Khovanov homology, an invariant of links in R 3 , is a graded homology theory that categorifies the Jones polynomial in the sense that the graded Euler characteristic of the homology is the Jones polynomial. Asaeda et al. ['Categorification of the Kauffman bracket skein module of I-bundles over surfaces', Algebr. ] generalised this construction by defining a double graded homology theory that categorifies the Kauffman bracket skein module of links in I-bundles over surfaces, except for the surface RP 2 , where the construction fails due to strange behaviour of links when projected to the nonorientable surface RP 2 . This paper categorifies the missing case of the twisted I-bundle over RP 2 , RP 2 ×I ≈ RP 3 \ { * }, by redefining the differential in the Khovanov chain complex in a suitable manner.2010 Mathematics subject classification: primary 57M27; secondary 57M25, 57R56.
Using computational techniques we tabulate prime knots up to five crossings in the solid torus and the infinite family of lens spaces L(p, q). For these knots we calculate the second and third skein module and establish which prime knots in the solid torus are amphichiral. Most knots are distinguished by the skein modules. For the handful of cases where the skein modules fail to detect inequivalent knots, we calculate and compare the hyperbolic structures of the knot complements. We were unable to resolve a handful of 5-crossing cases for p ≥ 13.
We classify non-affine, prime knots in the solid torus up to 6 crossings. We establish which of these are amphicheiral: almost all knots with symmetric Jones polynomial are amphicheiral, but in a few cases we use stronger invariants, such as HOMFLYPT and Kauffman skein modules, to show that some such knots are not amphicheiral. Examples of knots with the same Jones polynomial that are different in the HOMFLYPT skein module are presented. It follows from our computations, that the wrapping conjecture is true for all knots up to 6 crossings.
In this paper, we consider the following nonlinear Kirchhoff type problem:where a, b > 0 are constants, the nonlinearity f is superlinear at infinity with subcritical growth and V is continuous and coercive. For the case when f is odd in u we obtain infinitely many sign-changing solutions for the above problem by using a combination of invariant sets method and the Ljusternik-Schnirelman type minimax method. To the best of our knowledge, there are only few existence results for this problem. It is worth mentioning that the nonlinear term may not be 4-superlinear at infinity, in particular, it includes the power-type nonlinearity |u| p−2 u with p ∈ (2, 4].
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