Non-slice 3-stranded pretzel knots

In a recent study, Greene-Jabuka and Lecuona made significant progress in confirming the slice-ribbon conjecture for 3-stranded pretzel knots. However, they found an exception to this conjecture in an infinite family of knots denoted as $P(a,-a-2,-\frac{(a+1)^2}{2})$, where $a$ is an odd integer greater than 1. Lecuona and Miller further investigated these knots and showed that they are not slice unless $a$ satisfies the congruence condition $a\equiv 1, 11, 37, 47, 59 \pmod{60}$. In this note by Min Hoon Kim, Changhee Lee, and Minkyoung Song, the authors delve deeper into the remaining knots in the family $P(a,-a-2,-\frac{(a+1)^2}{2})$ that have not been proven to be non-slice. They present new findings that demonstrate that four-fifths of these remaining knots are indeed not slice. The study builds upon previous research on the slice-ribbon conjecture for general pretzel knots and links. The authors refer to works by Loncar [Lon14], Bryan [Bry17], Akbulut et al. [AKPR18], and Khandhawit et al. [KST20] as examples of other interesting results in this area. The main focus of this note is to address Lecuona's conjecture regarding the remaining knots in the family $P(a,-a-2,-\frac{(a+1)^2}{2})$. Lecuona proposed that none of these knots are topologically slice (i.e., they do not bound a locally flat embedded disk in four-dimensional space). This conjecture is particularly intriguing because it would imply that the slice-ribbon conjecture holds true for all 3-stranded pretzel knots. Lecuona previously showed that his conjecture is valid for most values of $a$, except when $a$ satisfies certain congruence conditions. Miller also contributed to this line of research by proving Lecuona's conjecture for a specific congruence condition. However, the case where $a\equiv 1, 11, 37, 47, 59 \pmod{60}$ remained open. The main result of this note is the confirmation that four-fifths of the remaining knots in the family $P(a,-a-2,-\frac{(a+1)^2}{2})$ are not topologically slice. This finding provides further evidence towards Lecuona's conjecture and contributes to the classification of all 3-stranded pretzel knots. The authors conclude by mentioning Theorem 4.1 from a related study by Wang et al. [WYF20], which provides additional insights into the structure and properties of these knots.

Confirming the Slice-Ribbon Conjecture for 3-Stranded Pretzel Knots

In a recent study, Greene-Jabuka and Lecuona made significant progress in confirming the slice-ribbon conjecture for 3-stranded pretzel knots. However, they found an exception to this conjecture in an infinite family of knots denoted as $P(a,-a-2,-\frac{(a+1)^2}{2})$, where $a$ is an odd integer greater than 1. In this note by Min Hoon Kim, Changhee Lee, and Minkyoung Song, the authors delve deeper into the remaining knots in this family that have not been proven to be non-slice. They present new findings that demonstrate that four-fifths of these remaining knots are indeed not slice.

Background

The slice-ribbon conjecture states that any knot or link can be sliced (i.e., it bounds a locally flat embedded disk) if and only if it is ribbon (i.e., it has a band presentation). This conjecture was first proposed by Fox in 1961 and has since been studied extensively by mathematicians such as Greene–Jabuka [GJ19], Lecuona [Lec20], Miller [Mil21], Bryan [Bry17], Akbulut et al. [AKPR18], Khandhawit et al. [KST20] and Wang et al.[WYF20].

Greene–Jabuka and Lecuona made significant progress towards proving this conjecture for 3-stranded pretzel knots with their research paper published in 2019. They confirmed the validity of the slice ribbon conjecture for all but one infinite family of 3 stranded pretzel knots denoted as $P(a,-a−2,-\frac{(a+1)^2}{2})$. The authors then proposed a further investigation into these remaining knots to determine whether they are topologically slice (i.e., do not bound a locally flat embedded disk).

Research Findings

Lecuona showed that his conjecture is valid for most values of $a$, except when $a$ satisfies certain congruence conditions ($a \equiv 1, 11, 37, 47, 59 \pmod{60}$). Miller also contributed to this line of research by proving Lecuona's conjecture for one specific congruence condition ($a \equiv 11 \pmod{60}$). However, the case where $a\equiv 1, 11, 37 ,47 ,59 \pmod {60}$ remained open until now. In their study Kim et al confirm that four fifths of these remaining knots are indeed not topologically slice thus providing further evidence towards Lecuona's original proposal regarding these particular families of three stranded pretzel knot being non slicable .

The main result from Kim et al’s study provides additional insights into how we classify all three stranded pretzel knot types which could lead to further discoveries about other more complex knot types such as 4 strand or 5 strand pretzels .

Conclusion

Kim et al’s research confirms four fifths of the remaining cases within the family P ( a , - a - 2 , - $\frac { ( a + 1 ) ^ 2 } { 2 } )$. This finding contributes significantly towards understanding more complex knot types while providing evidence towards Lecouana’s original proposal regarding non slicability within certain families . Further studies may help us gain insight into other aspects such as how properties like genus affects slicability which could lead to even more interesting results about higher order strands .