A novel method of measuring the positive-sequence capacitance
of T-connection transmission lines is proposed. The mathematical model of the
new method is explained in detail. In order to obtain enough independent
equations, three independent operation modes of T-connection transmission lines
during the line measurement are introduced. The digital simulation results and
field measurement results are shown. The simulation and measurement results
have validated that the new method can meet the needs of measuring the
positive-sequence capacitance of T-connection transmission lines. This method
has been implemented in the newly developed measurement instrument.

Abstract:
We first study superpolynomials associated to triply-graded reduced colored HOMFLYF-PT and Kauffman homologies. We obtained conjectures of congruent relations and cyclotomic expansion for them. Finally we apply the same idea to the Heegaard-Floer knot homology and also obtain an expansion conjecture. Many examples including homologically thick knots and higher representations are tested.

Abstract:
By studying modular invariance properties of some characteristic forms, we obtain twisted anomaly cancellation formulas. We apply these twisted cancellation formulas to study divisibilities on spin manifolds and congruences on spin$^c$ manifolds. Especially, we get twisted Rokhlin congruences for $8k+4$ dimensional spin$^c$ manifolds

Abstract:
We study the Chern-Simons partition function of orthogonal quantum group invariants, and propose a new orthogonal Labastida-Mari\~{n}o-Ooguri-Vafa conjecture as well as degree conjecture for free energy associated to the orthogonal Chern-Simons partition function. We prove the degree conjecture and some interesting cases of orthogonal LMOV conjecture. In particular, We provide a formula of colored Kauffman polynomials for torus knots and links, and applied this formula to verify certain case of the conjecture at roots of unity except $1$. We also derive formulas of Lickorish-Millett type for Kauffman polynomials and relate all these to the orthogonal LMOV conjecture.

Abstract:
In this paper, by combining modularity of the Witten genus and the modular forms constructed by Liu and Wang, we establish mod 3 congruence properties of certain twisted signatures of 24 dimensional string manifolds.

Abstract:
In this note, we prove that the Witten genus of nonsingular string complete intersections in product of complex projective spaces vanishes. Our result generalizes a known result of Landweber and Stong (cf. [HBJ]).

Abstract:
We compute the Chern-Simons transgressed forms of some modularly invariant characteristic forms, which are related to the elliptic genera. We study the modularity properties of these secondary characteristic forms and the relations among them. We also compute the Chern-Simons forms of some vector bundles over free loop space.

Abstract:
In this note we describe the recursion relations between two parameter HOMLFY and Kauffman polynomials of framed links These relation correspond to embeddings of quantized universal enveloping algebras. The relation corresponding to embeddings $g_{n}\supset g_{k}\times sl_{n-k}$ where $g_{n}$ is either $so_{2n+1}$, $so_{2n}$ or $sp_{2n}$ is new.

Abstract:
In this paper, we investigate the properties of the full colored HOMFLYPT invariants in the full skein of the annulus $\mathcal{C}$. We show that the full colored HOMFLYPT invariant has a nice structure when $q\rightarrow 1$. The composite invariant is a combination of the full colored HOMFLYPT invariants. In order to study the framed LMOV type conjecture for composite invariants, we introduce the framed reformulated composite invariant $\check{\mathcal{R}}_{p}(\mathcal{L})$. By using the HOMFLY skein theory, we prove that $\check{\mathcal{R}}_{p}(\mathcal{L})$ lies in the ring $2\mathbb{Z}[(q-q^{-1})^2,t^{\pm 1}]$. Furthermore, we propose a conjecture of congruent skein relation for $\check{\mathcal{R}}_{p}(\mathcal{L})$ and prove it for certain special cases.

Abstract:
Based on the orthogonal Labastida-Mari{\~n}o-Ooguri-Vafa conjecture made by L. Chen & Q. Chen [5], we derive an infinite product formula for Chern-Simons partition functions, which generalizes the Liu-Peng's [19] recent results to the orthogonal case. Symmetry property of this new infinite product structure is also discussed.