论文标题
在不可定向的$ 4 $基因的圆环结中
On a Refinement of the Non-Orientable $4$-genus of Torus Knots
论文作者
论文摘要
Batson在制定了Milnor猜想的不可方向的类似物时，Batson开发了一种优雅的结构，在$ s^3 $中的给定圆环结中，在$ b^4 $中产生了光滑的不可方向的跨度表面。虽然Lobb表明Batson的表面并不总是最小化不可取向的$ 4 $ genus，但我们证明它们总是在共享其正常欧拉数字的表面中确实会最小化。我们还完全确定了不可定向表面的正常欧拉数和第一个betti数的对，其边界位于一类的圆环结中，蝙蝠的表面是不可定向的$ 4 $ genus minimizizers。
In formulating a non-orientable analogue of the Milnor Conjecture on the $4$-genus of torus knots, Batson developed an elegant construction that produces a smooth non-orientable spanning surface in $B^4$ for a given torus knot in $S^3$. While Lobb showed that Batson's surfaces do not always minimize the non-orientable $4$-genus, we prove that they always do minimize among surfaces that share their normal Euler number. We also completely determine the possible pairs of normal Euler number and first Betti number for non-orientable surfaces whose boundary lies in a class of torus knots for which Batson's surfaces are non-orientable $4$-genus minimizers.
