Tangle move
For given -tangles
and
(cf. also Tangle), the tangle move, or more specifically the
-move, is substitution of the tangle
in the place of the tangle
in a link (or tangle). The simplest tangle
-move is a crossing change. This can be generalized to
-moves (cf. Montesinos–Nakanishi conjecture or [a5]),
-moves (cf. Fig.a1), and
-rational moves, where a rational
-tangle is substituted in place of the identity tangle [a6] (Fig.a2 illustrates a
-rational move).
A -rational move preserves the space of Fox
-colourings of a link or tangle (cf. Fox
-colouring). For a fixed prime number
, there is a conjecture that any link can be reduced to a trivial link by
-rational moves (
).
Kirby moves (cf. Kirby calculus) can be interpreted as tangle moves on framed links.
Figure: t130020a
Figure: t130020b
Habiro -moves [a2] are prominent in the theory of Vassiliev–Gusarov invariants of links and
-manifolds. The simplest and most extensively studied Habiro move (beyond the crossing change) is the
-move on a
-tangle (cf. Fig.a3). One can reduce every knot into the trivial knot by
-moves [a4].
Figure: t130020c
References
[a1] | T. Harikae, Y. Uchida, "Irregular dihedral branched coverings of knots" M. Bozhüyük (ed.) , Topics in Knot Theory , NATO ASI Ser. C , 399 , Kluwer Acad. Publ. (1993) pp. 269–276 |
[a2] | K. Habiro, "Claspers and finite type invariants of links" Geometry and Topology , 4 (2000) pp. 1–83 |
[a3] | R. Kirby, "Problems in low-dimensional topology" W. Kazez (ed.) , Geometric Topology (Proc. Georgia Internat. Topology Conf., 1993) , Studies in Adv. Math. , 2 , Amer. Math. Soc. /IP (1997) pp. 35–473 |
[a4] | H. Murakami, Y. Nakanishi, "On a certain move generating link homology" Math. Ann. , 284 (1989) pp. 75–89 |
[a5] | J.H. Przytycki, "![]() |
[a6] | Y. Uchida, S. Suzuki (ed.) , Knots '96, Proc. Fifth Internat. Research Inst. of MSJ , World Sci. (1997) pp. 109–113 |
Tangle move. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Tangle_move&oldid=18153