Parametrizing some interesting knotted surfaces
We discuss methods to construct a polynomial parametrization of some interesting knotted surfaces (knotted spheres, knotted tori and knotted planes) and provide examples.
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arXiv subjects
Rama Mishra
Publications and source records attributed to Rama Mishra.
Knots and non-orientable surfaces in 3-manifolds
In this article, we propose a new approach for describing and understanding knots and links in a 3-manifold through the use of an embedded non-orientable surface. Specifically, we define a plat-like representation based on this non-orientable surface. The method applies to manifolds of the form $M=\mathcal H\cup_φ \mathcal C(U)$ where $\mathcal H$ is a handlebody, $\mathcal C(U)$ is the mapping cylinder of the orientating two sheeted covering of a non-orientable closed surface $U$ and $φ:\partial \mathcal H\to \partial \mathcal C(U)$ is an attaching homeomorphism. We show that, by fixing such a splitting any link in the manifold can be represented as a plat-like closure of an element of the surface braid group of $\partial \mathcal H$. Manifolds of this type were extensively studied by J.H. Rubinstein \cite{rubinstein1978one}, where it is shown that any 3-manifold $M$, with a non-vanishing $H_2(M,\frac{\mathbb{Z}}{2\mathbb{Z}})$ will admit such a splitting. Thus the method is quite general. We provide explicit examples of such embeddings in lens spaces $L(2k,q)$ and the trivial circle bundles over orientable closed surfaces, $Σ\times S^1$
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Polynomially knotted 2-spheres
We review the polynomial parameterization of classical knots and prove the analogous results for long $2$ knots. We also construct polynomial parameterizations for certain classes of knotted spheres (such as spun and twist spun of the classical knots).
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Non-triviality of welded knots and ribbon torus-knots
In this paper we study welded knots and their invariants. We focus on generating examples of non-trivial knotted ribbon tori as the tube of welded knots that are obtained from classical knot diagrams by welding some of the crossings. Non-triviality is shown by determining the fundamental group of the concerned welded knot. Sample examples under consideration are the standard diagrams of the family of $(2,q)$ torus knots and the twist knots. Standard diagrams of knots from Rolfsen's tables with $6$ crossings are also discussed which are not in the family of torus and twist knots.
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Knots in $\mathbb{R}P^3$
This paper studies knots in three dimensional projective space. Our technique is to associate a virtual link to a link in projective space so that equivalent projective links go to equivalent virtual links (modulo a special flype move). We apply techniques in virtual knot theory to obtain a Jones polynomial for projective links. We show that this is equivalent to the known Jones polynomial defined by Drobotukhina for them. We apply virtual Khovanov homology and the virtual Rasmussen invariant of Dye, Kaestner, and Kauffman to projective links. We compare this cohomology theory with the Khovanov type theory developed by Manolescu and Willis for projective knots. We show that these theories are essentially equivalent.
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Plat closures of spherical braids in $\mathbb{R}P^3$
We define plat closure for spherical braids to obtain links in $\mathbb{R}P^3$ and prove that all links in $\mathbb{R}P^3$ can be realized in this manner. Given a spherical braid $β$ of $2n$ strands in $\mathbb{R}P^3$ we associate a permutation $h_β$ on $n$ elements called \textit{residual permutation}. We prove that the number of components of the plat closure link of a spherical braid $β$ is same as the number of disjoint cycles in $h_β$. We also present a set of moves on spherical braids in the same spirit as the classical Markov moves on braids. The completeness of this set of moves to capture the entire isotopy classes of the plat closure links is still to be explored.
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Geometry of knots in real projective $3$-space
This paper discusses some geometric ideas associated with knots in real projective 3-space $\mathbb{R}P^3$. These ideas are borrowed from classical knot theory. Since knots in $\mathbb{R}P^3$ are classified into three disjoint classes, - affine, class-$0$ non-affine and class-$1$ knots, it is natural to wonder in which class a given knot belongs to. In this paper we attempt to answer this question. We provide a structure theorem for these knots which helps in describing their behaviour near the projective plane at infinity. We propose a procedure called {\it space bending surgery}, on affine knots to produce several examples of knots. We later show that this operation can be extended on an arbitrary knot in $\mathbb{R}P^3$. We also define a notion of \say{ genus} for knots in $\mathbb{R}P^3$ and study some of its properties. We prove that this genus detects knottedness in $\mathbb{R}P^3$ and gives some criteria for a knot to be affine and of class-$1$. We also prove a \say{non-cancellation} theorem for space bending surgery using the properties of genus. We produce examples of class-$0 $ non-affine knots with genus $1$. And finally we study the notion of companionship of knots in $\mathbb{R}P^3$ and using that we provide a geometric criteria for a knot to be affine. Thus we highlight that, $\mathbb{R}P^3$ admits a knot theory with a truly different flavour than that of $S^3$ or $\mathbb{R}^3$.
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On the geometry of two state models for the colored Jones polynomial
Using the flow property of the R-matrix defining the colored Jones polynomial, we establish a natural bijection between the set of states on the part arc-graph of a link projection and the set of states on a corresponding bichromatic digraph, called arc-graph, as defined by Garoufalidis and Loebl. We use this to give a new and essentially elementary proof for a knot state-sum formula of Garoufalidis and Loebl. We will show that the state-sum contributions of states on the part arc-graph defined by the universal R-matrix of $U_q(sl(2,\mathbb{C}))$ correspond, under our bijection of sets of states, to the contributions in the formula of Garoufalidis and Loebl. This will show that the two state models are in fact not essentially distinct. Our approach will also extend the formula of Garoufalidis and Loebl to links. This requires some additional non-trivial observations concerning the geometry of states on part arc-graphs. We will discuss in detail the computation of the arc-graph state-sum, in particular for 3-braid closures.
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Colored HOMFLY-PT for hybrid weaving knot $\hat{W}_{3}(m,n)$
Weaving knots $W(p, n)$ of type $(p, n)$ denote an infinite family of hyperbolic knots which have not been addressed by the knot theorists as yet. Unlike the well-known $(p,n)$ torus knots, we do not have a closed-form expression for HOMFLY-PT and the colored HOMFLY-PT for $W(p,n)$. In this paper, we confine to a hybrid generalization of $W(3,n)$ which we denote as $\hat{W}_3(m,n)$ and obtain a closed-form expression for HOMFLY-PT using the Reshitikhin and Turaev method involving $\mathcal R$-matrices. Further, we also compute $[r]$-colored HOMFLY-PT for $W(3,n)$. Surprisingly, we observe that trace of the product of two dimensional $\hat{\mathcal{R}}$-matrices can be written in terms of an infinite family of Laurent polynomials $\mathcal{V}_{n,t}[q]$ whose absolute coefficients has an interesting relation to the Fibonacci numbers $\mathcal{F}_{n}$. We also computed reformulated invariants and the BPS integers in the context of topological strings. From our analysis, we propose that certain refined BPS integers for weaving knot $W(3,n)$ can be explicitly derived from the coefficients of Chebyshev polynomials of the first kind.
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Spaces of polynomial knots in low degree
We show that all knots up to $6$ crossings can be represented by polynomial knots of degree at most $7$, among which except for $5_2, 5_2^*, 6_1, 6_1^*, 6_2, 6_2^*$ and $6_3$ all are in their minimal degree representation. We provide concrete polynomial representation of all these knots. Durfee and O'Shea had asked a question: Is there any $5$ crossing knot in degree $6$? In this paper we try to partially answer this question. For an integer $d\geq2$, we define a set $\mathcal{\tilde{P}}_d$ to be the set of all polynomial knots given by $t\mapsto\big(f(t),g(t),h(t)\big)$ such that $\text{deg}(f)=d-2$, $\text{deg}(g)=d-1$ and $\text{deg}(h)=d$. This set can be identified with a subset of $\mathbb{R}^{3d}$ and thus it is equipped with the natural topology which comes from the usual topology $\mathbb{R}^{3d}$. In this paper we determine a lower bound on the number of path components of $\mathcal{\tilde{P}}_d$ for $d\leq 7$. We define a path equivalence for polynomial knots in the space $\mathcal{\tilde{P}}_d$ and show that it is stronger than the topological equivalence.
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Some spaces of polynomial knots
In this paper we study the topology of three different kinds of spaces associated to polynomial knots of degree at most $d$, for $d\geq2$. We denote these spaces by $\mathcal{O}_d$, $\mathcal{P}_d$ and $\mathcal{Q}_d$. For $d\geq3$, we show that the spaces $\mathcal{O}_d$ and $\mathcal{P}_d$ are path connected and the space $\mathcal{O}_d$ has the same homotopy type as $S^2$. Considering the space $\mathcal{P}=\bigcup_{d\geq2}\mathcal{O}_d$ of all polynomial knots with the inductive limit topology, we prove that it too has the same homotopy type as $S^2$. We also show that if two polynomial knots are path equivalent in $\mathcal{Q}_d$, then they are topologically equivalent. Furthermore, the number of path components in $\mathcal{Q}_d$ are in multiples of eight.
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Hecke algebra trace algorithm and some conjectures on weaving knots
Computing polynomial invariants for knots and links using braid representations relies heavily on finding the trace of Hecke algebra elements. There is no easy method known for computing the trace and hence it becomes difficult to compute the known polynomial invariants of knots using their braid representations. In this paper, we provide an algorithm to compute the trace of the Hecke algebra representation of any braid. We simplify this algorithm and write a Mathematica program to compute the invariants such as Alexander polynomial, Jones polynomial, HOMFLY-PT polynomial and Khovanov homology of a very special family of knots and links $W(n,m)$ known as weaving knots by expressing them as closure of weaving braids. We also explore on the relationship between the topological and geometric invariants of this family of alternating and hyperbolic knots (links) by generating data for the subfamilies $W(3,m)$, $W(4,m)$, $W(5,m)$ and $W(6,m)$ of weaving knots.
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Polynomial Invariants, Knot Homologies, and Higher Twist Numbers of Weaving Knots $W(3,n)$
We investigate several conjectures in geometric topology by assembling computer data obtained by studying weaving knots, a doubly infinite family $W(p,n)$ of examples of hyperbolic knots. In particular, we compute some important polynomial knot invariants, as well as knot homologies, for the subclass $W(3,n)$ of this family. We use these knot invariants to conclude that all knots $W(3,n)$ are fibered knots and provide estimates for some geometric invariants of these knots. Finally, we study the asymptotics of the ranks of their Khovanov homology groups. Our investigations provide evidence for our conjecture that, asymptotically as $n$ grows large, the ranks of Khovanov homology groups of $W(3,n)$ are normally distributed.
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The Jones Polynomial and Khovanov Homology of Weaving Knots $W(3,n)$
In this paper we compute the signature for a family of knots $W(k,n)$, the weaving knots of type $(k,n)$. By work of E.~S.~Lee the signature calculation implies a vanishing theorem for the Khovanov homology of weaving knots. Specializing to knots $W(3,n)$, we develop recursion relations that enable us to compute the Jones polynomial of $W(3,n)$. Using additional results of Lee, we compute the ranks of the Khovanov Homology of these knots. At the end we provide evidence for our conjecture that, asymptotically, the ranks of Khovanov Homology of $W(3,n)$ are {\it normally distributed}.
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Constructing real rational knots by gluing
We show that the problem of constructing a real rational knot of a reasonably low degree can be reduced to an algebraic problem involving the pure braid group: expressing an associated element of the pure braid group in terms of the standard generators of the pure braid group. We also predict the existence of a real rational knot in a degree that is expressed in terms of the edge number of its polygonal representation.
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On 3-super bridge knots
It is known that there are only finitely many knots with super bridge index 3. Jin and Jeon have provided a list of possible such candidates. However, they conjectured that the only knots with super bridge index 3 are trefoil and the figure eight knot. In this paper, we prove that the $5_2$ knot and the $6_2$ knot are also 3-super bridge knots by providing a polynomial representation of these knots in degree $6.$ This also answers a question asked by Durfee and O'Shea in their paper on polynomial knots: is there any 5-crossing knot in degree 6?
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Nodal Parity Invariants of Knotted Rigid Vertex Graphs
This paper introduces new invariants of rigid vertex graph embeddings by using non-local combinatorial information that is available at each graphical node. The new non-local information that we use in this paper involves parity in the Gauss code of the underlying graph. We apply these methods to graphs in classical and virtual knot theory, and we give formulations for applications to the topology of protein folding.
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Polynomial representation for long knots
We discuss the polynomial representation for long knots and elaborate on how to obtain them with a bound on degrees of the defining polynomials, for any knot-type.
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