arXiv subjects
Publications and source records attributed to Adam S. Sikora.
We propose conjectural generalizations of the Fermat-Catalan conjecture, the Tijdeman-Zagier conjecture, and of the Fermat Last Theorem, in which powers are replaced by products of integers. We also formulate a new explicit version of the abc conjecture.
We show that the Kauffman bracket skein module of a closed Seifert fibered 3-manifold $M$ is finitely generated over $\mathbb Z[A^{\pm 1}]$ if and only if $M$ is irreducible and non-Haken. We analyze in detail the character varieties $X(M)$ of such manifolds and show that under mild conditions they are reduced. We compute the Kauffman bracket skein modules for these $3$-manifolds (over $\mathbb Q(A)$) and show that their dimensions coincide with $|X(M)|.$
The proof of Witten's finiteness conjecture established that the Kauffman bracket skein modules of closed $3$-manifolds are finitely generated over $\mathbb Q(A)$. In this paper, we develop a novel method for computing these skein modules. We show that if the skein module $S(M,\mathbb Q[A^{\pm 1}])$ of $M$ is tame (e.g. finitely generated over $\mathbb Q[A^{\pm 1}]$), and the $SL(2,\mathbb C)$-character variety is reduced, then the dimension $\dim_{\mathbb Q(A)}\, S(M, \mathbb Q(A))$ is the number of closed points in this character variety. This, in particular, verifies a conjecture in the literature that relates the dimension $\dim_{\mathbb Q(A)}\, S(M, \mathbb Q(A))$ to the Abouzaid-Manolescu $SL(2,\mathbb C)$-Floer theoretic invariants, for large families of 3-manifolds. We also prove a criterion for reduceness of character varieties of closed $3$-manifolds and use it to compute the skein modules of Dehn fillings of $(2,2n+1)$-torus knots and of the figure-eight knot. The later family gives the first instance of computations of skein modules for closed hyperbolic 3-manifolds. We also prove that the skein modules of rational homology spheres have dimension at least $1$ over $\mathbb Q(A)$.
We develop a theory of stated SL(n)-skein modules, $S_n(M,N),$ of 3-manifolds $M$ marked with intervals $N$ in their boundaries. They consist of linear combinations of $n$-webs with ends in $N$, considered up to skein relations inspired by the relations of the Reshetikhin-Turaev theory. We prove that cutting $M$ along a disk resulting in a $3$-manifold $M'$ yields a homomorphism $S_n(M)\to S_n(M')$. That result allows to analyze the skein modules of $3$-manifolds through the skein modules of their pieces. The theory of stated skein modules is particularly rich for thickened surfaces $M=\Sigma \times (-1,1),$ in whose case, $S_n(M)$ is an algebra, denoted by $S_n(\Sigma).$ We prove that the skein algebra of the ideal bigon is $O_q(SL(n))$ and that it provides simple geometric interpretations of the product, coproduct, counit, the antipode, and the cobraided structure on $O_q(SL(n)).$ Additionally, we show that a splitting of a thickened bigon near a marking defines a $O_q(SL(n))$-comodule structure on $S_n(M),$ or dually, an $U_q(sl_n)$-module structure. Furthermore, we show that the skein algebra of surfaces $\Sigma_1, \Sigma_2$ glued along two sides of a triangle is isomorphic with the braided tensor product $S_n(\Sigma_1)\underline{\otimes} S_n(\Sigma_2)$ of Majid. These results allow for a geometric interpretation of further concepts in the theory of quantum groups, for example, of the braided products and of Majid's transmutation operation. We prove that the factorization homology of surfaces with coefficients in $Rep\, U_q(sl_n)$ is equivalent to the category of left modules over $S_n(\Sigma)$. We also discuss the relation with the quantum moduli spaces of Alekseev-Schomerus. Finally, we show that for surfaces $\Sigma$ with boundary, $S_n(\Sigma)$ is a free module with a basis induced from the Kashiwara-Lusztig canonical bases.
In this paper, we apply Kauffman bracket skein algebras to develop a theory of skein adequate links in thickened surfaces. We show that any alternating link diagram on a surface is skein adequate. We apply our theory to establish the first and second Tait conjectures for adequate links in thickened surfaces. Our notion of skein adequacy is broader and more powerful than the corresponding notions of adequacy previously considered for link diagrams in surfaces. For a link diagram $D$ on a surface $\Sigma$ of minimal genus $g(\Sigma)$, we show that $${\rm span}([D]_\Sigma) \leq 4c(D) + 4 |D|-4g(\Sigma),$$ where $[D]_\Sigma$ is its skein bracket, $|D|$ is the number of connected components of $D$, and $c(D)$ is the number of crossings. This extends a classical result of Kauffman, Murasugi, and Thistlethwaite. We further show that the above inequality is an equality if and only if $D$ is weakly alternating. This is a generalization of a well-known result for classical links due to Thistlethwaite. Thus the skein bracket detects the crossing number for weakly alternating links. As an application, we show that the crossing number is additive under connected sum for adequate links in thickened surfaces.
We study systems of $2$-tangle equations which play an important role in the analysis of enzyme actions on DNA strands. We show that every system of framed tangle equations has at most one framed rational solution. Furthermore, we show that the Jones Unknot conjecture implies that if a system of tangle equations has a rational solution then that solution is unique among all $2$-tangles. This result potentially opens a door to a purely topological disproof of the Jones Unknot conjecture. We introduce the notion of the Kauffman bracket ratio $\{T\}_q\in \mathbb Q(q)$ of any $2$-tangle $T$ and we conjecture that for $q=1$ it is the slope of meridionally incompressible surfaces in $D^3-T$. We prove that conjecture for algebraic $T$. We also prove that for rational $T$, the brackets $\{T\}_q$ coincide with the $q$-rationals of Morier-Genoud-Ovsienko. Additionally, we relate systems of tangle equations to the Cosmetic Surgery Conjecture and the Nugatory Crossing Conjecture.
We proved by computer enumeration that the Jones polynomial distinguishes the unknot for knots up to 22 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. That made computations up to 21 crossings possible on a single processor desktop computer. We explain these strategies in this paper. We also provide total numbers of algebraic tangles up to 18 crossings and of Conway polyhedra up to 22 vertices. We encountered new unknot diagrams with no crossing-reducing pass moves in our search. We report one such diagram in this paper.
Extending upon our previous work, we verify the Jones Unknot Conjecture for all knots up to $24$ crossings. We describe the method of our approach and analyze the growth of the computational complexity of its different components.
The $SU_3$-skein algebra of a surface $F$ is spanned by isotopy classes of certain framed graphs in $F\times I$ called $3$-webs subject to the skein relations encapsulating relations between $U_q(sl(3))$-representations. These skein algebras are quantizations of the $SL(3)$-character varieties of surfaces. It is expected that their theory parallels that of the Kauffman bracket skein algebras. We make the first step towards developing that theory by proving that the reduced $SU_3$-skein algebra of any surface of finite type is finitely generated. We achieve that result by developing a theory of canonical forms of webs in surfaces. Specifically, we show that for any ideal triangulation of $F$ every reduced $3$-web can be uniquely decomposed into unions of pyramid formations of hexagons and disjoint arcs in the faces of the triangulation with possible additional "crossbars" connecting their edges along the ideal triangulation. We show that such canonical position is unique up to "crossbar moves". That leads us to an associated system of coordinates for webs in triangulated surfaces (counting intersections of the web with the edges of the triangulation and their rotation numbers inside of the faces of the triangulation) which determine a reduced web uniquely. Finally, we relate our skein algebras to $\cal A$-varieties of Fock-Goncharov and to $\text{Loc}_{SL(3)}$-varieties of Goncharov-Shen. We believe that our coordinate system for webs is a manifestation of a (quantum) mirror symmetry conjectured by Goncharov-Shen.
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.
The Jones unknot conjecture states that the Jones polynomial distinguishes the unknot from nontrivial knots. We prove it for knots up to 23 crossings.
We show that the Kauffman bracket skein algebra of any oriented surface F (possibly with marked points in its boundary) has no zero divisors and that its center is generated by knots parallel to the unmarked components of the boundary of F. Furthermore, we show that skein algebras are Noetherian and Ore. Our proofs rely on certain filtrations of skein algebras induced by pants decompositions of surfaces. We prove some basic algebraic properties of the associated graded algebras along the way.
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. However, we also prove that any generic SO(2n,C)-representation of a free group can be distinguished from all non-equivalent representations by trace functions and by a single generalized trace function.
We describe the relation between G-character varieties, $X_G(Γ)$, and $G/H$-character varieties, where $H$ is a finite, central subgroup of $G.$ In particular, we find finite generating sets of coordinate rings $C[X_{G/H}(Γ)]$ 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, $F_2.$ In the second part of the paper, we prove several properties of SO(2n,C)-character varieties. This is a particularly interesting class of character varieties because unlike for all other classical groups G, the coordinate rings $C[X_{G}(Γ)]$ are generally not generated by trace functions $τ_γ$, for $γ\in Γ$, for G=SO(2n,C). In fact, we prove that the coordinate ring $C[X_{SO(2n,C)}(Γ)]$ is not even generated by "generalized trace functions," $τ_{γ,V},$ for all $γ\in Γ$ and all representations $V$ of $SO(2n,C)$ for $n=2$ and groups $Γ$ of corank $\geq 2$.
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). Consequently, this irreducible component has the "Goldman" symplectic form for N=2, for which the combinatorial formulas for Goldman bracket hold.
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.
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.
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. Let M be an orientable 3-manifold with a connected boundary F of genus > 1 and let X_G^g(F) be the subset of the G -character variety of F composed of conjugacy classes of good representations. By a theorem of Goldman, X_G^g(F) is a holomorphic symplectic manifold. We prove that the set of good G-representations of pi_1(F) which extend to representations of pi_1(M) is an isotropic submanifold of X_G^g(F). If these representations correspond to reduced points of the G-character variety of M then this submanifold is Lagrangian.