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TACNODE, Point of Osculation, Osculation Point, Double Cusp - the Third in the Series of Ak-Curve Sin- O.O T 1 ~ T2 Gularities

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TACNODE, Point of Osculation, Osculation Point, Double Cusp - the Third in the Series of Ak-Curve Sin- O.O T 1 ~ T2 Gularities T TACNODE, point of osculation, osculation point, double cusp - The third in the series of Ak-curve sin- o.o T 1 ~ T2 gularities. The point (0,0) is a tacnode of the curve X 4 __ y2 • 0 in R 2. J I The first of the Ak-curve singularities are: an ordi- TId T 1 * T 2 nary double point, also called a node or crunode; the Fig. 1. cusp, or spinode; the tacnode; and the ramphoid cusp. They are exemplified by the curves X k+l - y2 = 0 Several special families of tangles have been consid- for k = 1,2,3,4. ered, including the rational tangles, the algebraic The terms 'crunode' and 'spinode' are seldom used tangles and the periodic tangles (see Rotor). The n- nowadays (2000). braid group is a subgroup of the monoid of n-tangles (cf. See also Node; Cusp. also Braided group). One has also considered framed tangles and graph tangles. The category of tangles, with References boundary points as objects and tangles as morphisms, is [1] ABHYANKAR, S.S.: Algebraic geometry for scientists and en- gineers, Amer. Math. Soc., 1990, p. 3; 60. important in developing quantum invariants of links and [2] DIMCA, A.: Topics on real and complex singularities, Vieweg, 3-manifolds (e.g. Reshetikhin-Turaev invariants). Tan- 1987. gles are also used to construct topological quantum field [3] GRIFFITHS, PH., AND HARRIS, J.: Principles of algebraic ge- theories. ometry, Wiley, 1978, p. 293; 507. [4] WALKER,R.J.: Algebraic curves, Princeton Univ. Press, 1950, References Reprint: Dover 1962. [1] BONAHON, P., AND SIEBENMANN, L.: Geometric splittings of M. Hazewinkel classical knots and the algebraic knots of Conway, Vol. 75 of MSC 1991:14H20 Lecture Notes, London Math. Soc., to appear. [2] CONWAY, J.H.: 'An enumeration of knots and links', in J. LEECH (ed.): Computational Problems in Abstract Alge- TANGLE, relative link - A one-dimensional manifold bra, Pergamon Press, 1969, pp. 329-358. properly embedded in a 3-ball, D a. [3] LOZANO, M.: 'Arcbodies', Math. Proc. Cambridge Philos. Soe. 94 (1983), 253-260. Two tangles are considered equivalent if they are am- bient isotopic with their boundary fixed. An n-tangle Jozef Przytycki has 2n points on the boundary; a link is a 0-tangle. MSC 1991:57M25 The term arcbody is used for a one-dimensional mani- fold properly embedded in a 3-dimensional manifold. TANGLE MOVE - For given n-tangles 2/"1 and T2 Tangles can be represented by their diagrams, i.e. (cf. also Tangle), the tangle move, or more specifically regular projections into a 2-dimensional disc with ad- the (T1,T2)-move, is substitution of the tangle T2 in ditional over- and under-information at crossings. Two the place of the tangle T1 in a link (or tangle). The tangle diagrams represent equivalent tangles if they are simplest tangle 2-move is a crossing change. This can related by Reidemeister moves (cf. Reidemeister the- be generalized to n-moves (cf. Montesinos-Nakanishi orem). The word 'tangle' is often used to mean a tangle conjecture or [5]), (m, q)-moves (cf. Fig. 1), and p/q- diagram or part of a link diagram. rational moves, where a rational 2-tangle is substituted The set of n-tangles forms a monoid; the identity in place of the identity tangle [6] (Fig. 2 illustrates a tangle and composition of tangles is illustrated in Fig. 1. 13/5-rational move). TANGLE MOVE A p/q-rational move preserves the space of Fox p- TAU METHOD, r method A method initially for- colourings of a link or tangle (cf. Fox n-colouring). mulated as a tool for the approximation of special func- For a fixed prime number p, there is a conjecture that tions of mathematical physics (cf. also Special func- any link can be reduced to a trivial link by p/q-rational tions), which could be expressed in terms of simple dif- moves (Iql _< p/2). ferential equations. It developed into a powerful and ac- Kirby moves (cf. Kirby calculus) can be interpreted curate tool for the numerical solution of complex differ- as tangle moves on framed links. ential and functional equations. A main idea in it is to approximate the solution of a given problem by solving exactly an approximate problem. ... q half twists Lanczos ~ formulation of the tau method. In [17], C. J~"-J'~"~"" "~'~ (m,q)-move Lanczos remarked that truncation of the series solution m half twists of a differential equation is, in some way, equivalent to introducing a perturbation term in the right-hand side Fig. 1. of the equation. Conversely, a polynomial perturbation term can be used to produce a truncated series, that is, a polynomial solution. 13/5-move Assume one wishes to solve by means of a power se- ries expansion the simple linear differential equation (cf. also Linear differential operator) Dy(x):=y'(x)+y(x)=0, O<x<l, Fig. 2. v(0) = 1, Habiro Cn-moves [1] are prominent in the theory of which defines y(x) = exp(-x). To find the coefficients Vassiliev Gusarov invariants of links and 3-manifolds. of a formal series expansion of y(x), one substitutes the The simplest and most extensively studied Habiro move series in the equation and generates a system of alge- (beyond the crossing change) is the A-move on a 3- braic equations for the coefficients: jaj + aj-1 = 0 for tangle (cf. Fig. 3). One can reduce every knot into the j = 1, 2,..., solving it in terms of a0. The value of a0 is trivial knot by A-moves [4]. fixed using the initial condition. To find a finite expan- sion, say of order n, one needs to make all coefficients aj with j > n equal to zero. This is achieved by adding a term of the form rx n to the right-hand side of the -move differential equation. One has (n + 1)an+l + an = % so that a,,+l, and all the coefficients following it, will be equal to zero if one chooses as = r. The same condi- Fig. 3. tion follows by substituting a segment of degree n of the References series expansion of y(x) = exp(-x) into the equation. [1] HABIRO, K.: 'Claspers and finite type invariants of links', Ge- If the solution of the perturbed differential equation is ometry and Topology 4 (2000), 1-83. regarded as an approximation to that of the original [2] HARIKAE, T., AND UCHIDA, Y.: 'Irregular dihedral branched coverings of knots', in M. BOZH/SY/)K (ed.): Topics in Knot equation with, say, a right-hand side equal to zero, it Theory, Vol. 399 of NATO ASI Ser. C, Kluwer Acad. Publ., seems natural to replace it by the best uniform ap- 1993, pp. 269-276. proxlmation of zero over the same interval J, which is [3] KIRBY, R.: 'Problems in low-dimensional topology', in a Chebyshev polynomial T2 (x) of degree n, defined over W. KAZEZ (ed.): Geometric Topology (Proc. Georgia Inter- J (cf. also Chebyshev polynomials). nat. Topolo9y Conf., 1993), Vol. 2 of Studies in Adv. Math., Amer. Math. Soc./IP, 1997, pp. 35-473. Therefore, to find an accurate polynomial approxima- [4] MURAKAMI, H., AND NAKANISHI, Y.: 'On a certain move gen- tion of y(x), Lanczos proposed solving exactly the more erating link homology', Math. Ann. 284 (1989), 75-89. complex perturbed problem (the tau problem): [5] PRZYTYCKI, J.H.: '3-coloring and other elementary invariants of knots': Knot Theory, Vol. 42, Banach Center Publ., 1998, Dye(x) = rT,~ (x), pp. 275-295. with the same initial conditions as before. The polyno- [6] UCHTDA, Y., in S. SUZUKI (ed.): Knots '96, Proc. Fifth Inter- nat. Research Inst. of MS J, World Sei., 1997, pp. 109 113. mial y*(x) is called the tau method approximation of Jozef Przytycki y(x) over the given interval J. MSC 1991:57M25 This tau problem can be solved for the unknown co- efficients of y*(x) using several alternative procedures. 396 TAU METHOD One of them is described above, that is, to set up and A sequence of canonical polynomials defined as sim- solve a system of linear algebraic equations linking the ply as DQn(x) := x n for all n = 0, 1,..., need not al- unknown coefficients of Dy* (x) with those of 7T~ (x). In ways exist or need not be unique. An algebraic and algo- this process one can assume that yn(X) itself can be ex- rithmic theory of the tau method, initially constructed pressed in either powers of x, or in Chebyshev, Legendre for elements D of the class D of linear differential op- or other polynomials. The first choice was Lanczos' orig- erators of arbitrary integer order, with polynomial or inal choice, and he explicitly indicated the possibility of rational coefficients (essentially the tools a computer choosing the others. handles) was discussed by E.L. Ortiz in [24]. In this The second choice is a tau method, often [8] called the work, canonical polynomials are defined as realizations Chebyshev method (or Legendre method) and, also, the of classes of equivalence of polynomials, for which the spectral method. This last formulation of the tau method algebraic kernel of the differential operator is the mod- has been extensively used and applied, since 1971, to ulus. These classes have gaps in their index sequence. complex problems in fluid dynamics by S.A. Orsag [11]. Elements D E D are then uniquely associated with re- There are at least three other approaches to the tau presentatives of such classes of canonical sequences.
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