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Pan Peng

Publications and source records attributed to Pan Peng.

63 records · Page 4Linked to original sources
arXiv2015

Congruent skein relations for colored HOMFLY-PT invariants and colored Jones polynomials

Colored HOMFLY-PT invariant, the generalization of the colored Jones polynomial, is one of the most important quantum invariants of links. This paper is devoted to investigating the basic structures of the colored HOMFLY-PT invariants of links. By using the HOMFLY-PT skein theory, firstly, we show that the (reformulated) colored HOMFLY-PT invariants actually lie in the ring $\mathbb{Z}[(q-q^{-1})^2,t^{\pm 1}]$. Secondly, we establish some symmetric formulas for colored HOMFLY-PT invariants of links, which include the rank-level duality as an easy consequence. Finally, motivated by the Labastida-Mariño-Ooguri-Vafa conjecture for framed links, we propose congruent skein relations for (reformulated) colored HOMFLY-PT invariants which are the generalizations of the skein relation for classical HOMFLY-PT polynomials. Then we study the congruent skein relation for colored Jones polynomials. In fact, we obtain a succinct formula for the case of knot. As an application, we prove a vanishing result for Reshetikhin-Turaev invariants of a family of 3-manifolds. Finally we study the congruent skein relations for $SU(n)$ quantum invariants.

math.GT↗
arXiv2015

Testing Cluster Structure of Graphs

We study the problem of recognizing the cluster structure of a graph in the framework of property testing in the bounded degree model. Given a parameter $\varepsilon$, a $d$-bounded degree graph is defined to be $(k, ϕ)$-clusterable, if it can be partitioned into no more than $k$ parts, such that the (inner) conductance of the induced subgraph on each part is at least $ϕ$ and the (outer) conductance of each part is at most $c_{d,k}\varepsilon^4ϕ^2$, where $c_{d,k}$ depends only on $d,k$. Our main result is a sublinear algorithm with the running time $\widetilde{O}(\sqrt{n}\cdot\mathrm{poly}(ϕ,k,1/\varepsilon))$ that takes as input a graph with maximum degree bounded by $d$, parameters $k$, $ϕ$, $\varepsilon$, and with probability at least $\frac23$, accepts the graph if it is $(k,ϕ)$-clusterable and rejects the graph if it is $\varepsilon$-far from $(k, ϕ^*)$-clusterable for $ϕ^* = c'_{d,k}\frac{ϕ^2 \varepsilon^4}{\log n}$, where $c'_{d,k}$ depends only on $d,k$. By the lower bound of $Ω(\sqrt{n})$ on the number of queries needed for testing graph expansion, which corresponds to $k=1$ in our problem, our algorithm is asymptotically optimal up to polylogarithmic factors.

cs.DS↗
arXiv2015

Testing Small Set Expansion in General Graphs

We consider the problem of testing small set expansion for general graphs. A graph $G$ is a $(k,ϕ)$-expander if every subset of volume at most $k$ has conductance at least $ϕ$. Small set expansion has recently received significant attention due to its close connection to the unique games conjecture, the local graph partitioning algorithms and locally testable codes. We give testers with two-sided error and one-sided error in the adjacency list model that allows degree and neighbor queries to the oracle of the input graph. The testers take as input an $n$-vertex graph $G$, a volume bound $k$, an expansion bound $ϕ$ and a distance parameter $\varepsilon>0$. For the two-sided error tester, with probability at least $2/3$, it accepts the graph if it is a $(k,ϕ)$-expander and rejects the graph if it is $\varepsilon$-far from any $(k^*,ϕ^*)$-expander, where $k^*=Θ(k\varepsilon)$ and $ϕ^*=Θ(\frac{ϕ^4}{\min\{\log(4m/k),\log n\}\cdot(\ln k)})$. The query complexity and running time of the tester are $\widetilde{O}(\sqrt{m}ϕ^{-4}\varepsilon^{-2})$, where $m$ is the number of edges of the graph. For the one-sided error tester, it accepts every $(k,ϕ)$-expander, and with probability at least $2/3$, rejects every graph that is $\varepsilon$-far from $(k^*,ϕ^*)$-expander, where $k^*=O(k^{1-ξ})$ and $ϕ^*=O(ξϕ^2)$ for any $0<ξ<1$. The query complexity and running time of this tester are $\widetilde{O}(\sqrt{\frac{n}{\varepsilon^3}}+\frac{k}{\varepsilon ϕ^4})$. We also give a two-sided error tester with smaller gap between $ϕ^*$ and $ϕ$ in the rotation map model that allows (neighbor, index) queries and degree queries.

cs.DS↗
arXiv2013

Detecting and Characterizing Small Dense Bipartite-like Subgraphs by the Bipartiteness Ratio Measure

We study the problem of finding and characterizing subgraphs with small \textit{bipartiteness ratio}. We give a bicriteria approximation algorithm \verb|SwpDB| such that if there exists a subset $S$ of volume at most $k$ and bipartiteness ratio $θ$, then for any $0<ε<1/2$, it finds a set $S'$ of volume at most $2k^{1+ε}$ and bipartiteness ratio at most $4\sqrt{θ/ε}$. By combining a truncation operation, we give a local algorithm \verb|LocDB|, which has asymptotically the same approximation guarantee as the algorithm \verb|SwpDB| on both the volume and bipartiteness ratio of the output set, and runs in time $O(ε^2θ^{-2}k^{1+ε}\ln^3k)$, independent of the size of the graph. Finally, we give a spectral characterization of the small dense bipartite-like subgraphs by using the $k$th \textit{largest} eigenvalue of the Laplacian of the graph.

cs.DS↗
arXiv2011

The Small-Community Phenomenon in Networks

We investigate several geometric models of network which simultaneously have some nice global properties, that the small diameter property, the small-community phenomenon, which is defined to capture the common experience that (almost) every one in our society belongs to some meaningful small communities by the authors (2011), and that under certain conditions on the parameters, the power law degree distribution, which significantly strengths the results given by van den Esker (2008), and Jordan (2010). The results above, together with our previous progress in Li and Peng (2011), build a mathematical foundation for the study of communities and the small-community phenomenon in various networks. In the proof of the power law degree distribution, we develop the method of alternating concentration analysis to build concentration inequality by alternatively and iteratively applying both the sub- and super-martingale inequalities, which seems powerful, and which may have more potential applications.

math.PR↗
arXiv2010

On a proof of the Labastida-Marino-Ooguri-Vafa conjecture

We outline a proof of a remarkable conjecture of Labastida-Mari{ñ}o-Ooguri-Vafa about certain new algebraic structures of quantum link invariants and the integrality of infinite family of new topological invariants. Our method is based on the cut-and-join analysis and a special rational ring characterizing the structure of the Chern-Simons partition function.

math.GT↗
arXiv2010

New Structure of Knot Invariants

Based on the proof of Labastida-Mari{ñ}o-Ooguri-Vafa conjecture \cite{lmov}, we derive an infinite product formula for Chern-Simons partition functions, the generating function of quantum $\fsl_N$ invariants. Some symmetry properties of the infinite product will also be discussed.

math.GT↗
arXiv2009

Proof of the Labastida-Marino-Ooguri-Vafa Conjecture

Based on large N Chern-Simons/topological string duality, in a series of papers, J.M.F. Labastida, M. Marino, H. Ooguri and C. Vafa conjectured certain remarkable new algebraic structure of link invariants and the existence of infinite series of new integer invariants. In this paper, we provide a proof of this conjecture. Moreover, we also show these new integer invariants vanish at large genera.

math.QA↗