The WashCycle Blog has posted details of Inauguration Day bicycle access. The area shaded in red will be pedestrians only, but the area in blue will be bicycles, pedestrians and buses only (no automobile traffic). The Red markers are bridges that will be closed to bikes.
The bicycle markers are points where the Washington Area Bicyclists Association will be offering guarded bicycle parking.
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Taking your own bike into Washington on transit is always problematic. Amtrak only allows folding bicycles, boxed full size bicycles can be checked a day in advance of travel for $15 (only the early AM train has a baggage car). Chinatown buses are your only real option to bring a full size bike to DC.
In DC Metrorail will be banning bikes from Jan 17th to Jan 2oth but Metrobuses do have bike racks.