Here are sections from Topeka's Traffic Regulations that are pertinent to bicycling. Read the original regulations in the city's Traffic and Vehicles Codebook (pdf).
Sec. 142-6. Display of license plates.
(a) Every vehicle, motor vehicle and motorized bicycle parked, stopped, standing or operated on the streets, public areas, park grounds, public or private parking lots or other publicly accessible areas of the city shall display a state license plate as provided by the laws of the state.
Sec. 142-182. Bicycle trail.
(a) The areas described in this section shall be reserved for exclusive use by bicycles and/or pedestrians:
(1) The top of the levee system known as the Oakland Unit of the Topeka Flood Protection Project, in such areas specifically designated and marked by appropriate signs, such signs to be installed by the city.
(2) The Shunga Bicycle/Pedestrian Trail, in such areas specifically designated and marked by appropriate signs at the trail entrances; such signs to be installed by the city.
(b) No motor vehicles shall be driven in the area designated by this section except authorized maintenance vehicles of the city. Bicycles shall be ridden only upon the top of the levee system and on designated areas of the Shunga Bicycle/Pedestrian Trail. Bicycles shall not be ridden on the face of the levee.
(Code 1981, § 43-589)
Sec. 142-454. Use of skateboards, roller skates or bicycles prohibited.
It shall be unlawful for any person other than a law enforcement officer in the performance of his official duties to use or operate any skateboard, roller skates or bicycle at any time within any municipal parking stations owned and/or operated by the city. A person convicted of violating this section shall be deemed guilty of a misdemeanor and punished in accordance with city ordinances.
(Ord. No. 16594, § 1(43-462), 5-11-93)
ARTICLE IX. BICYCLES
DIVISION 1.
GENERALLY
Sec. 142-611. Riding on sidewalks regulated.
(a) No person shall ride a bicycle upon a sidewalk within the central business district.
(b) The traffic engineer is authorized to erect signs on any sidewalk or roadway prohibiting the riding of bicycles thereon by any person, and when such signs are in place, it shall be unlawful for any person to disobey such sign.
(c) Whenever any person is riding a bicycle upon a sidewalk, such person shall yield the right-of-way to any pedestrian, and shall give audible signal before overtaking and passing such pedestrian.
(d) No person riding a bicycle on a sidewalk shall exceed a speed of five miles per hour.
(e) This section shall not apply to a law enforcement officer engaged in the performance of his official duties.
(Ord. No. 16594, § 2(43-586), 5-11-93)