Tolvanen picked up an assist in Tuesday's 3-2 loss versus Colorado.
Tolvanen inked a two-year, $6.95 million contract with the Kraken on Friday, according to Elliotte Friedman of Sportsnet.
Tolvanen (undisclosed) will draw into the lineup Thursday against Minnesota, per the media site.
Tolvanen (undisclosed) won't play Tuesday versus Winnipeg.
Tolvanen notched an assist, two shots on goal, two hits and two blocked shots in Tuesday's 5-4 overtime loss to the Golden Knights.
Tolvanen scored a power-play goal on four shots in Saturday's 2-1 loss to the Oilers.
Tolvanen's (undisclosed) availability for Friday's game against St. Louis is expected to be a game-time decision, per Mike Benton of the Kraken Audio Network.
Tolvanen scored a goal on four shots and added five hits in Thursday's 4-2 loss to the Oilers.
Tolvanen posted an assist in Thursday's 4-1 win over the Capitals.
Tolvanen notched two assists, one on the power play, in Tuesday's 5-2 win over the Sabres.