Use integers or fractions, but nothing with decimal points, to avoid this problem.

There are other, but much higher, limits on the ability to calculate exact values without decimal points, but you may never encounter those in your work.

Try using 1/10 (or, say SetPrecision[0.1, 100]) instead of 0.1 to avoid doing the entire computation at machine precision. Also the definition of t should probably use t[k_, i_, j_] instead of t[k, i, j_]. Then it returns positive numbers

In[14]:= N[{p[2], p[4]}, 20]

Out[14]= {4.9209276886444687077*10^-2094, 9.8369344496002929467*10^-2091}