Adam Sikora - Univ. of Maryland at College Park

Adam Sikora
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Adam Sikora
Univ. of Maryland at College Park
College Park
United States

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Pub Categories

Mathematics - Analysis of PDEs (34)
Mathematics - Geometric Topology (11)
Mathematics - Quantum Algebra (7)
Mathematics - Functional Analysis (7)
Mathematics - Representation Theory (6)
Mathematics - Algebraic Geometry (3)
Mathematics - Classical Analysis and ODEs (3)
Mathematics - Spectral Theory (2)
Mathematics - Symplectic Geometry (1)
Mathematics - Algebraic Topology (1)
Mathematics - Metric Geometry (1)
Mathematics - Differential Geometry (1)

Publications Authored By Adam Sikora

Let $(X,d,\mu)$ be a doubling metric measure space endowed with a Dirichlet form $\mathscr{E}$ deriving from a "carr\'e du champ". Assume that $(X,d,\mu,\mathscr{E})$ supports a scale-invariant $L^2$-Poincar\'e inequality. In this article, we study the following properties of harmonic functions, heat kernels and Riesz transform for $p\in (2,\infty]$: (i) $(G_p)$: $L^p$-boundedness of the gradient of the associated heat semigroup; (ii) $(RH_p)$: $L^p$-reverse H\"older inequality for the gradient of harmonic functions; (iii) $(R_p)$: $L^p$-boundedness of the Riesz transform ($p<\infty$); (iv) $(GBE)$: a generalized Bakry-\'Emery condition. Read More

We describe a simple but surprisingly effective technique of obtaining spectral multiplier results for abstract operators which satisfy the finite propagation speed property for the corresponding wave equation propagator. We show that, in this setting, spectral multipliers follow from resolvent type estimates. The most notable point of the paper is that our approach is very flexible and can be applied even if the corresponding ambient space does not satisfy the doubling condition or if the semigroup generated by an operator is not uniformly bounded. Read More

We proved by computer enumeration that the Jones polynomial distinguishes the unknot for knots up to 21 crossings. Following an approach of Yamada, we generated knot diagrams by inserting algebraic tangles into Conway polyhedra, computed their Jones polynomials by a divide-and-conquer method, and tested those with trivial Jones polynomials for unknottedness with the computer program SnapPy. We employed numerous novel strategies for reducing the computation time per knot diagram and the number of knot diagrams to be considered. Read More

On a complete non-compact Riemannian manifold satisfying the volume doubling property, we give conditions on the negative part of the Ricci curvature that ensure that, unless there are harmonic one-forms, the Gaussian heat kernel upper estimate on functions transfers to one-forms. These conditions do no entail any constraint on the size of the Ricci curvature, only on its decay at infinity. Read More

Let G be a connected reductive affine algebraic group. In this short note we define the "variety of G-characters" of a finitely generated group F and show that the quotient of the G-character variety of F by the action of the trace preserving outer automorphisms of G normalizes the variety of G-characters when F is a free group, free abelian group, or a surface group. Read More

We show that the Kauffman bracket skein algebra of any oriented surface F has no zero-divisors and that its center is generated by knots parallel to the boundary of F. Furthermore, we generalize the notion of skein algebras to skein algebras of marked surfaces and we prove analogous results them. Our proofs rely on certain filtrations of skein algebras induced by pants decompositions of surfaces and by the associated Dehn-Thurston intersection numbers. Read More

This paper comprises two parts. In the first, we study $L^p$ to $L^q$ bounds for spectral multipliers and Bochner-Riesz means with negative index in the general setting of abstract self-adjoint operators. In the second we obtain the uniform Sobolev estimates for constant coefficients higher order elliptic operators $P(D)-z$ and all $z\in {\mathbb C}\backslash [0, \infty)$, which give an extension of the second order results of Kenig-Ruiz-Sogge \cite{KRS}. Read More

We prove that the coordinate rings of SO(2n,C)-character varieties are not generated by trace functions nor generalized trace functions for $n\geq 2$ and all groups Gamma of corank $\geq 2.$ Furthermore, we give examples of non-conjugate completely reducible representations undistinguishable by generalized trace functions. Hence, SO(2n,C)-character varieties are not varieties of characters. Read More

The unit sphere $\mathbb{S}$ in $\mathbb{C}^n$ is equipped with the tangential Cauchy-Riemann complex and the associated Laplacian $\Box_b$. We prove a H\"ormander spectral multiplier theorem for $\Box_b$ with critical index $n-1/2$, that is, half the topological dimension of $\mathbb{S}$. Our proof is mainly based on representation theory and on a detailed analysis of the spaces of differential forms on $\mathbb{S}$. Read More

We investigate spectral multipliers, Bochner-Riesz means and convergence of eigenfunction expansion corresponding to the Schr\"odinger operator with anharmonic potential ${\mathcal L}=-\frac{d^2}{dx^2}+|x|$. We show that the Bochner-Riesz profile of the operator ${\mathcal L}$ completely coincides with such profile of the harmonic oscillator ${\mathcal H}=-\frac{d^2}{dx^2}+x^2$. It is especially surprising because the Bochner-Riesz profile for the one-dimensional standard Laplace operator is known to be essentially different and the case of operators ${\mathcal H}$ and ${\mathcal L}$ resembles more the profile of multidimensional Laplace operators. Read More

We establish that the Riesz transforms of all orders corresponding to the Gru\v{s}in operator $H_N=-\nabla_{x}^2-|x|^{2N}\,\nabla_{y}^2$, and the first-order operators $(\nabla_{x},x^\nu\,\nabla_{y})$ where $x\in \Ri^n$, $y\in\Ri^m$, $N\in\Ni_+$, and $\nu\in\{1,\ldots,n\}^N$, are bounded on $L_p(\Ri^{n+m})$ for all $p\in\langle1,\infty\rangle$ and are also weak-type $(1,1)$. Moreover, the transforms of order less than or equal to $N+1$ corresponding to $H_N$ and the operators $(\nabla_{x}, |x|^N\nabla_{y})$ are bounded on $L_p(\Ri^{n+m})$ for all $p\in\langle1,\infty\rangle$. But all transforms of order $N+2$ are bounded if and only if $p\in\langle1,n\rangle$. Read More

On doubling metric measure spaces endowed with a strongly local regular Dirichlet form, we show some characterisations of pointwise upper bounds of the heat kernel in terms of global scale-invariant inequalities that correspond respectively to the Nash inequality and to a Gagliardo-Nirenberg type inequality when the volume growth is polynomial. This yields a new proof and a generalisation of the well-known equivalence between classical heat kernel upper bounds and relative Faber-Krahn inequalities or localized Sobolev or Nash inequalities. We are able to treat more general pointwise estimates, where the heat kernel rate of decay is not necessarily governed by the volume growth. Read More

We examine the validity of the Poincar\'e inequality for degenerate, second-order, elliptic operators $H$ in divergence form on $L_2(\Ri^{n}\times\Ri^{m})$. We assume the coefficients are real symmetric and $a_1H_\delta\geq H\geq a_2H_\delta$ for some $a_1,a_2>0$ where $H_\delta$ is a generalized Gru\v{s}in operator, \[ H_\delta=-\nabla_{x_1}\,|x_1|^{(2\delta_1,2\delta_1')}\,\nabla_{x_1}-|x_1|^{(2\delta_2,2\delta_2')}\,\nabla_{x_2}^2 \;. \] Here $x_1\in\Ri^n$, $x_2\in\Ri^m$, $\delta_1,\delta_1'\in[0,1\rangle$, $\delta_2,\delta_2'\geq0$ and $|x_1|^{(2\delta,2\delta')}=|x_1|^{2\delta}$ if $|x_1|\leq 1$ and $|x_1|^{(2\delta,2\delta')}=|x_1|^{2\delta'}$ if $|x_1|\geq 1$. Read More

We analyze degenerate, second-order, elliptic operators $H$ in divergence form on $L_2(\Ri^{n}\times\Ri^{m})$. We assume the coefficients are real symmetric and $a_1H_\delta\geq H\geq a_2H_\delta$ for some $a_1,a_2>0$ where \[ H_\delta=-{\nabla}_{x_1}\cdot(c_{\delta_1, \delta'_1}(x_1)\,\nabla_{x_1})-c_{\delta_2, \delta'_2}(x_1)\,\nabla_{x_2}^2 \;. \] Here $x_1\in\Ri^n$, $x_2\in\Ri^m$ and $c_{\delta_i, \delta'_i}$ are positive measurable functions such that $c_{\delta_i, \delta'_i}(x)$ behaves like $|x|^{\delta_i}$ as $x\to0$ and $|x|^{\delta_i'}$ as $x\to\infty$ with $\delta_1,\delta_1'\in[0,1\rangle$ and $\delta_2,\delta_2'\geq0$. Read More

We describe the relation between G-character varieties, $X_G(\Gamma)$, and $G/H$-character varieties, where $H$ is a finite, central subgroup of $G.$ In particular, we find finite generating sets of coordinate rings $C[X_{G/H}(\Gamma)]$ for classical groups $G$ and $H$ as above. Using this approach we find an explicit description of $C[X_{SO(4,C)}(F_2)]$ for the free group on two generators, $F_2. Read More

Let $M$ be a manifold with ends constructed in \cite{GS} and $\Delta$ be the Laplace-Beltrami operator on $M$. In this note, we show the weak type $(1,1)$ and $L^p$ boundedness of the Hardy-Littlewood maximal function and of the maximal function associated with the heat semigroup $\M_\Delta f(x)=\sup_{t> 0} |\exp (-t\Delta)f(x)| $ on $L^p(M)$ for $1 < p \le \infty$. The significance of these results comes from the fact that $M$ does not satisfies the doubling condition. Read More

Let $L$ be a non-negative self adjoint operator acting on $L^2(X)$ where $X$ is a space of homogeneous type. Assume that $L$ generates a holomorphic semigroup $e^{-tL}$ whose kernels $p_t(x,y)$ satisfy generalized $m$-th order Gaussian estimates. In this article, we study singular and dyadically supported spectral multipliers for abstract self-adjoint operators. Read More

We describe weighted Plancherel estimates and sharp Hebisch-M\"uller-Stein type spectral multiplier result for a new class of Grushin type operators. We also discuss the optimal exponent for Bochner-Riesz summability in this setting. Read More

We prove that for every reductive group G with a maximal torus T and the Weyl group W there is a natural normalization map chi from T^N/W to an irreducible component of the G-character variety of Z^N. We prove that chi is an isomorphism for all classical groups. Additionally, we prove that even though there are no irreducible representations in the above mentioned irreducible component of the character variety for non-abelian G, the tangent spaces to it coincide with H^1(Z^N, Ad rho). Read More

We study the Grushin operators acting on $\R^{d_1}_{x'}\times \R^{d_2}_{x"}$ and defined by the formula \[ L=-\sum_{\jone=1}^{d_1}\partial_{x'_\jone}^2 - (\sum_{\jone=1}^{d_1}|x'_\jone|^2) \sum_{\jtwo=1}^{d_2}\partial_{x"_\jtwo}^2. \] We obtain weighted Plancherel estimates for the considered operators. As a consequence we prove $L^p$ spectral multiplier results and Bochner-Riesz summability for the Grushin operators. Read More

We consider abstract non-negative self-adjoint operators on $L^2(X)$ which satisfy the finite speed propagation property for the corresponding wave equation. For such operators we introduce a restriction type condition which in the case of the standard Laplace operator is equivalent to $(p,2)$ restriction estimate of Stein and Tomas. Next we show that in the considered abstract setting our restriction type condition implies sharp spectral multipliers and endpoint estimates for the Bochner-Riesz summability. Read More

We find finite, reasonably small, generator sets of the coordinate rings of G-character varieties of finitely generated groups for all classical groups G. This result together with the method of Grobner basis gives an algorithm for describing character varieties by explicit polynomial equations. Additionally, we describe finite sets of generators of the fields of rational functions on G-character varieties for all exceptional algebraic groups G. Read More

We develop a theory of sets with distributive products (called shelves and multi-shelves) and of their homology. We relate the shelf homology to the rack and quandle homology. Read More

The classical Stein-Tomas restriction theorem is equivalent to the statement that the spectral measure $dE(\lambda)$ of the square root of the Laplacian on $\RR^n$ is bounded from $L^p(\RR^n)$ to $L^{p'}(\RR^n)$ for $1 \leq p \leq 2(n+1)/(n+3)$, where $p'$ is the conjugate exponent to $p$, with operator norm scaling as $\lambda^{n(1/p - 1/p') - 1}$. We prove a geometric generalization in which the Laplacian on $\RR^n$ is replaced by the Laplacian, plus suitable potential, on a nontrapping asymptotically conic manifold, which is the first time such a result has been proven in the variable coefficient setting. It is closely related to, but stronger than, Sogge's discrete $L^2$ restriction theorem, which is an $O(\lambda^{n(1/p - 1/p') - 1})$ estimate on the $L^p \to L^{p'}$ operator norm of the spectral projection for a spectral window of fixed length. Read More

Let $\Omega$ be an open subset of $\Ri^d$ with $0\in \Omega$. Further let $H_\Omega=-\sum^d_{i,j=1}\partial_i\,c_{ij}\,\partial_j$ be a second-order partial differential operator with domain $C_c^\infty(\Omega)$ where the coefficients $c_{ij}\in W^{1,\infty}_{\rm loc}(\bar\Omega)$ are real, $c_{ij}=c_{ji}$ and the coefficient matrix $C=(c_{ij})$ satisfies bounds $00$ where $\mu(s)=\int^s_0dt\,c(t)^{-1/2}$ then we establish that $H_\Omega$ is $L_1$-unique, i. Read More

Let $M^\circ$ be a complete noncompact manifold and $g$ an asymptotically conic Riemaniann metric on $M^\circ$, in the sense that $M^\circ$ compactifies to a manifold with boundary $M$ in such a way that $g$ becomes a scattering metric on $M$. Let $\Delta$ be the positive Laplacian associated to $g$, and $P = \Delta + V$, where $V$ is a potential function obeying certain conditions. We analyze the asymptotics of the spectral measure $dE(\lambda) = (\lambda/\pi i) \big(R(\lambda+i0) - R(\lambda - i0) \big)$ of $P_+^{1/2}$, where $R(\lambda) = (P - \lambda^2)^{-1}$, as $\lambda \to 0$, in a manner similar to that done previously by the second author and Vasy, and by the first two authors. Read More

Let $L$ be a non-negative self adjoint operator acting on $L^2(X)$ where $X$ is a space of homogeneous type. Assume that $L$ generates a holomorphic semigroup $e^{-tL}$ whose kernels $p_t(x,y)$ have Gaussian upper bounds but possess no regularity in variables $x$ and $y$. In this article, we study weighted $L^p$-norm inequalities for spectral multipliers of $L$. Read More

Let $\Omega$ be an open subset of $\Ri^d$ and $H_\Omega=-\sum^d_{i,j=1}\partial_i c_{ij} \partial_j$ a second-order partial differential operator on $L_2(\Omega)$ with domain $C_c^\infty(\Omega)$ where the coefficients $c_{ij}\in W^{1,\infty}(\Omega)$ are real symmetric and $C=(c_{ij})$ is a strictly positive-definite matrix over $\Omega$. In particular, $H_\Omega$ is locally strongly elliptic. We analyze the submarkovian extensions of $H_\Omega$, i. Read More

We show that the $L^p$ boundedness, $p>2$, of the Riesz transform on a complete non-compact Riemannian manifold with upper and lower Gaussian heat kernel estimates is equivalent to a certain form of Sobolev inequality. We also characterize in such terms the heat kernel gradient upper estimate on manifolds with polynomial growth. Read More

Let $H$ be the symmetric second-order differential operator on $L_2(\Ri)$ with domain $C_c^\infty(\Ri)$ and action $H\varphi=-(c \varphi')'$ where $ c\in W^{1,2}_{\rm loc}(\Ri)$ is a real function which is strictly positive on $\Ri\backslash\{0\}$ but with $c(0)=0$. We give a complete characterization of the self-adjoint extensions and the submarkovian extensions of $H$. In particular if $\nu=\nu_+\vee\nu_-$ where $\nu_\pm(x)=\pm\int^{\pm 1}_{\pm x} c^{-1}$ then $H$ has a unique self-adjoint extension if and only if $\nu\not\in L_2(0,1)$ and a unique submarkovian extension if and only if $\nu\not\in L_\infty(0,1)$. Read More

Let $S$ be the submarkovian semigroup on $L_2({\bf R}^d)$ generated by a self-adjoint, second-order, divergence-form, elliptic operator $H$ with $W^{1,\infty}$ coefficients $c_{kl}$. Further let $\Omega$ be an open subset of ${\bf R}^d$. Under mild conditions we prove that $S$ leaves $L_2(\Omega)$ invariant if, and only if, it is invariant under the flows generated by the vector fields $\sum_{l=1}^d c_{kl} \partial_l$ for all $k$. Read More

We study properties of irreducible and completely reducible representations of finitely generated groups Gamma into reductive algebraic groups G in in the context of the geometric invariant theory of the G-action on Hom(Gamma,G) by conjugation. In particular, we study properties of character varieties, X_G(Gamma)=Hom(Gamma,G)//G. We describe the tangent spaces to X_G(Gamma) in terms of first cohomology groups of Gamma with twisted coefficients, generalizing the well known formula. Read More

We study multivariable spectral multipliers $F(L_1,L_2)$ acting on Cartesian product of ambient spaces of two self-adjoint operators $L_1$ and $L_2$. We prove that if $F$ satisfies H\"ormander type differentiability condition then the operator $F(L_1,L_2)$ is of Calder\'on-Zygmund type. We apply obtained results to the analysis of quasielliptic operators acting on product of some fractal spaces. Read More

Let G be a simple complex algebraic group and g its Lie algebra. We show that the g-Witten-Reshetikhin-Turaev quantum invariants determine a deformation-quantization, C_q[X_G(torus)], of the coordinate ring of the G-character variety of the torus. We prove that this deformation is in the direction of the Goldman's bracket. Read More

Let $S=\{S_t\}_{t\geq0}$ be the submarkovian semigroup on $L_2(\Ri^d)$ generated by a self-adjoint, second-order, divergence-form, elliptic operator $H$ with Lipschitz continuous coefficients $c_{ij}$. Further let $\Omega$ be an open subset of $\Ri^d$. Under the assumption that $C_c^\infty(\Ri^d)$ is a core for $H$ we prove that $S$ leaves $L_2(\Omega)$ invariant if, and only if, it is invariant under the flows generated by the vector fields $Y_i=\sum^d_{j=1}c_{ij}\partial_j$. Read More

We study the boundedness on $L^p$ of the Riesz transform $\nabla L^{-1/2}$, where $L$ is one of several operators defined on $\R$ or $\R_+$, endowed with the measure $r^{d-1} dr$, $d > 1$, where $dr$ is Lebesgue measure. For integer $d$, this mimics the measure on Euclidean $d$-dimensional space, and in this case our setup is equivalent to looking at the Laplacian acting on radial functions on Euclidean space or variations of Euclidean space such as the exterior of a sphere (with either Dirichlet or Neumann boundary conditions), or the connected sum of two copies of $\R^d$. In this way we illuminate some recent results on the Riesz transform on asymptotically Euclidean manifolds. Read More

We develop a theory of confluence of graphs. We describe an algorithm for proving that a given system of reduction rules for abstract graphs and graphs in surfaces is locally confluent. We apply this algorithm to show that each simple Lie algebra of rank at most 2, gives rise to a confluent system of reduction rules of graphs (via Kuperberg's spiders) in an arbitrary surface. Read More

Let L be a generator of a semigroup satisfying the Gaussian upper bounds. In this paper, we study further a new BMO_L space associated with L which was introduced recently by Duong and Yan. We discuss applications of the new BMO_L spaces in the theory of singular integration such as BMO_L estimates and interpolation results for fractional powers, purely imaginary powers and spectral multipliers of self adjoint operators. Read More

We prove that in presence of $L^2$ Gaussian estimates, so-called Davies-Gaffney estimates, on-diagonal upper bounds imply precise off-diagonal Gaussian upper bounds for the kernels of analytic families of operators on metric measure spaces. Read More

We analyze degenerate, second-order, elliptic operators $H$ in divergence form on $L_2({\bf R}^{n}\times{\bf R}^{m})$. We assume the coefficients are real symmetric and $a_1H_\delta\geq H\geq a_2H_\delta$ for some $a_1,a_2>0$ where \[ H_\delta=-\nabla_{x_1} c_{\delta_1, \delta'_1}(x_1) \nabla_{x_1}-c_{\delta_2, \delta'_2}(x_1) \nabla_{x_2}^2 . \] Here $x_1\in{\bf R}^n$, $x_2\in{\bf R}^m$ and $c_{\delta_i, \delta'_i}$ are positive measurable functions such that $c_{\delta_i, \delta'_i}(x)$ behaves like $|x|^{\delta_i}$ as $x\to0$ and $|x|^{\delta_i'}$ as $x\to\infty$ with $\delta_1,\delta_1'\in[0,1>$ and $\delta_2,\delta_2'\geq0$. Read More

It is shown that the theory of real symmetric second-order elliptic operators in divergence form on $\Ri^d$ can be formulated in terms of a regular strongly local Dirichlet form irregardless of the order of degeneracy. The behaviour of the corresponding evolution semigroup $S_t$ can be described in terms of a function $(A,B) \mapsto d(A ;B)\in[0,\infty]$ over pairs of measurable subsets of $\Ri^d$. Then \[ |(\phi_A,S_t\phi_B)|\leq e^{-d(A;B)^2(4t)^{-1}}\|\phi_A\|_2\|\phi_B\|_2 \] for all $t>0$ and all $\phi_A\in L_2(A)$, $\phi_B\in L_2(B)$. Read More

We establish the short-time asymptotic behaviour of the Markovian semigroups associated with strongly local Dirichlet forms under very general hypotheses. Our results apply to a wide class of strongly elliptic, subelliptic and degenerate elliptic operators. In the degenerate case the asymptotics incorporate possible non-ergodicity. Read More

Let $S=\{S_t\}_{t\geq0}$ be the semigroup generated on $L_2(\Ri^d)$ by a self-adjoint, second-order, divergence-form, elliptic operator $H$ with Lipschitz continuous coefficients. Further let $\Omega$ be an open subset of $\Ri^d$ with Lipschitz continuous boundary $\partial\Omega$. We prove that $S$ leaves $L_2(\Omega)$ invariant if, and only if, the capacity of the boundary with respect to $H$ is zero or if, and only if, the energy flux across the boundary is zero. Read More

We consider properties of second-order operators $H = -\sum^d_{i,j=1} \partial_i \, c_{ij} \, \partial_j$ on $\Ri^d$ with bounded real symmetric measurable coefficients. We assume that $C = (c_{ij}) \geq 0$ almost everywhere, but allow for the possibility that $C$ is singular. We associate with $H$ a canonical self-adjoint viscosity operator $H_0$ and examine properties of the viscosity semigroup $S^{(0)}$ generated by $H_0$. Read More

We generalize our previous work on categorification of Kauffman bracket skein module of surfaces, by extending our homology to tangles in cylinders over surfaces, F x [0,1]. Our homology of 0-tangles and 1-tangles in D^3 coincides (up to normalization) with Khovanov link homology and the reduced Khovanov link homology. We prove the basic properties of our homology. Read More

Khovanov defined graded homology groups for links L in R^3 and showed that their polynomial Euler characteristic is the Jones polynomial of L. Khovanov's construction does not extend in a straightforward way to links in I-bundles M over surfaces F not D^2 (except for the homology with Z/2 coefficients only). Hence, the goal of this paper is to provide a nontrivial generalization of his method leading to homology invariants of links in M with arbitrary rings of coefficients. Read More

For any n>1 we define an isotopy invariant, _n, for a certain set of n-valent ribbon graphs Gamma in R^3, including all framed oriented links. We show that our bracket coincides with the Kauffman bracket for n=2 and with the Kuperberg's bracket for n=3. Furthermore, we prove that for any n, our bracket of a link L is equal, up to normalization, to the SU_n-quantum invariant of L. Read More

For an abstract self-adjoint operator $L$ and a local operator $A$ we study the boundedness of the Riesz transform $AL^{-\alpha}$ on $L^p$ for some $\alpha >0$. A very simple proof of the obtained result is based on the finite speed propagation property for the solution of the corresponding wave equation. We also discuss the relation between the Gaussian bounds and the finite speed propagation property. Read More

The Kauffman-Harary conjecture states that for any reduced alternating diagram K of a knot with a prime determinant p, every non-trivial Fox p-coloring of K assigns different colors to its arcs. We generalize the conjecture by stating it in terms of homology of the double cover of S^3 branched along a link. In this way we extend the scope of the conjecture to all prime alternating links of arbitrary determinants. Read More