A determinant formula for the Jones polynomial of pretzel knots

This paper presents an algorithm to construct a weighted adjacency matrix of a plane bipartite graph obtained from a pretzel knot diagram. The determinant of this matrix after evaluation is shown to be the Jones polynomial of the pretzel knot by way of perfect matchings (or dimers) of this graph. The weights are Tutte's activity letters that arise because the Jones polynomial is a specialization of the signed version of the Tutte polynomial. The relationship is formalized between the familiar spanning tree setting for the Tait graph and the perfect matchings of the plane bipartite graph above. Evaluations of these activity words are related to the chain complex for the Champanerkar-Kofman spanning tree model of reduced Khovanov homology.Comment: 19 pages, 12 figures, 2 table

Involutions of knots that fix unknotting tunnels

Let K be a knot that has an unknotting tunnel tau. We prove that K admits a strong involution that fixes tau pointwise if and only if K is a two-bridge knot and tau its upper or lower tunnel.Comment: 9 pages, 3 figure

Alexander quandle lower bounds for link genera

We denote by Q_F the family of the Alexander quandle structures supported by finite fields. For every k-component oriented link L, every partition P of L into h:=|P| sublinks, and every labelling z of such a partition by the natural numbers z_1,...,z_n, the number of X-colorings of any diagram of (L,z) is a well-defined invariant of (L,P), of the form q^(a_X(L,P,z)+1) for some natural number a_X(L,P,z). Letting X and z vary in Q_F and among the labellings of P, we define a derived invariant A_Q(L,P)=sup a_X(L,P,z). If P_M is such that |P_M|=k, we show that A_Q(L,P_M) is a lower bound for t(L), where t(L) is the tunnel number of L. If P is a "boundary partition" of L and g(L,P) denotes the infimum among the sums of the genera of a system of disjoint Seifert surfaces for the L_j's, then we show that A_Q(L,P) is at most 2g(L,P)+2k-|P|-1. We set A_Q(L):=A_Q(L,P_m), where |P_m|=1. By elaborating on a suitable version of a result by Inoue, we show that when L=K is a knot then A_Q(K) is bounded above by A(K), where A(K) is the breadth of the Alexander polynomial of K. However, for every g we exhibit examples of genus-g knots having the same Alexander polynomial but different quandle invariants A_Q. Moreover, in such examples A_Q provides sharp lower bounds for the genera of the knots. On the other hand, A_Q(L) can give better lower bounds on the genus than A(L), when L has at least two components. We show that in order to compute A_Q(L) it is enough to consider only colorings with respect to the constant labelling z=1. In the case when L=K is a knot, if either A_Q(K)=A(K) or A_Q(K) provides a sharp lower bound for the knot genus, or if A_Q(K)=1, then A_Q(K) can be realized by means of the proper subfamily of quandles X=(F_p,*), where p varies among the odd prime numbers.Comment: 36 pages; 16 figure

Complexity of links in 3-manifolds

We introduce a natural-valued complexity c(X) for pairs X=(M,L), where M is a closed orientable 3-manifold and L is a link contained in M. The definition employs simple spines, but for well-behaved X's we show that c(X) equals the minimal number of tetrahedra in a triangulation of M containing L in its 1-skeleton. Slightly adapting Matveev's recent theory of roots for graphs, we carefully analyze the behaviour of c under connected sum away from and along the link. We show in particular that c is almost always additive, describing in detail the circumstances under which it is not. To do so we introduce a certain (0,2)-root for a pair X, we show that it is well-defined, and we prove that X has the same complexity as its (0,2)-root. We then consider, for links in the 3-sphere, the relations of c with the crossing number and with the hyperbolic volume of the exterior, establishing various upper and lower bounds. We also specialize our analysis to certain infinite families of links, providing rather accurate asymptotic estimates.Comment: 24 pages, 6 figure

Tangent-point self-avoidance energies for curves

We study a two-point self-avoidance energy $E_q$ which is defined for all rectifiable curves in $R^n$ as the double integral along the curve of $1/r^q$. Here $r$ stands for the radius of the (smallest) circle that is tangent to the curve at one point and passes through another point on the curve, with obvious natural modifications of this definition in the exceptional, non-generic cases. It turns out that finiteness of $E_q(\gamma)$ for $q\ge 2$ guarantees that $\gamma$ has no self-intersections or triple junctions and therefore must be homeomorphic to the unit circle or to a closed interval. For $q>2$ the energy $E_q$ evaluated on curves in $R^3$ turns out to be a knot energy separating different knot types by infinite energy barriers and bounding the number of knot types below a given energy value. We also establish an explicit upper bound on the Hausdorff-distance of two curves in $R^3$ with finite $E_q$-energy that guarantees that these curves are ambient isotopic. This bound depends only on $q$ and the energy values of the curves. Moreover, for all $q$ that are larger than the critical exponent $2$, the arclength parametrization of $\gamma$ is of class $C^{1,1-2/q}$, with H\"{o}lder norm of the unit tangent depending only on $q$, the length of $\gamma$, and the local energy. The exponent $1-2/q$ is optimal.Comment: 23 pages, 1 figur

Evaluation of the risk of malignancy index in preoperative diagnosis of ovarian masses

Background: Ovarian cancer possesses a challenge to screening tests due to its anatomical location, poor natural history, lack of specific lesion, symptoms and signs and low prevalence. Authors shall be considering RMI 2 and RMI 4 (forms of RMI) and comparing them with histopathology report to derive the sensitivity, specificity and other parameters of these tests.Methods: A prospective   study was conducted from September 2016- September 2017 at Mazumdar Shaw Hospital, Narayana Hrudayalaya, Bangalore.73 patients met the inclusion criteria. RMI 2   and RMI4 were calculated for all the patients and these scores were compared to the final histopathology reports.Results: In present study of 73 patients RMI2 showed a sensitivity of 86.6%, specificity of 86.5 %, Positive predictive value of 81.25% and negative predictive value of 90.24 %. Whereas RMI4 showed a sensitivity of 86.6%, specificity of 86.5 %, Positive predictive value of 83.87 and negative predictive value of 90.48 %. These results are comparable to other studies conducted.  The risk of malignancy index 2 and 4 are also almost comparable with each other and so either can be used for determining the risk of malignancy in patients with adnexal masses. These results were derived in an Indian population across all age groups showing that authors can apply this low-cost method even in resource limited settings.Conclusions: Authors found that Risk of malignancy index is a simple and affordable method to determine the likelihood of a patient having adnexal mass to be malignant. This can thus help save the resources and make the services available at grass root level

Higher Order Terms in the Melvin-Morton Expansion of the Colored Jones Polynomial

We formulate a conjecture about the structure of `upper lines' in the expansion of the colored Jones polynomial of a knot in powers of (q-1). The Melvin-Morton conjecture states that the bottom line in this expansion is equal to the inverse Alexander polynomial of the knot. We conjecture that the upper lines are rational functions whose denominators are powers of the Alexander polynomial. We prove this conjecture for torus knots and give experimental evidence that it is also true for other types of knots.Comment: 21 pages, 1 figure, LaTe

Conjugate Generators of Knot and Link Groups

This note shows that if two elements of equal trace (e.g., conjugate elements) generate an arithmetic two-bridge knot or link group, then the elements are parabolic. This includes the figure-eight knot and Whitehead link groups. Similarly, if two conjugate elements generate the trefoil knot group, then the elements are peripheral.Comment: 10 pages, submitted to Journal of Knot Theory and Its Ramification

Contractions of Low-Dimensional Lie Algebras

Theoretical background of continuous contractions of finite-dimensional Lie algebras is rigorously formulated and developed. In particular, known necessary criteria of contractions are collected and new criteria are proposed. A number of requisite invariant and semi-invariant quantities are calculated for wide classes of Lie algebras including all low-dimensional Lie algebras. An algorithm that allows one to handle one-parametric contractions is presented and applied to low-dimensional Lie algebras. As a result, all one-parametric continuous contractions for the both complex and real Lie algebras of dimensions not greater than four are constructed with intensive usage of necessary criteria of contractions and with studying correspondence between real and complex cases. Levels and co-levels of low-dimensional Lie algebras are discussed in detail. Properties of multi-parametric and repeated contractions are also investigated.Comment: 47 pages, 4 figures, revised versio