# Tangle move

For given \$n\$-tangles \$T_1\$ and \$T_2\$ (cf. also Tangle), the tangle move, or more specifically the \$(T_1,T_2)\$-move, is substitution of the tangle \$T_2\$ in the place of the tangle \$T_1\$ in a link (or tangle). The simplest tangle \$2\$-move is a crossing change. This can be generalized to \$n\$-moves (cf. Montesinos–Nakanishi conjecture or [a5]), \$(m,q)\$-moves (cf. Fig.a1), and \$(p/q)\$-rational moves, where a rational \$(p/q)\$-tangle is substituted in place of the identity tangle [a6] (Fig.a2 illustrates a \$(13/5)\$-rational move).

A \$(p/q)\$-rational move preserves the space of Fox \$p\$-colourings of a link or tangle (cf. Fox \$n\$-colouring). For a fixed prime number \$p\$, there is a conjecture that any link can be reduced to a trivial link by \$(p/q)\$-rational moves (\$|q| \le p/2\$).

Kirby moves (cf. Kirby calculus) can be interpreted as tangle moves on framed links.

Figure: t130020a

Figure: t130020b

Habiro \$C_n\$-moves [a2] are prominent in the theory of Vassiliev–Gusarov invariants of links and \$3\$-manifolds. The simplest and most extensively studied Habiro move (beyond the crossing change) is the \$\Delta\$-move on a \$3\$-tangle (cf. Fig.a3). One can reduce every knot into the trivial knot by \$\Delta\$-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 Zbl 0941.57015 [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, "\$3\$-coloring and other elementary invariants of knots" , Knot Theory , 42 , Banach Center Publ. (1998) pp. 275–295 [a6] Y. Uchida, S. Suzuki (ed.) , Knots '96, Proc. Fifth Internat. Research Inst. of MSJ , World Sci. (1997) pp. 109–113
How to Cite This Entry:
Tangle move. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Tangle_move&oldid=53963
This article was adapted from an original article by Jozef Przytycki (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article