The Effect of Irreducible Dislocation on Functional Outcomes in Knees With Multiligament Injuries: A Matched-Cohort Analysis

Background: Most cases of bicruciate knee dislocation (KD) with associated posteromedial disruption (KD-IIIM) are reducible, but some cannot be reduced by closed reduction because of soft tissue incarceration. Purpose: To compare the clinical characteristics and functional outcomes of KD-IIIM injuries in patients with or without incarceration of soft tissue requiring open or arthroscopic reduction. Study Design: Cohort study; Level of evidence, 3. Methods: This retrospective cohort study of patients with KD was conducted between January 2013 and December 2017 at a single large institution. We applied a 1:2 matching ratio between patients with irreducible KD-IIIM injuries (irreducible group; n = 14) and those with reducible KD-IIIM injuries (control group; n = 28). There were 13 patients in the irreducible group and 25 in the control group who completed follow-up (≥2 years) and were included in our analysis. The efficacy of treatment in patients with KD was evaluated based on range of motion, the Tegner score, the Lysholm score, and the International Knee Documentation Committee (IKDC) score. Results: At the end of follow-up, the mean Tegner score was 4.5 (range, 4-6), the mean Lysholm score was 79.2 (range, 60-95), and the mean IKDC score was 78.6 (range, 60.9-95.4) in the irreducible group. The respective results in the control group were 4.6 (range, 3-8), 83.1 (range, 39-100), and 80.6 (range, 42.5-96.6). These scores did not differ significantly between the 2 groups. Similarly, mean range of motion was similar between groups (irreducible, 118.1°; control, 124.8°). In the irreducible group, the acute subgroup showed significantly higher Lysholm and IKDC scores than the chronic subgroup, while the acute and chronic subgroups in the control group showed no significant differences in these respective outcome scores. Conclusion: In the present study, the treatment of irreducible KD led to similar functional outcomes compared with reducible KD. However, the treatment of chronic irreducible KD led to worse outcomes compared with acute irreducible KD, and therefore, urgent reduction is recommended in these patients.

Lattices From Tight Frames and Vertex Transitive Graphs

We show that real tight frames that generate lattices must be rational. In the case of irreducible group frames, we show that the corresponding lattice is always strongly eutactic. We use this observation to describe a construction of strongly eutactic lattices from vertex transitive graphs. We show that such lattices exist in arbitrarily large dimensions and discuss some examples arising from complete graphs, product graphs, as well as some other notable examples of graphs. In particular, some well-known root lattices and those related to them can be recovered this way. We discuss various properties of this construction and also mention some potential applications of lattices generated by incoherent systems of vectors.

THE SOLUTION OF THE THREE-DIMENSIONAL TWISTED GROUP LATTICES

We define new lattices called d-dimensional twisted group lattices. They are similar to ordinary lattices except that abstract screw dislocations are present at the centers of all plaquettes. Some physical aspects are enumerated. We consider the statistical mechanics system of free propagation on the three-dimensional twisted group lattice. For this case, the partition function is explicitly computed by finding the irreducible group representations.

A Geometrical Study of the Alternating and Symmetric Groups

Let a finite group Τ be represented as an irreducible group of order N of linear substitutions on n variables,The variables may be chosen so that the substitutions of the group leave invariant the Hermitian form