论文标题
马祖尔打结的强斜率猜想和交叉数字
The Strong Slope Conjecture and crossing numbers for Mazur doubles of knots
论文作者
论文摘要
Garoufalidis提出的斜率猜想断言，有色琼斯多项式的程度决定了边界斜率及其改进，是Kalfagianni和Tran提出的强斜率猜想，并断言，在程度上，线性项在程度上决定了满足坡度猜想的必要表面的拓扑。在某些假设下，我们表明，如果原始结的话，Mazur的打结会满足强烈的斜率猜想。因此，通过有限的电缆序列获得的任何结，无扭曲的W-与W> 0的列为Whitehead双打，连接的总和和Mazur双打b--毫无用处的打结或圆环结都能满足强斜率的猜想。另一方面，可能值得一提的是，在这些假设下，如果存在一个小于-1/4的斜坡的结，则其Mazur double会为强斜率猜想提供反例，或者具有与任何结节琼斯表面无关的琼斯表面。 在Kalfagianni和Lee的工作之后，我们还使用结果表明，具有微不足道的Writhe的足够结K的Mazur双倍的交叉数为9C（K）+2或9C（K）（K）+3。
The Slope Conjecture proposed by Garoufalidis asserts that the degree of the colored Jones polynomial determines a boundary slope, and its refinement, the Strong Slope Conjecture proposed by Kalfagianni and Tran asserts that the linear term in the degree determines the topology of an essential surface that satisfies the Slope Conjecture. Under certain hypotheses, we show that Mazur doubles of knots satisfy the Strong Slope Conjecture if the original knot does. Consequently, any knot obtained by a finite sequence of cabling, untwisted w--generalized Whitehead doublings with w > 0, connected sums and Mazur doublings of B--adequate knots or torus knots satisfies the Strong Slope Conjecture. On the other hand, it may be worth mentioning that under these hypotheses, if there exists a knot with a Jones slope less than -1/4, then its Mazur double would either provide a counterexample to the Strong Slope Conjecture or have a Jones surface that is unrelated to any Jones surface of the knot. Following work of Kalfagianni and Lee, we also use our results to show that the Mazur double of an adequate knot K with trivial writhe has crossing number either 9c(K)+2 or 9c(K)+3.
