Presenting the cork to the host in a restaurant can be traced back during the 1800s. Since patrons in the restaurant do not have any ways of knowing if they have received the exact wine that they ordered, so as the retailers of wine, the wine companies have decided to mark their brand logo to the cork.
Now, to simply put, the significance of presenting the cork to the host in a restaurant is:to let the host see if the wine is branded and is from the correct producer and vintage as what you ordered.to let the host inspect the integrity of the cork.
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Upon the approval of the server even the somm will be open the bottle of the win and place it in the cork on the table and the coaster. As the first sip is poured the cork is there just to confirm that the branding matches the label.