We consider the fractal dimensions d(k) of the k-connected part of percolation clusters in two dimensions, generalizing the cluster (k=1) and backbone (k=2) dimensions. The codimensions (x) over tilde (k)=2-d(k) describe the asymptotic decay of the probabilities P(r,R)similar to(r/R)((x) over tilde k) that an annulus of radii rmuch less than1 and Rmuch greater than1 is traversed by k disjoint paths, all living on the percolation clusters. Using a transfer matrix approach, we obtain numerical results for (x) over tilde (k), kless than or equal to6. They are well fitted by the ansatz (x) over tilde (k)=1/12k(2)+1/48k+(1-k)C, with C=0.0181+/-0.0006. [References: 14]
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