## 1 Introduction  
The countermovement jump (CMJ) is a vital skill used frequently in many sports and recreational activities [1]. CMJ is usually combined with an arm-swing, which plays a crucial role in improving performance. Numerous previous studies have focused on the effect of arm-swing on the CMJ. It hasbeen widely reported that jump height could be improved by 14-28% when using arm-swing [2; 3; 4]. Three theories, \"transmission of force\" [5], \"pull\" [3], and \"joint torque augmentation\" [1], have been proposed to explain the effect of the arm-swing on vertical jump height.  
These theories can also be applied when discussing the role of the core in the transmission of force and energy during takeoff. Previous studies [6; 7; 8; 9] have indicated that core strength is critical during energy transfer. [PERSON] et al. [7] defined core strength as: \"the ability to control the position and motion of the trunk over the pelvis and leg to allow optimum production, transfer, and control of force and motion to the terminal segment.\" Core strength also plays a vital role in force transmission between the upper and lower body [10; 11; 12]. It has been suggested that the core musculature is responsible for keeping the stability of the trunk and is crucial in transferring energy between the larger torso and smaller extremities in sport activities [8; 9].  
Therefore, core strength appears to play an essential role in energy and force transmissions, crucial in vertical jumps, especially with arm-swing. However, previous research has reported inconsistent conclusions on the relationship between core strength and vertical jump [13]. Some studies observed positive results [7; 14; 15; 16] that core strength could enhance vertical jump performance, but others disagreed [17; 18; 19]. A variety of different core strength tests or training have been employed in previous studies. The \"principle of dynamic correspondence\" was applied to explain how training exercise improves performance and it was claimed that specific training modalities must be in harmony with those parameters of movement that characterize the structure of competition technique [20]. Only when core strength training is consistent with the kinetic and kinematic structures of the CMJ will the core strength effectively improve CMJ performance [12; 20; 21]. Thus, the confounding conclusions of core strength are mainly due to the neglect of the \"specificity\" of core strength to the sport event [6; 8; 12; 22]. Although a recent study from authors [12] has disclosed the \"specificity\" of core stability by investigating the relationship between five core stability-related measurements and CMJ height, there is no study devoted to the relationship between different core strength types and exercise performance.  
To detect the relationships between different types of core strength and CMJ performance, in order to help coaching and athlete development, we examined 12 core strength measurements, including dynamic strength at different speeds and static core strength at different positions. The study aimed to investigate whether the dynamic and static core strength measurements have different correlations with CMJ height with and without arm-swing. We hypothesized that CMJ heights, both with and without arm-swing, have significant but different relationships with all 12 core strength measurements.