Khovanov homotopy type, periodic links and localizations.

Given an m-periodic link L ⊂ S 3 , we show that the Khovanov spectrum X L constructed by Lipshitz and Sarkar admits a group action. We relate the Borel cohomology of X L to the equivariant Khovanov homology of L constructed by the second author. The action of Steenrod algebra on the cohomology of X L gives an extra structure of the periodic link. Another consequence of our construction is an alternative proof of the localization formula for Khovanov homology, obtained first by Stoffregen and Zhang. By applying the Dwyer-Wilkerson theorem we express Khovanov homology of the quotient link in terms of equivariant Khovanov homology of the original link.

The method for measurement of the three-dimensional scoliosis angle from standard radiographs.

BACKGROUND: Three-dimensional idiopathic scoliosis cannot be accurately assessed with the aid of a single plane parameter - the Cobb angle. We propose a novel method for evaluating the three-dimensional (3D) pattern of scoliosis based on two X-rays (PA and lateral). The proposed method consists of the measurements of the angles between the upper endplate of the upper-end vertebra and the lower endplate of the lower-end vertebra (3D scoliosis angle). METHODS: The 3D-angles of thirty scoliosis curves were measured with either computed tomography (CT) or digitally reconstructed radiographs (DRRs): PA and lateral. CT was used as a reference. In the case of CT, the 3D angle was calculated based on the coordinates of three points situated on the upper endplate and those of three points situated on the lower endplate of the scoliosis curve. In the case of the DRR, the 3D angle was calculated using the four-angle method: the angles formed by the endplates of the curve with the transverse plane. The results were tested with the Student's t-test, and the agreement of measurements was tested with the intraclass correlation coefficient. RESULTS: There was no significant difference between the 3D-angle measurements obtained with DRRs versus CT, p > 0.05. There was, however, a significant difference between the 3D-scoliosis angle and the Cobb angle measurements performed based on the X-rays. The reproducibility and reliability of 3D angle measurements were high. CONCLUSIONS: Based on two standard radiographs, PA and lateral, it is possible to calculate the 3D scoliosis angle. The proposed method facilitates 3D-scoliosis assessment without the use of sophisticated devices. Considering the 3D nature of AIS, the 3D parameters of the spine may help to apply a more effective treatment and estimate a more precise prognosis for patient with scoliosis.