The creation of unwound or unknotted loops of DNA is facilitated by topoisomerases, which alter the conformation of DNA by cleaving the phosphate backbone on one or both the strands. A major unexplored question has been the possibility of topoisomerases failing to unpack supercoiled DNA on any one of the several knots, thereby failing to initiate the unwinding process. Computing the unknotting number enables biologists to estimate the smallest amount of times the topoisomerase must act to unpack DNA. In this paper, we define a new variant called the fault-tolerant unknotting number t ( K ) of the unknotting number u ( K ) for a knot K . Besides, t ( K ) is related to the unknotting and knotting phenomena of DNA. We determine t ( D ) for some families of knot diagrams and conclude that the difference between u ( D ) and t ( D ) for a knot diagram D can be arbitrarily large. We also conjectured the exis...

中文翻译：

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节的不解链数的变体在生物分子拓扑学中的应用

拓扑异构酶促进了DNA的解链或无链环的产生，拓扑异构酶通过切割一条或两条链上的磷酸骨架来改变DNA的构象。一个主要的尚未探索的问题是拓扑异构酶可能无法解开多个结中任一结上的超螺旋DNA，从而无法启动解链过程。计算未知数使生物学家能够估计拓扑异构酶解聚DNA所必须起作用的最小次数。在本文中，我们定义了一个新的变体，称为结K的不知道结点数u（K）的容错不知道结点数t（K）。此外，t（K）与DNA的不打结现象有关。我们确定了某些结图族的t（D），并得出结论，结图D的u（D）和t（D）之间的差可以任意大。我们还推测了存在...