论文标题
在$ q $形式的farey和$ q $ formed的真实二次非理性数字上的同源解释
On $q$-deformed Farey sum and a homological interpretation of $q$-deformed real quadratic irrational numbers
论文作者
论文摘要
Bapat，Becker和Licata通过常规的持续分数引入了左右$ Q $ $ Q $的理性数字，他们对左右$ Q $ $ Q $ $ Q $ QUEN-FORMATION的合理数字进行了同源解释。在本文中，我们专注于负的持续分数，定义为剩下的$ q $成型的负持续分数。我们提供了一个计算基于$ q $ q $ q $ q $ quy-qunformed的理性数字的$ q $ fory的公式。我们使用此公式为左$ q $ formed的理性数字与相应理性结的琼斯多项式之间的关系提供了组合证明，这是Bapat，Becker和Licata使用同源技术证明的。最后，我们结合了他们的工作和$ q $ formed的farey sum，并对$ q $ defformed farey sum进行了同源解释。我们还提供了一种方法，可以找到真正的二次非理性数字和同源代数之间的关系。
The left and right $q$-deformed rational numbers were introduced by Bapat, Becker and Licata via regular continued fractions, and they gave a homological interpretation for left and right $q$-deformed rational numbers. In the present paper, we focus on negative continued fractions and defined left $q$-deformed negative continued fractions. We give a formula for computing the $q$-deformed Farey sum of the left $q$-deformed rational numbers based on it. We use this formula to give a combinatorial proof of the relationship between the left $q$-deformed rational number and the Jones polynomial of the corresponding rational knot which was proved by Bapat, Becker and Licata using a homological technique. Finally, we combine their work and the $q$-deformed Farey sum, and give a homological interpretation of the $q$-deformed Farey sum. We also give an approach to finding a relationship between real quadratic irrational numbers and homological algebra.
