1. Bring sugar, 1 cup water, and 3/4 of the apple peel to a boil in a medium saucepan over medium-high heat. (Reserve the remaining apple peel for garnish.) Cook, stirring occasionally, until sugar has dissolved, about 5 minutes. Transfer syrup to a medium bowl set over an ice bath; let stand, stirring occasionally, until cool. Pour through a fine sieve into another bowl; discard solids.

2. Cut apple into chunks; press through an electric juicer into a small bowl. Pour through a fine sieve into another bowl; discard any solids, and set juice aside

3. Cut remaining peel into thin, 4-inch-long strips. Tie each strip into a knot; set aside in a bowl of water with lemon juice.

4. Fill a cocktail shaker with ice. Add vodka, 2 ounces syrup, and 1/3 cup apple juice; shake. Divide between 2 martini glasses; garnish with apple knots.