd like to see how it goes in the manga: Ch.22 P.31 - P.33, then Ch.22 P.38 - P.42, and finally Ch.23 P.43 - Ch.24 P.7.

I haven't read the manga myself but based on the anime, I can say that Tenjin allowed Yato to break his promise since it was Hiyori's own wish to remain with him.
The last and foremost evidence of her ties not being cut to the far shore is the fact that she can still turn into her spirit form (with the tail).
In conclusion: If he had indeed cut the ties, she would have returned to being a normal human who:

Would have been unable to convert to her spirit form
Would have lost all her memories of Yato
Would not have been able to see Yato

None of the above happened, so luckily, Tenjin let the promise slide.