Only if you are Sara Palin. (I'm not sure you understand the distance involved)
Seriously though, it's actually probably cheaper to lay it across the equator than try to put a cable across the bearing straight. The ocean is pretty turbulent in the straight, it's pretty turbulent any time you get closer to the poles. There were articles the other day that global warming has opened up so much water this summer north of canada that they've had 15' (3m) waves. You need calm water (including underwater currents of which there is a big one in the straight) when dropping miles of cable to the bottom of the ocean.
Consider for a moment you're in water a mile deep laying cable. The total suspended amount of cable from the boat to the bottom of the ocean is more than 2 miles, and with the armoring and other features may weigh several thousand pounds and have enough cross-sectional area that a couple mile per hour current passing over the cable could capsize your ship, drag the ship under or shear the cable in half. The forces they deal with when laying the cables is HUGE without waves and strong ocean currents. They need massive boats, calm waters and very careful monitoring to put these cables down. If you tear the cable in half while laying it you've got to find the end, drag it and several miles of cable off the bottom, cut the end and resplice everything including the power and armoring, water proof the splices and then relay the whole thing along with continuing to lay new cable.
Where possible they run the cable over islands to make the runs through the ocean shorter. But trying to lay the cable across the bearing straight would be a major challenge. It would be an even bigger challenge once you got it across the bearing straight to get it anywhere because it's about 2000 miles from anything on either end and it would come ashore in some of the most hostile territory on earth (such as all the ground being permafrost which means you can't really bury the cable).