In this paper, we study the chord index of virtual knots, which can be thought of as an extension of the chord parity. We show how to use the chord index to enhance the quandle coloring invariants. The notion of indexed quandle is introduced, which generalizes the quandle idea. Some applications of this new invariant is discussed. We also study how to define a generalized chord index via a fixed finite biquandle. Finally, the chord index and its applications in twisted knot theory are discussed.

中文翻译：

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和弦索引、它的定义、应用和概括

在本文中，我们研究了虚拟结的和弦指数，它可以被认为是和弦奇偶性的延伸。我们展示了如何使用和弦索引来增强 quandle 着色不变量。引入了索引 quandle 的概念，它概括了 quandle 的想法。讨论了这个新不变量的一些应用。我们还研究了如何通过固定的有限双弦来定义广义和弦指数。最后讨论了弦指数及其在扭结理论中的应用。