

Unknot Equivalence  [Fred Curtis  FebMay 2001]  
These working notes were motivated by my horror at the described length and complexity of Haken's 1961 unknotequivalence algorithm in [Adams94]. [Hass99] indicates no known nontrivial lower bounds to the complexity of the problem at the time of writing. 
[Synopsis] [Chronology / Working] [References]
Deciding if a given knot diagram is the unknot, i.e. deciding if the diagram can be untangled into a simple loop, is part of the harder, more general task of deciding if two knot diagrams represent the same knot.  
Let
U_{0}
denote
the
trivial
(0crossing)
diagram
of
the
unknot. Let +I denote addition of a crossing to a knot diagram by a type I Reidemeister move. Let I denote removal of a crossing by a type I move. Similarly, let +II (II) denote addition (removal) of a pair of crossings by a type II Reidemeister move. Let III denote a type III Reidemeister move. For a knot diagram D, let Dm denote the set of the diagrams which may result when a move m is applied to D. Let DM denote the set of the diagrams which may result when a move from the set M is applied to D. Let DM* denote the set of diagrams which may result when a finite sequence of moves from the set M is applied to D. Examples: all possible diagrams of a knot with diagram K may be written as K{+I,I,+II,II,III}* ; this page is largely concerned with the set of all unknot diagrams U_{0}{+I,I,+II,II,III}*. Write a particular element of DM* as Dt_{1}...t_{n} to mean the result of applying transformations t_{1},...,t_{n} to D. Each transformation t_{i} consists of a move m_{i} from M plus information to describe where in Dt_{1}...t_{i1} the move m_{i} should be applied. Principle of Move Transposition (PMT) : If a diagram D...T_{1}T_{2}... contains two adjacent transformation sequences T_{1} and T_{2} which change disjoint regions then the diagram is unchanged by transposing T_{1} and T_{2}, i.e. D...T_{1}T_{2}... = D...T_{2}T_{1}... 

Theorem 1: U_{0}{+I}* = U_{0}{+I,I}* [proof]
Theorem 2: U_{0}{+II}* = U_{0}{+II,II}* [proof]
The predicate ArbReduce_{0} captures the reduction of a diagram to U_{0} by successive removal of crossings via arbitrary moves from a set M. Define ArbReduce_{0}(S, M) for a set M of crossingremoval moves to mean for any diagram D in the set S,
Theorem 1 implies ArbReduce_{0}(U_{0}{+I}*, {I}).
Theorem 2 implies ArbReduce_{0}(U_{0}{+II}*, {II}).
Conjecture 6 : ArbReduce_{0}(U_{0}{+II}*, {I,II})
A
Nasty
Unknot

Based
on Fig 1.29 from [Adams94]  Goeritz's unknot  The
Monster

Based
on
file 'monsterinit.kps' from [KnotPlot]  Cousin
It

Based
on
file 'ksinit.kps' from [KnotPlot]  
Nemesis

An
unknot
of
Thistlethwaite 
9^{th} Feb 2001
11th Feb 2001

12th Feb 2001
13^{th} Feb 2001
14th Feb 2001
16th Feb 2001
Theorem 1: U_{0}{+I}* = U_{0}{+I,I}*Proof:
 U_{0}{+I}* is a subset of U_{0}{+I,I}* since {+I} is a subset of {+I,I}.
 For any D = U_{0}t_{1}t_{2}...t_{n} in U_{0}{+I,I}*,
 If all t_{i} are +I then D is in U_{0}{+I}*.
 If least one t_{i} is I then let t_{k} be the first. U_{0}t_{1}t_{2}...t_{k1} has exactly k  1 crossings. The crossing removed by t_{k} has either
 exactly one loop, generated by some t_{j} where j < k.
D = U_{0}t_{1}...t_{j1}t_{j}t_{k}t_{j+1}...t_{k1}t_{k+1}...t_{n} (t_{k} moved by PMT)
= U_{0}t_{1}...t_{j1}t_{j+1}...t_{k1}t_{k+1}...t_{n} (t_{j}, t_{k} cancel) two loops, in which case U_{0}t_{1}t_{2}...t_{k1} has exactly one crossing, so k = 2, U_{0}t_{1}t_{2} = U_{0} and D = U_{0}t_{1}t_{2}t_{3}...t_{n} = U_{0}t_{3}...t_{n}.
In each case, D may be expressed as U_{0}t'_{1}...t'_{n2} containing one less I move. Successive removal of I moves yields D is in U_{0}{+I}*.
In each case D is in U_{0}{+I}*, so U_{0}{+I,I}* is a subset of U_{0}{+I}*
17th Feb 2001
19th Feb 2001
20th Feb 2001
21st Feb 2001
Theorem 2: U_{0}{+II}* = U_{0}{+II,II}*Proof:
 U_{0}{+II}* is a subset of U_{0}{+II,II}* since {+II} is a subset of {+II,II}.
 For any D = U_{0}t_{1}t_{2}...t_{n} in U_{0}{+II,II}*,
 If all t_{i} are +II then D is in U_{0}{+II}*.
 If least one t_{i} is II then let t_{k} be the first. U_{0}t_{1}t_{2}...t_{k1} has exactly 2(k  1) crossings. The pair of crossings removed by t_{k} were either
 generated by a single +II move t_{j} where j < k.
D = U_{0}t_{1}...t_{j1}t_{j}t_{k}t_{j+1}...t_{k1}t_{k+1}...t_{n} (t_{k} moved by PMT)
= U_{0}t_{1}...t_{j1}t_{j+1}...t_{k1}t_{k+1}...t_{n} (t_{j}, t_{k} cancel) generated by two distinct +II moves, say t_{i} and t_{j} where i < j < k. Then
D = U_{0}t_{1}...t_{j1}t_{j}t_{k}t_{j+1}...t_{k1}t_{k+1}...t_{n} (t_{k} moved by PMT  see fig 2.1)
= U_{0}t_{1}...t_{j1}t_{j+1}...t_{k1}t_{k+1}...t_{n} (t_{j}, t_{k} cancel  see fig 2.2)In each case, D may be expressed as U_{0}t'_{1}...t'_{n2} containing one less II move. Successive removal of II moves yields D is in U_{0}{+II}*.
In each case D is in U_{0}{+II}*, so U_{0}{+II,II}* is a subset of U_{0}{+II}*
Fig 2.1  The region affected by t_{k} is shown dashed U_{0}...t_{i} U_{0}...t_{i}...t_{j1} U_{0}...t_{i}...t_{j} U_{0}...t_{i}...t_{j}...t_{k1} U_{0}...t_{i}...t_{j}...t_{k}
Fig 2.2  The region affected by t_{k} is shown dashed U_{0}...t_{i} U_{0}...t_{i}...t_{j1} U_{0}...t_{i}...t_{j1}t_{j} U_{0}...t_{i}...t_{j1}t_{j}t_{k}
= U_{0}...t_{i}...t_{j1}U_{0}...t_{i}...t_{j1}t_{j}t_{k}t_{j+1}...t_{k1}
= U_{0}...t_{i}...t_{j1}t_{j+1}...t_{k1}
22nd Feb 2001
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Early Mar 2001
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Early May 2001
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8th Dec 2001
"But where is the ambiguity? Over there, in a box. [...] But is the truth, as Hitchcock observes, in the box? No, there isn't room, the ambiguity has put on weight."  Monty Python's Flying Circus  Series 2 Episode 11
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