We study an ensemble of polymers that strongly interact with the surrounding medium. One end of the polymers is fixed. but otherwise there are no restrictions on their possible Conformations, allowing also self-Crossings. We study in this paper by a Flory argument and by numerical methods the statistical properties of these in the low-temperature limit. We find that these properties strongly depend on the disorder of the surrounding medium, in contrast to the direc.ted polymer case. The statistical mechanics of polymers (ii has been a field of intense study over a number of years now, both due to their interesting properties per se, and as result of their technological importance. The problem of a self-ai'oidiJ1g polymer in a quenched disordered environment has proven to be a highly nontrivial subject within this field, and is not yet completely understood (2). Fairly recently, Kardar and Zhang (3) have solved a simplified version of this problem, substituting the self-avoidedness of the polymers by assuming that they are directed, I-e- that their conformations are all in a preferred direction without overhangs. Their conclusion is that the behavior of the polymers are govemed by a strong-disorder fixed point leading to universal critical behavior different from that of the free self-avoiding polymers. This problem has been studied earlier by a number of authors (4-7), both numerically and analytically. We study in particuliar the low temperature limit of such polymers, whith one end at a fixed position, both analytically and numerically. This problem has also been addressed in references (6) and (7). Imagine a polymer interacting with a d-dimensional random medium through a local interaction energy ~ ix) with no long-range correlations. The polymer also has a bare line tension y. The line tension can be caused by entropic and/or elastic effects. The polymer has a