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For given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130020/t1300201.png" />-tangles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130020/t1300202.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130020/t1300203.png" /> (cf. also [[Tangle|Tangle]]), the tangle move, or more specifically the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130020/t1300204.png" />-move, is substitution of the tangle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130020/t1300205.png" /> in the place of the tangle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130020/t1300206.png" /> in a link (or tangle). The simplest tangle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130020/t1300207.png" />-move is a crossing change. This can be generalized to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130020/t1300209.png" />-moves (cf. [[Montesinos–Nakanishi conjecture|Montesinos–Nakanishi conjecture]] or [[#References|[a5]]]), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130020/t13002011.png" />-moves (cf. Fig.a1), and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130020/t13002013.png" />-rational moves, where a rational <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130020/t13002014.png" />-tangle is substituted in place of the identity tangle [[#References|[a6]]] (Fig.a2 illustrates a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130020/t13002015.png" />-rational move).
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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 [[#References|[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 [[#References|[a6]]] (Fig.a2 illustrates a $(13/5)$-rational move).
  
A <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130020/t13002016.png" />-rational move preserves the space of Fox <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130020/t13002017.png" />-colourings of a link or tangle (cf. [[Fox-n-colouring|Fox <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130020/t13002018.png" />-colouring]]). For a fixed prime number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130020/t13002019.png" />, there is a conjecture that any link can be reduced to a trivial link by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130020/t13002020.png" />-rational moves (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130020/t13002021.png" />).
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A $(p/q)$-rational move preserves the space of Fox $p$-colourings of a link or tangle (cf. [[Fox-n-colouring|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|Kirby calculus]]) can be interpreted as tangle moves on framed links.
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Kirby moves (cf. [[Kirby calculus]]) can be interpreted as tangle moves on framed links.
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/t130020a.gif" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/t130020a.gif" />
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Figure: t130020b
 
Figure: t130020b
  
Habiro <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130020/t13002023.png" />-moves [[#References|[a2]]] are prominent in the theory of Vassiliev–Gusarov invariants of links and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130020/t13002024.png" />-manifolds. The simplest and most extensively studied Habiro move (beyond the crossing change) is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130020/t13002026.png" />-move on a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130020/t13002027.png" />-tangle (cf. Fig.a3). One can reduce every knot into the trivial knot by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130020/t13002028.png" />-moves [[#References|[a4]]].
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Habiro $C_n$-moves [[#References|[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 [[#References|[a4]]].
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/t130020c.gif" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/t130020c.gif" />
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====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  K. Habiro,  "Claspers and finite type invariants of links"  ''Geometry and Topology'' , '''4'''  (2000)  pp. 1–83</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  H. Murakami,  Y. Nakanishi,  "On a certain move generating link homology"  ''Math. Ann.'' , '''284'''  (1989)  pp. 75–89</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  J.H. Przytycki,  "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130020/t13002029.png" />-coloring and other elementary invariants of knots" , ''Knot Theory'' , '''42''' , Banach Center Publ.  (1998)  pp. 275–295</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  Y. Uchida,  S. Suzuki (ed.) , ''Knots '96, Proc. Fifth Internat. Research Inst. of MSJ'' , World Sci.  (1997)  pp. 109–113</TD></TR></table>
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<table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  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</TD></TR>
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<TR><TD valign="top">[a2]</TD> <TD valign="top">  K. Habiro,  "Claspers and finite type invariants of links"  ''Geometry and Topology'' , '''4'''  (2000)  pp. 1–83</TD></TR>
 +
<TR><TD valign="top">[a3]</TD> <TD valign="top">  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</TD></TR>
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<TR><TD valign="top">[a4]</TD> <TD valign="top">  H. Murakami,  Y. Nakanishi,  "On a certain move generating link homology"  ''Math. Ann.'' , '''284'''  (1989)  pp. 75–89</TD></TR>
 +
<TR><TD valign="top">[a5]</TD> <TD valign="top">  J.H. Przytycki,  "$3$-coloring and other elementary invariants of knots" , ''Knot Theory'' , '''42''' , Banach Center Publ.  (1998)  pp. 275–295</TD></TR>
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<TR><TD valign="top">[a6]</TD> <TD valign="top">  Y. Uchida,  S. Suzuki (ed.) , ''Knots '96, Proc. Fifth Internat. Research Inst. of MSJ'' , World Sci.  (1997)  pp. 109–113</TD></TR>
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</table>
 +
 
 +
{{TEX|done}}

Revision as of 17:50, 30 December 2016

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
[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=40108
This article was adapted from an original article by Jozef Przytycki (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article