• Honam Mathematical Journal
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• v.26 no.2
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• pp.237-246
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• 2004

In this paper, three kinds of minimal digital pseudotori $DT_6$, $DT^{\prime}_{18}$, $DT^{{\prime}{\prime}}_{26}$, which are derived from the minimal simple 4- and 8-curves, $MSC_4$ and $MSC^{\prime}_8$, are shown and are proved not to be digitally ${\kappa}$-homotopy equivalent to each other, where ${\kappa}{\in}\{6,\;18,\;26\}$. Furthermore, the digital topological properties of the minimal digital ${\kappa}$-pseudotori are investigated in the digital homotopical point of view, where ${\kappa}{\in}\{6,\;18,\;26\}$.

• Honam Mathematical Journal
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• v.26 no.4
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• pp.411-421
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• 2004

A digital $(k_0,\;k_1)$-homotopy is induced from digital $(k_0,\;k_1)$-continuity with the n kinds of $k_i$-adjacency relations in ${\mathbb{Z}}^n$, $i{\in}\{0,\;1\}$. The k-fundamental group, ${\pi}^k_1(X,\;x_0)$, is derived from the pointed digital k-homotopy, $k{\in}\{3^n-1(n{\geq}2),\;3^n-{\sum}^{r-2}_{k=0}C^n_k2^{n-k}-1(2{\leq}r{\leq}n-1(n{\geq}3)),\;2n(n{\geq}1)\}$. In this paper two kinds of digital 8-pseudotori stemmed from the minimal simple closed 4-curve and the minimal simple closed 8-curve with 8-contractibility or without 8-contractibility, e.g., $DT_8$ and $DT^{\prime}_8$, are introduced and their digital topological properties are studied by the calculation of the k-fundamental groups, $k{\in}\{8,\;32,\;64,\;80\}$.