Eigenvalue hypothesis for multi-strand braids

Computing polynomial form of the colored HOMFLY-PT for non-arborescent knots obtained from three or more strand braids is still an open problem. One of the efficient methods suggested for the three-strand braids relies on the eigenvalue hypothesis which uses the Yang-Baxter equation to express the answer through the eigenvalues of the ${\cal R}$-matrix. In this paper, we generalize the hypothesis to higher number of strands in the braid where commuting relations of non-neighbouring $\mathcal{R}$ matrices are also incorporated. By solving these equations, we determine the explicit form for $\mathcal{R}$-matrices and the inclusive Racah matrices in terms of braiding eigenvalues (for matrices of size up to 6 by 6). For comparison, we briefly discuss the highest weight method for four-strand braids carrying fundamental and symmetric rank two $SU_q(N)$ representation. Specifically, we present all the inclusive Racah matrices for representation $[2]$ and compare with the matrices obtained from eigenvalue hypothesis.Comment: 23 page

Fingertip reconstruction: study of functional and cosmetic results of various local flaps and grafting techniques in patients attending a tertiary care center in North India

Background: Finger injuries are becoming more common with the increasing use of mechanical, industrial, and household appliances. There are six main types of repair for this injury. The replacement of skin loss in digital injuries, particularly on the volar aspects, is an important part of hand-injury management. It would be of value to know the best type of cover, especially for sensory reinnervation as well as for subjective evaluation, cosmetic appearance and freedom from complications. The objective of this study was to measure incidence of age, sex, site, pattern of injury as well as to evaluate and compare the functional and cosmetic results of the various local flap and grafting techniques in finger reconstruction distal to metacarpophalangeal joint.Methods: In a prospective study between May 2014 to October 2018 consisting of 50 patients (43 males, 7 females), aged between 2 years and 54 years, with soft tissue loss distal to metacarpophalangeal joint were treated with either free split-thickness skin grafts, palmar flaps, cross-finger flap or pedicle flaps from the abdomen. The follow-up period ranged from 10 days to 15 months from the date of surgery. Subjective and objective evaluation was done and graded as excellent, good, fair and poor.Results: In this series there were 2 good and 6 fair results for split skin grafting; 9 good and 6 fair results for v-y plasty; 5 excellent, 12 good and 4 fair results for cross finger flap; 1 good and 1 fair result for radial artery based flap; 1 excellent and 1 good for first metacarpal artery based flap and 2 good results for abdominal flap. Overall 6(12%) had excellent, 27(54%) had good and 17(34%) had fair results. None of them had poor results.Conclusions: The group of patients with cross-finger flaps had less subjective complaints and more normal objective testing results than the other groups. When an extensive tactile pad avulsion exists, the cross-finger pedicle flap offers the best long-term result with fewer secondary problems. The exclusive use of any one method cannot be advocated since each serves an useful purpose under proper conditions

Indications and effectiveness of arthroscopic lavage and debridement in osteoarthritis knee

Background: Degenerative joint disease will afflict most of the people if they live long enough. The majority of patients with osteoarthritis present to orthopaedic surgeons seeking relief of pain and associated restoration of function. Various intraarticular and periarticular structures may be the cause of pain. Improvement in osteoarthritis by arthroscopic lavage and debridement is seen due to various mechanisms.Methods: In a prospective study from July 2016 to December 2018, 35 patients with primary osteoarthritis knee were assessed for effectiveness of arthroscopic lavage and debridement in relieving symptoms of osteoarthritis of knee and to determine the indications of arthroscopy in osteoarthritis of knee. Assessment was done using variables as age of the patient, body mass index, varus deformity, radiographic grading and arthroscopic grading.Results: A declining trend was seen on follow up over time; 91.4% excellent to good results seen at one month follow up, 76.1% at six months, 49.93% at twelve months, 37.5%, at eighteen months, 23.07% at twenty-four months and 28.5% at thirty months. Results at six month follow up when compared, were better for age less than 50 years (88.8% Vs 73.1% in >50 years age), normal weight patients with BMI 18.5 to 25(94.5% Vs 58.5% in overweight), varus angulation 100), radiological grade I and II (95-100% Vs 45-50% in grade III and IV) and arthroscopic grade I and II (94-100% Vs 0-77% in grade III and IV).Conclusions: Arthroscopic lavage and debridement is an effective method of treatment for osteoarthritis knee in patients with grade I and grade II osteoarthritis having symptoms of pain and locking due to loose bodies or degenerative meniscal tears