The paper computes the A-polynomial of twist knots, along with the A-polynomial of another infinite family of knots. The A-polynomial of a knot was first introduced by Cooper, Culler, Gillet, Long, and Shalen. It is a sophisticated invariant, difficult to compute, but at the same time very powerful at encoding topological and geometric information about the knot. Before, except for particular examples computed with computer assistance, only the case of torus knots has been understood. The paper enhances our knowledge and understanding of the A-polynomial with its computation for twist knots and for another infinite family of knots.
The result of highest interest is Theorem 1. (i) For \(n\neq -1,0,1,2\) the A-polynomial \(A_{J(2,2n)} \) of the \(2n\)th twist knot \(J(2,2n)\) is given recursively by \[ A_{J(2,2n)}(L,M)=xA_{J(2,2n-2n/| n| )}(L,M)-yA_{J(2,2n-4n/| n| )}(L,M), \] where \[ x=-L+L^2+2LM^2+M^4+2LM^4+L^2M^4+2LM^6+M^8-LM^8,\quad y=M^4(L+M^2)^4, \] and with initial conditions \[ \begin{aligned} A_{J(2,4)} (L,M)&=-L^2+L^3+2L^2M^2+LM^4+2L^2M^4-LM^6-L^2M^8\\ &+2LM^{10}+L^2M^{10}+2LM^{12}+M^{14}-LM^{14},\\ A_{J(2,2)}(L,M)&=L+M^6,\\ A_{J(2,0)}&=1,\\ A_{J(2,-2)}(L,M)&=-L+LM^2+M^4+2LM^4+L^2M^4+LM^6-LM^8. \end{aligned} \] (ii) The A-polynomial \(A_{J(2,2n+1)}(L,M)\) of the (\(2n+1\))th twist knot \(J(2,2n+1)\) is given by \(M^8nA_{J(2n,-2n)}(L,M^{-1})\) if \(n>0\) and \(M^{-8n-2}A_{J(2,-2n)}(L,M^{-1})\) if \(n<0\). The proof uses the presentation of the fundamental group of these knots which arises by viewing them as 2-bridge knots. A careful analysis of SL\((2,\mathbb C)\) representations and the use of the trace identity for SL\((2,\mathbb C)\) yield the recursive relations. The authors further prove that the A-polynomials of twist knots are irreducible and compute their Newton polygons.