Let K be a genus g alternating knot with Alexander polynomial Δ_(K)(T)=Σ_(i=-g)^(g)a_(i)T^(i).We show that if |a_(g)|=|a_(g-1)|,then K is the torus knot T_(2g+1,±2).This is a special case of the Fox Trapezoidal...Let K be a genus g alternating knot with Alexander polynomial Δ_(K)(T)=Σ_(i=-g)^(g)a_(i)T^(i).We show that if |a_(g)|=|a_(g-1)|,then K is the torus knot T_(2g+1,±2).This is a special case of the Fox Trapezoidal Conjecture.The proof uses Ozsvath and Szabo's work on alternating knots.展开更多
The relationship between a link diagram and its corresponding planar graph is briefly reviewed. A necessary and sufficient condition is given to detect when a planar graph corresponds to a knot. The relationship betwe...The relationship between a link diagram and its corresponding planar graph is briefly reviewed. A necessary and sufficient condition is given to detect when a planar graph corresponds to a knot. The relationship between planar graph and almost planar Seifert surface is discussed. Using planar graph, we construct an alternating amphicheiral prime knot with crossing number n for any even number n 〉 4. This gives an affirmative answer to problem 1.66（B） on Kirby＇s problem list .展开更多
In this paper, by the twist-crossing number of knots, we give an upper bound on the Euler characteristic of a kind of essential surfaces in the complements of alternating knots and almost alternating knots, which impr...In this paper, by the twist-crossing number of knots, we give an upper bound on the Euler characteristic of a kind of essential surfaces in the complements of alternating knots and almost alternating knots, which improves the estimation of the Euler characteristic of the essential surfaces with boundaries under certain conditions. Furthermore, we give the genus of the essential surfaces.展开更多
文摘Let K be a genus g alternating knot with Alexander polynomial Δ_(K)(T)=Σ_(i=-g)^(g)a_(i)T^(i).We show that if |a_(g)|=|a_(g-1)|,then K is the torus knot T_(2g+1,±2).This is a special case of the Fox Trapezoidal Conjecture.The proof uses Ozsvath and Szabo's work on alternating knots.
文摘The relationship between a link diagram and its corresponding planar graph is briefly reviewed. A necessary and sufficient condition is given to detect when a planar graph corresponds to a knot. The relationship between planar graph and almost planar Seifert surface is discussed. Using planar graph, we construct an alternating amphicheiral prime knot with crossing number n for any even number n 〉 4. This gives an affirmative answer to problem 1.66（B） on Kirby＇s problem list .
基金The NSF (11071106) of Chinathe Program (LR2011031) for Liaoning Excellent Talents in University
文摘In this paper, by the twist-crossing number of knots, we give an upper bound on the Euler characteristic of a kind of essential surfaces in the complements of alternating knots and almost alternating knots, which improves the estimation of the Euler characteristic of the essential surfaces with boundaries under certain conditions. Furthermore, we give the genus of the essential surfaces.
