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Combinatorial Game Theory, Well-Tempered Scoring Games, and a Knot Game

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Combinatorial Game Theory, Well-Tempered Scoring Games, and a Knot Game Combinatorial Game Theory, Well-Tempered Scoring Games, and a Knot Game Will Johnson June 9, 2011 Contents 1 To Knot or Not to Knot 4 1.1 Some facts from Knot Theory . 7 1.2 Sums of Knots . 18 I Combinatorial Game Theory 23 2 Introduction 24 2.1 Combinatorial Game Theory in general . 24 2.1.1 Bibliography . 27 2.2 Additive CGT specifically . 28 2.3 Counting moves in Hackenbush . 34 3 Games 40 3.1 Nonstandard Definitions . 40 3.2 The conventional formalism . 45 3.3 Relations on Games . 54 3.4 Simplifying Games . 62 3.5 Some Examples . 66 4 Surreal Numbers 68 4.1 Surreal Numbers . 68 4.2 Short Surreal Numbers . 70 4.3 Numbers and Hackenbush . 77 4.4 The interplay of numbers and non-numbers . 79 4.5 Mean Value . 83 1 5 Games near 0 85 5.1 Infinitesimal and all-small games . 85 5.2 Nimbers and Sprague-Grundy Theory . 90 6 Norton Multiplication and Overheating 97 6.1 Even, Odd, and Well-Tempered Games . 97 6.2 Norton Multiplication . 105 6.3 Even and Odd revisited . 114 7 Bending the Rules 119 7.1 Adapting the theory . 119 7.2 Dots-and-Boxes . 121 7.3 Go . 132 7.4 Changing the theory . 139 7.5 Highlights from Winning Ways Part 2 . 144 7.5.1 Unions of partizan games . 144 7.5.2 Loopy games . 145 7.5.3 Mis`eregames . 146 7.6 Mis`ereIndistinguishability Quotients . 147 7.7 Indistinguishability in General . 148 II Well-tempered Scoring Games 155 8 Introduction 156 8.1 Boolean games . 156 8.2 Games with scores . 158 8.3 Fixed-length Scoring Games . 160 9 Well-tempered Scoring Games 168 9.1 Definitions . 168 9.2 Outcomes and Addition . 171 9.3 Partial orders on integer-valued games . 176 9.4 Who goes last? . 180 9.5 Sides . 183 9.6 A summary of results so far . 186 2 10 Distortions 188 10.1 Order-preserving operations on Games . 188 10.2 Compatibility with Equivalence . 191 10.3 Preservation of i-games . 196 11 Reduction to Partizan theory 203 11.1 Cooling and Heating . 203 11.2 The faithful representation . 208 11.3 Describing everything in terms of G . 212 12 Boolean and n-valued games 216 12.1 Games taking only a few values . 216 12.2 The faithful representation revisited: n = 2 or 3 . 217 12.3 Two-valued games . 220 12.4 Three-valued games . 227 12.5 Indistinguishability for rounded sums . 230 12.6 Indistinguishability for min and max . 243 III Knots 252 13 To Knot or Not to Knot 253 13.1 Phony Reidemeister Moves . 255 13.2 Rational Pseudodiagrams and Shadows . 259 13.3 Odd-Even Shadows . 265 13.4 The other games . 268 13.5 Sums of Rational Knot Shadows . 271 13.6 Computer Experiments and Additional Thoughts . 273 A Bibliography 278 3 Chapter 1 To Knot or Not to Knot Here is a picture of a knot: Figure 1.1: A Knot Shadow Unfortunately, the picture doesn't show which strand is on top and which strand is below, at each intersection. So the knot in question could be any one of the following eight possibilities. Figure 1.2: Resolutions of Figure 1.1 4 Ursula and King Lear decide to play a game with Figure 1.1. They take turns alternately resolving a crossing, by choosing which strand is on top. If Ursula goes first, she could move as follows: King Lear might then respond with For the third and final move, Ursula might then choose to move to Figure 1.3: The final move Now the knot is completely identified. In fact, this knot can be untied as follows, so mathematically it is the unknot: 5 Because the final knot was the unknot, Ursula is the winner - had it been truly knotted, King Lear would be the winner. A picture of a knot like the ones in Figures 1.2 and 1.3 is called a knot diagram or knot projection in the field of mathematics known as Knot The- ory. The generalization in which some crossings are unresolved is called a pseudodiagram - every diagram we have just seen is an example. A pseudo- diagram in which all crossing are unresolved is called a knot shadow. While knot diagrams are standard tools of knot theory, pseudodiagrams are a recent innovation by Ryo Hanaki for the sake of mathematically modelling electron microscope images of DNA in which the elevation of the strands is unclear, like the following1: Figure 1.4: Electron Microscope image of DNA Once upon a time, a group of students in a Research Experience for 1Image taken from http://www.tiem.utk.edu/bioed/webmodules/dnaknotfig4.jpg on July 6, 2011. 6 Undergraduates (REU) at Williams College in 2009 were studying properties of knot pseudodiagrams, specifically the knotting number and trivializing number, which are the smallest number of crossings which one can resolve to ensure that the resulting pseudodiagram corresponds to a knotted knot, or an unknot, respectively. One of the undergraduates2 had the idea of turning this process into a game between two players, one trying to create an unknot and one trying to create a knot, and thus was born To Knot or Not to Knot (TKONTK), the game described above. In addition to their paper on knotting and trivialization numbers, the students in the REU wrote an additional Shakespearean-themed paper A Midsummer Knot's Dream on To Knot or Not to Knot and a couple of other knot games, with names like \Much Ado about Knotting." In their analysis of TKONTK specifically, they considered starting positions of the following sort: For these positions, they determined which player wins under perfect play: • If the number of crossings is odd, then Ursula wins, no matter who goes first. • If the number of crossings is even, then whoever goes second wins. They also showed that on a certain large class of shadows, the second player wins. 1.1 Some facts from Knot Theory In order to analyze TKONTK, or even to play it, we need a way to tell whether a given knot diagram corresponds to the unknot or not. Unfortu- 2Oliver Pechenik, according to http://www.math.washington.edu/~reu/papers/ current/allison/UWMathClub.pdf 7 nately this problem is very non-trivial, and while algorithms exist to answer this question, they are very complicated. One fundamental fact in knot theory is that two knot diagrams corre- spond to the same knot if and only if they can be obtained one from the other via a sequence of Reidemeister moves, in addition to mere distortions (isotopies) of the plane in which the knot diagram is drawn. The three types of Reidemeister moves are 1. Adding or removoing a twist in the middle of a straight strand. 2. Moving one strand over another. 3. Moving a strand over a crossing. These are best explained by a diagram: Figure 1.5: The three Reidemeister Moves Given this fact, one way to classify knots is by finding properties of knot diagrams which are invariant under the Reidemeister moves. A number of 8 surprising knot invariants have been found, but none are known to be com- plete invariants, which exactly determine whether two knots are equivalent. Although this situation may seem bleak, there are certain families of knots in which we can test for unknottedness easily. One such family is the family of alternating knots. These are knots with the property that if you walk along them, you alternately are on the top or the bottom strand at each successive crossing. Thus the knot weaves under and over itself perfectly. Here are some examples: The rule for telling whether an alternating knot is the unknot is simple: color the regions between the strands black and white in alternation, and connect the black regions into a graph. Then the knot is the unknot if and only if the graph can be reduced to a tree by removing self-loops. For instance, is not the unknot, while 9 is. Now it turns out that any knot shadow can be turned into an alternating knot - but in only two ways. The players are unlikely to produce one of these two resolutions, so this test for unknottedness is not useful for the game. Another family of knots, however, works out perfectly for TKONTK. These are the rational knots, defined in terms of the rational tangles.A tangle is like a knot with four loose ends, and two strands. Here are some examples: The four loose ends should be though of as going off to infinity, since they can't be pulled in to unknot the tangle. We consider two tangles to be equivalent if you can get from one to the other via Reidemeister moves. A rational tangle is one built up from the following two 10 via the following operations: Now it can easily be seen by induction that if T is a rational tangle, then T is invariant under 180o rotations about the x, y, or z axes: 11 Because of this, we have the following equivalences, In other words, adding a twist to the bottom or top of a rational tangle has the same effect, and so does adding a twist on the right or the left. So we can actually build up all rational tangles via the following smaller set of operations: 12 John Conway found a way to assign a rational number (or 1) to each rational tangle, so that the tangle is determined up to equivalence by its number.
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