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Register a new userOn the Harary-Kauffman Conjecture and Turks Head Knots
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The m,n Turks Head Knot, THK(m,n), is an alternating (m,n) torus knot. We prove the Harary-Kauffman conjecture for all THK(m,n) except for the case where m geq 5 is odd and n geq 3 is relatively prime to m. We also give evidence in support of the conjecture in that case. Our proof rests on the observation that none of these knots have prime determinant except for THK(m,2) when P_m is a Pell prime.
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We show that the Conway polynomials of Fibonacci links are Fibonacci polynomials modulo 2. We deduce that, when $ n otequiv 0 Mod 4$ and $(n,j) eq (3,3),$ the Fibonacci knot $ cF_j^{(n)} $ is not a Lissajous knot.
This paper bounds the computational cost of computing the Kauffman bracket of a link in terms of the crossing number of that link. Specifically, it is shown that the image of a tangle with $g$ boundary points and $n$ crossings in the Kauffman bracket skein module is a linear combination of $O(2^g)$ basis elements, with each coefficient a polynomial with at most $n$ nonzero terms, each with integer coefficients, and that the link can be built one crossing at a time as a sequence of tangles with maximum number of boundary points bounded by $Csqrt{n}$ for some $C.$ From this it follows that the computation of the Kauffman bracket of the link takes time and memory a polynomial in $n$ times $2^{Csqrt{n}}.$
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Given a compact oriented 3-manifold M in S^3 with boundary, an (M,2n)-tangle T is a 1-manifold with 2n boundary components properly embedded in M. We say that T embeds in a link L in S^3 if T can be completed to L by a 1-manifold with 2n boundary components exterior to M. The link L is called a closure of T. We define the Kauffman bracket ideal of T to be the ideal I_T generated by the reduced Kauffman bracket polynomials of all closures of T. If this ideal is non-trivial, then T does not embed in the unknot. We give an algorithm for computing a finite list of generators for the Kauffman bracket ideal of any (S^1 x D^2, 2)-tangle, also called a genus-1 tangle, and give an example of a genus-1 tangle with non-trivial Kauffman bracket ideal. Furthermore, we show that if a single-component genus-1 tangle S can be obtained as the partial closure of a (B^3, 4)-tangle T, then I_T = I_S.
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