Knot Theory Week 2: TricolorabilityXiaoyu Qiao, E. L.January 20, 2015A central problem in knot theory is concerned with telling different knots apart. We introducethe notion of what it means for two knots to be “the same” or “different,” and how we maydistinguish one kind of knot from another.1Knot EquivalenceDefinition. Two knots are equivalent if one can be transformed into another by stretching ormoving it around without tearing it or having it intersect itself.Below is an example of two equivalent knots with different regular projections (Can you seewhy?).Two knots are equivalent if and only if the regular projection of one knot can be transformedinto that of the other knot through a finite sequence of Reidemeister moves.Then, how do we know if two knots are different? For example, how can we tell that the trefoilknot and the figure-eight knot are actually not the same?An equivalent statement to the biconditional above would be: two knots are not equivalent ifand only if there is no finite sequence of Reidemeister moves that can be used to transform one intoanother. Since there is an infinite number of possible sequences of Reidemeister moves, we certainlycannot try all of them. We need a different method to prove that two knots are not equivalent: aknot invariant.1
2Knot InvariantA knot invariant is a function that assigns a quantity or a mathematical expression to each knot,which is preserved under knot equivalence. In other words, if two knots are equivalent, then theymust be assigned the same quantity or expression. However, the converse is not true; if two knotsare assigned the same invariant, it does not necessarily mean that they are equivalent. Differentknots may have the same knot invariant, but depending on how “good” the particular invariant is,we may end up with fewer nonequivalent knots that are assigned the same value.What we do know is that if two knots are assigned different values, then it must imply thatthey are nonequivalent knots.A trivial example of a knot invariant is the constant invariant. We assign a constant a to allknots. This is a valid invariant, since “knots K1 and K2 are equivalent” implies “K1 and K2 havethe same invariant, namely, a.” However, this cannot provide any information on whether any twoknots are different, since there is only one possible category, the one which every knot belongs to.Other known knot invariants include Tricolorability, Crossing number, Bridge number, Unknotting number, Linking number and Polynomials. However, we do not yet know whether there existsa perfect knot invariant which can tell apart all knots from each other, or even all knots from justthe unknot.In this write-up, we focus on tricolorability.3TricolorabilityA strand in a knot diagram is a continuous piece that goes from one undercrossing to the next.The number of strands is the same as the number of crossings.A knot is tricolorable if each strand of the knot diagram can be colored one of three colors, subjectto the following rules:1. At least two colors must be used; and2. At each crossing, the three incident strands are either all the same color or alldifferent colors.As shown above, the trefoil is tricolorable. Now try the three knots below; are they tricolorable?2
The left knot, known as the Granny Knot, is a composite knot obtained by taking the connectedsum of two identical trefoil knots; The middle knot is 74 Knot. The right one is figure-eightknot.The Granny Knot and the 74 Knot are tricolorable, but the figure-eight knot is not tricolorable.If we tried, we would run into a contradiction of the tricolorability rules, as the strand representedin black cannot be assigned any of red, green, or blue.Remark. Tricolorability is an invariant under Reidemeister moves.We show this by examining each of the three Reidemeister moves.Move I, which allows us to untwist a knot, or to remove one crossing, preserves tricolorabilityas shown above. The entire twist is assigned a single color (since only two strands are actually usedto represent a twist, we cannot use three distinct colors; therefore, they must have been the samecolor). After Move I, we can still keep the same color for the resulting strand. Conversely, if wehad a strand and wanted to introduce a crossing, then we could color all strands that meet at thecrossing to be the same color, and tricolorability will be preserved.With Move II, first suppose that we want to introduce two new crossings onto two separatestrands that have different colors assigned. Then, we could color each of the resulting three strandsdifferently by introducing the third color, as shown in the image above. Conversely, if we were to3
pull apart two strands and thus remove two crossings, then we could use the two colors that wereused before and remove the third color.If the two strands were of the same color, then we could assign the same color to the third strandthat results from introducing two crossings. Conversely, if we were to pull apart two strands of thesame color that formed two crossings, we could keep the same color as well, and tricolorability ispreserved.Now, if we have three strands, then sliding the top strand from one side of the crossing toanother using Move III does not alter tricolorability, as shown in the images above. There are fourpossible cases: three cases in which one of the three crossings have strands with the same color,and one case in which all of the crossings have strands with the same color.Now that we have shown that tricolorability is indeed a knot invariant, we have a useful fact: ifwe cannot color one regular projection of a knot using the tricolorability rules, then we also cannotdo so with other regular projections of the same knot.Remark. The unknot is not tricolorable.The simplest regular projection of the unknot, a closed loop with no crossings, is not tricolorable,since there is only a single strand and we cannot use at least two distinct colors.4
Therefore, we need not wonder whether this regular projection is tricolorable or not, if wealready know that this is the unknot, since tricolorability is a knot invariant. Either every regularprojection of a knot must be tricolorable, or every projection of a knot must be non-tricolorable.Remark. If a knot is tricolorable, then it is not equivalent to the unknot.Since the unknot is non-tricolorable, for any knot to be equivalent to the unknot, it must benon-tricolorable as well. Thus, a tricolorable knot is not equivalent to the unknot.However, if a knot is non-tricolorable, then do we know that it is equivalent to the unknot? Weillustrate the answer with the following example:We have previously shown that this is a non-tricolorable knot. However, it is also nontrivial (i.e.not equivalent to the unknot). Thus, knowing that two knots are both non-tricolorable (or, bothtricolorable) is not enough to actually tell whether those two knots are distinct. This is becausetricolorability can only provide a binary classification: the knots that are tricolorable, and the knotsthat are non-tricolorable.However, we can find the answer to our earlier question of proving that the trefoil is not equivalent to the figure-eight knot. Since the trefoil is tricolorable and the figure-eight knot is not, weknow that they must be distinct.Image sources:Tricolorability, e/notes/knots.pdf Knots: a handout for mathcircles5
The right one is gure-eight knot. The Granny Knot and the 7 4 Knot are tricolorable, but the gure-eight knot is not tricolorable. If we tried, we would run into a contradiction of the tricolorability rules, as the strand represented in black cannot be assigned any of red, green, or blue. Remark. Tricolorability is an invariant under .
8) Learn to tie the required knots: Daisies should learn 1 knot, Brownies 2, Juniors 3, Cadets 4, Seniors, 5 and Ambassadors all 6. Mooring Hitch Heaving Line Knot Square Knot Bowline Knot Overhand Knot Clove Hitch Knot Double Fisherman’s Knot 9) Give at least two situations that each knot
(prorated 13/week) week 1 & 2 156 week 3 130 week 4 117 week 5 104 week 6 91 week 7 78 week 8 65 week 9 52 week 10 39 week 11 26 week 12 13 17-WEEK SERIES* JOIN IN MEMBER PAYS (prorated 10.94/week) week 1 & 2 186.00 week 3 164.10 week 4 153.16 week 5 142.22 week 6 131.28 week 7 120.34
the trefoil knot, which has three crossings and is a very popular knot. The trefoil knot can be found in gure 2. Figure 1: Unknot Figure 2: Trefoil Knot Knot theory was rst developed in 1771 by Alexandre-Th eophile Vandermonde, while the true mathematical studies of knots began in the 19th century with Carl Friedrich Gauss.
The Square Knot is a very ancient knot and is also referred to as the Reef Knot or Hercules Knot. The Square Knot has been used for millennia by human kind for various purposes, including artwork, binding wounds, sailing, and textiles. This knot should not be used to tie two pieces of rope together nor be used in critical situations, as it
Clasped Hands Knot . .110 Diamond Ring Knot . .112 section 11 short and Long sinnets Eternity Knot . . . . . . . . .116 . equivalent lengths of paracord. xv Rope Parts Rope Loops Knot Parts Knot Movements starting end Running end Bight Lop crook counterclockwise Loop File Size: 628KB
18 3 BEND Square Knot aka Reef Knot Types of Knots Bend: Knot used to secure two ends of rope Binding: Knot used to secure objects together Decorative: Knot usually used solely as decoration such as wrappings, necklaces, key chains, etc. Hitch: Knot used to secure a rope to ano
The Blimp Knot also appears on the Decorative Knots page, but I included it here because it can be used as a "stopper knot" at the end of a rope or string. 6. Double Overhand Knot or ABOK #516 See the Overhand Knot (below). 7. Figure Eight Knot or ABOK #520 Tying a "stopper knot" at the end of the rope can help prevent the end from slipping
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