A Conjecture of Kauuman on Amphicheiral Alternating Knots
نویسندگان
چکیده
We give a counterexample to the following conjecture of Kauuman 2]: Conjecture Let K be an amphicheiral alternating knot. Then there exists a reduced alternating knot-diagram D of K, such that G(D) is isomorphic to G (D), where G(D) is a checkerboard-graph of D and G (D) its dual.
منابع مشابه
Colored Tutte Polynomials and Kauuman Brackets for Graphs of Bounded Tree Width
Jones polynomials and Kauuman polynomials are the most prominent invariants of knot theory. For alternating links, they are easily computable from the Tutte polynomials by a result of Thistlethwaite (1988), but in general one needs colored Tutte polynomials, as introduced by Bollobas and Riordan (1999). Knots and links can be presented as labeled planar graphs. The tree width of a link L is dee...
متن کاملInfinite Order Amphicheiral Knots
In 1977 Gordon [G] asked whether every class of order two in the classical knot concordance group can be represented by an amphicheiral knot. The question remains open although counterexamples in higher dimensions are now known to exist [CM]. This problem is more naturally stated in terms of negative amphicheiral knots, since such knots represent 2–torsion in concordance; that is, if K is negat...
متن کاملChirality and the Conway polynomial
In recent workwith J.Mostovoy and T. Stanford, the author found that for every natural number n, a certain polynomial in the coefficients of the Conway polynomial is a primitive integer-valued degree n Vassiliev invariant, but that modulo 2, it becomes degree n-1. The conjecture then naturally suggests itself that these primitive invariants are congruent to integer-valued degree n-1 invariants....
متن کاملChirality and the Conway Polynomial
We conjecture that for the Conway polynomial C(z) of an amphicheiral knot, the product C(z)C(iz)C(z 2) must be a square inside the ring of polynomials with Z 4 coefficients. This conjecture has two ancestors. One is a theorem of Kawauchi and Hartley, which says that the Conway polynomial of a strongly amphicheiral knot must decompose as φ(z)φ(−z). This implies our main conjecture, at least for ...
متن کاملOn rational sliceness of Miyazaki’s fibered, −amphicheiral knots
We prove that certain fibered, −amphicheiral knots are rationally slice. Moreover, we show that the concordance invariants ν and Υ(t) from Heegaard Floer homology vanish for a class of knots that includes rationally slice knots.
متن کامل