Coloring invariants for links in $Σ_g\times S^1$
Let $Σ_g$ be a closed oriented surface of genus $g$, in this paper we discuss how to define coloring invariants and its generalizations for links in $Σ_g\times S^1$.
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arXiv subjects
Hongzhu Gao
Publications and source records attributed to Hongzhu Gao.
Chord index for knots in thickened surfaces
In this note, we construct a chord index homomorphism from a subgroup of $H_1(Σ, \mathbb{Z})$ to the group of chord indices of a knot $K$ in $Σ\times I$. Some knot invariants derived from this homomorphism are discussed.
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General affine differential geometry of surfaces in affine space $A^3$, I: the elliptical case
In this paper we study the general affine differential geometry of surfaces in affine space $A^3$. For a regular elliptical surface we define a moving frame of minimal order and get the complete system of differential invariants. As an application we classify regular elliptical surfaces of constant curvatures up to affine congruence.
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Some remarks on the chord index
In this paper we discuss how to define a chord index via smoothing a real crossing point of a virtual knot diagram. Several polynomial invariants of virtual knots and links can be recovered from this general construction. We also explain how to extend this construction from virtual knots to flat virtual knots.
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The rational cohomology Hopf algebra of a generic Kac-Moody group
In this paper we determine the rational homotopy type of the classifying space of a generic Kac-Moody group by computing its rational cohomology ring. As an application we determine the rational homology Hopf algebra of the generic Kac-Moody group.
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Integral positive (negative) quandle cocycle invariants are trivial for knots
In this note we prove that for any finite quandle $X$ and any 2-cocycle $ϕ\in Z^2_{Q\pm}(X; \mathds{Z})$, the cocycle invariant $Φ^{\pm}_ϕ(K)$ is trivial for all knots $K$.
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Positive quandle homology and its applications in knot theory
Algebraic homology and cohomology theories for quandles have been studied extensively in recent years. With a given quandle 2(3)-cocycle one can define a state-sum invariant for knotted curves(surfaces). In this paper we introduce another version of quandle (co)homology theory, say positive quandle (co)homology. Some properties of positive quandle (co)homology groups are given and some applications of positive quandle cohomology in knot theory are discussed.
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A polynomial invariant of virtual links
In this paper, we define some polynomial invariants for virtual knots and links. In the first part we use Manturov's parity axioms to obtain a new polynomial invariant of virtual knots. This invariant can be regarded as a generalization of the odd writhe polynomial defined by the first author. The relation between this new polynomial invariant and the affine index polynomial is discussed. In the second part we introduce a polynomial invariant for long flat virtual knots. In the third part we define a polynomial invariant for 2-component virtual links. This polynomial invariant can be regarded as a generalization of the linking number.
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On region crossing change and incidence matrix
In a recent work of Ayaka Shimizu$^{[5]}$, she defined an operation named region crossing change on link diagrams, and showed that region crossing change is an unknotting operation for knot diagrams. In this paper, we prove that region crossing change on a 2-component link diagram is an unknotting operation if and only if the linking number of the diagram is even. Besides, we define an incidence matrix of a link diagram via its signed planar graph and its dual graph. By studying the relation between region crossing change and incidence matrix, we prove that a signed planar graph represents an $n$-component link diagram if and only if the rank of the associated incidence matrix equals to $c-n+1$, here $c$ denotes the size of the graph.
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