We extend the quandle cocycle invariant to oriented singular knots and links using algebraic structures called oriented singquandles and assigning weight functions at both regular and singular crossings. This invariant coincides with the classical cocycle invariant for classical knots, but provides extra information about singular knots and links. The new invariant distinguishes the singular granny knot from the singular square knot.

中文翻译：

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Cocycle 不变量和定向奇异结

我们使用称为定向 singquandles 的代数结构将 quandle cocycle 不变量扩展到定向奇异结和链接，并在常规和奇异交叉处分配权重函数。这个不变量与经典结的经典循环不变量相吻合，但提供了关于奇异结和链接的额外信息。新的不变量将奇异奶奶结与奇异方结区分开来。