The Mechaber writes that someone who neglected to count the omer for one full day may not continue counting the rest of the omer with a beracha. However, someone who is not sure if he counted, may continue counting with a beracha. Someone who realized during bein hashemashos that he has not yet counted should count then, because it is possible that it is still part of the previous day. But he should count without a beracha, since it is also possible that it is already the next day. There is a machlokes among the poskim as to whether he may continue counting future nights with a beracha. Rav Shlomo Zalman sides with those who say he cannot continue to count with a beracha. However, many contemporary poskim hold that he may continue counting with a beracha. (According to Rav Elyashiv, this is only the case if he remembers to count within 13.5 minutes after shkia. According to the Or Ltzion, he has up to 25 minutes after shkia to rectify the situation.)
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Hilchos Tzitzis 12 (page 47)
Rabbeinu Tam's Opinion on Torn Tzitzis
The Halacha of Torn Tzitzis
Location of "Kdei Aniva"
Rabbeinu Tam's opinion on torn tzitzis
According to Rabbeinu Tam, torn tzitzis remain kosher only if they are still kdei aniva and at least two of the four [doubled] strings are intact (i.e. both ends of two doubled strings are the full twelve godlin -- see Shulchan Aruch 11:4). According to the Chazon Ish, the two remaining strings only need to be one tefach on each side.
According to the Mechaber, the halacha follows the Rosh's opinion, which holds that as long as there remains kdei aniva on the four [doubled] strings, the tzitzis are kosher and a beracha can be recited. When possible, through, it is proper to be stringent like Rabbenu Tam (i.e. two complete strings). According to the Rama, the halacha follows the opinion of Rabbeinu Tam.
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Location of kdei aniva
The Rishonim discuss where the kdei aniva should be on the tzitzis. According to Rashi, it must be on the loose part of the strings. According to the Ri, it can even be in the braided section of the tzitzis. It is proper to follow Rashi's opinion in this matter, but one may rely on the Ri to wear the tzitzis without reciting a beracha if other kosher tzitzis are unavailable
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According to the Rosh, if tzitzis strings tore after they were placed on a garment they remain kosher if the length of kdei aniva is left on all four [doubled] strings. It is unclear whether the tzitzis are pasul if only one of the strings meets the requirement of kdei aniva.
The main opinion as to the length of kdei aniva is the length of string needed to wrap a torn string around all the torn strings. In a time of pressing need, it is adequate if it can wrap around just one other string.
A torn string that was reconnected with a permanent knot may be used for tzitzis. However, a string that tore and was disqualified after being placed on the garment cannot be repaired through tying.
Wearing tzitzis in a reshus harabim on Shabbos
Wearing a garment without proper tzitzis on Shabbos
Wearing a garment that is less than the shiur with tzitzis on Shabbos