We present a twisted version of the Alexander polynomial associated with a matrix rep- resentation of the knot group.Examples of two knots with the same Alexander module but different twisted Alexander polynomials are...We present a twisted version of the Alexander polynomial associated with a matrix rep- resentation of the knot group.Examples of two knots with the same Alexander module but different twisted Alexander polynomials are given.展开更多
We obtain an equation among invariants obtained from the Alexander module of an amphicheiral link. For special cases, it deduces necessary conditions on the Alexander polynomial. By using the present results and some ...We obtain an equation among invariants obtained from the Alexander module of an amphicheiral link. For special cases, it deduces necessary conditions on the Alexander polynomial. By using the present results and some known results, we show that the Alexander polynomial of an algebraically split component- preservingly （±）-amphicheiral link with even components is zero, and we determine prime amphieheiral links with at least 2 components and up to 9 crossings.展开更多
The Conway potential function （CPF） for colored links is a convenient version of the multi- variable Alexander-Conway polynomial. We give a skein characterization of CPF, much simpler than the one by Murakami. In pa...The Conway potential function （CPF） for colored links is a convenient version of the multi- variable Alexander-Conway polynomial. We give a skein characterization of CPF, much simpler than the one by Murakami. In particular, Conway＇s ＂smoothing of crossings＂ is not in the axioms. The proof uses a reduction scheme in a twisted group-algebra PnBn, where Bn is a braid group and Pn is a domain of multi-variable rational fractions. The proof does not use computer algebra tools. An interesting by-product is a characterization of the Alexander-Conway polynomial of knots.展开更多
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
文摘We present a twisted version of the Alexander polynomial associated with a matrix rep- resentation of the knot group.Examples of two knots with the same Alexander module but different twisted Alexander polynomials are given.
基金supported by National Natural Science Foundation of China (Grant No. 10801021/a010402)
文摘We obtain an equation among invariants obtained from the Alexander module of an amphicheiral link. For special cases, it deduces necessary conditions on the Alexander polynomial. By using the present results and some known results, we show that the Alexander polynomial of an algebraically split component- preservingly （±）-amphicheiral link with even components is zero, and we determine prime amphieheiral links with at least 2 components and up to 9 crossings.
文摘The Conway potential function （CPF） for colored links is a convenient version of the multi- variable Alexander-Conway polynomial. We give a skein characterization of CPF, much simpler than the one by Murakami. In particular, Conway＇s ＂smoothing of crossings＂ is not in the axioms. The proof uses a reduction scheme in a twisted group-algebra PnBn, where Bn is a braid group and Pn is a domain of multi-variable rational fractions. The proof does not use computer algebra tools. An interesting by-product is a characterization of the Alexander-Conway polynomial of knots.
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.
