اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدA splicing formula for the LMO invariant
89
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
We prove a splicing formula for the LMO invariant, which is the universal finite-type invariant of rational homology $3$-spheres. Specifically, if a rational homology $3$-sphere $M$ is obtained by gluing the exteriors of two framed knots $K_1 subset M_1$ and $K_2subset M_2$ in rational homology $3$-spheres, our formula expresses the LMO invariant of $M$ in terms of the Kontsevich-LMO invariants of $(M_1,K_1)$ and $(M_2,K_2)$. The proof uses the techniques that Bar-Natan and Lawrence developed to obtain a rational surgery formula for the LMO invariant. In low degrees, we recover Fujitas formula for the Casson-Walker invariant and we observe that the second term of the Ohtsuki series is not additive under standard splicing. The splicing formula also works when each $M_i$ comes with a link $L_i$ in addition to the knot $K_i$, hence we get a satellite formula for the Kontsevich-LMO invariant.
قيم البحث
اقرأ أيضاً
We prove a cabling formula for the concordance invariant $ u^+$, defined by the author and Hom. This gives rise to a simple and effective 4-ball genus bound for many cable knots.
A new topological operad is introduced, called the splicing operad. This operad acts on a broad class of spaces of self-embeddings N --> N where N is a manifold. The action of this operad on EC(j,M) (self embeddings R^j x M --> R^j x M with support i
n I^j x M) is an extension of the action of the operad of (j+1)-cubes on this space. Moreover the action of the splicing operad encodes Larry Siebenmanns splicing construction for knots in S^3 in the j=1, M=D^2 case. The space of long knots in R^3 (denoted K_{3,1}) was shown to be a free 2-cubes object with free generating subspace P, the subspace of long knots that are prime with respect to the connect-sum operation. One of the main results of this paper is that K_{3,1} is free with respect to the splicing operad action, but the free generating space is the much `smaller space of torus and hyperbolic knots TH subset K_{3,1}. Moreover, the splicing operad for K_{3,1} has a `simple homotopy-type as an operad.
We define and study a bigraded knot invariant whose Euler characteristic is the Alexander polynomial, closely connected to knot Floer homology. The invariant is the homology of a chain complex whose generators correspond to Kauffman states for a knot
diagram. The definition uses decompositions of knot diagrams: to a collection of points on the line, we associate a differential graded algebra; to a partial knot diagram, we associate modules over the algebra. The knot invariant is obtained from these modules by an appropriate tensor product.
By 2-twist-spinning the knotted graph that represents the knotted handlebody $5_2$, we obtain a knotted foam in 4-dimensional space with a non-trivial quandle cocycle invariant.
We extend the definition of Khovanov-Lee homology to links in connected sums of $S^1 times S^2$s, and construct a Rasmussen-type invariant for null-homologous links in these manifolds. For certain links in $S^1 times S^2$, we compute the invariant by
reinterpreting it in terms of Hochschild homology. As applications, we prove inequalities relating the Rasmussen-type invariant to the genus of surfaces with boundary in the following four-manifolds: $B^2 times S^2$, $S^1 times B^3$, $mathbb{CP}^2$, and various connected sums and boundary sums of these. We deduce that Rasmussens invariant also gives genus bounds for surfaces inside homotopy 4-balls obtained from $B^4$ by Gluck twists. Therefore, it cannot be used to prove that such homotopy 4-balls are non-standard.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
