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From Wikipedia, the free encyclopedia

In the mathematical field of knot theory, a chiral knot is a knot that is not equivalent to its mirror image. An oriented knot that is equivalent to its mirror image is an amphichiral knot, also called an achiral knot or amphicheiral knot. The chirality of a knot is a knot invariant. A knot's chirality can be further classified depending on whether or not it is invertible.

There are only five knot symmetry types, indicated by chirality and invertibility: fully chiral, reversible, positively amphichiral noninvertible, negatively amphichiral noninvertible, and fully amphichiral invertible.[1]

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  • Braids. Chapter 3 - The world of knots
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Transcription

Let's change topic for a while These sailor knots are not knotted! The knots can escape from the string ends To capture the knots we have to glue the string ends together Now these are knots Closing the ends of the objects on the right we get more complex knots, where two components can be linked together Without originality, we call these objects links If we can untie them, we obtain a very simple link, called unlink From now on, we will say knots referring to both knots and links Look at this knot. If we take its mirror image, it looks different but we can deform the knot on the left into the one on the right without cutting the string Then, we will consider them the same knot even if they look different at first These two knots look different, too. But they are tied in the same manner They have different shapes but represent the same knot One can be deformed into the other without ever cutting the strings This knot is called trefoil As before, we can ask: is the trefoil the same as its mirror image? This is a very technical and difficult problem The two knots are not the same. They are called left-hand and right-hand trefoil How can we distinguish different knots or recognize different drawings of the same knot? A way to approach such a problem is to relate the realm of knots, closed strings, to that of braids It is easy to see that when we have a braid, we can tie the strand ends together and we can cross the bridge to the realm of knots And vice versa? Can we cross the bridge in the other direction? A theorem of Alexander ensures that it is possible and gives an algorithm to do it We describe it, even if more efficient ones are known We choose an axis. We will make a reel around it We choose a starting point and walk along the knot turning clockwise around the axis At some time the knot can turn and we will be walking anti-clockwise We color all the anti-clockwise pieces red Now we move each red piece in turn to the other side of the axis and color them yellow again In the end we have turned our knot into a reel around the axis Walking along the knot we will always be going in the same sense around the axis We take a half plane with the axis as border and cut the knot along it We open the strands keeping the endpoints fixed on the half planes The strings can never touch each other And here is our braid! When we close it, we get a knot equivalent to the original one that is, we can deform one into the other without cutting the string Why make life so difficult? The knot on the left seems simpler! But in this way we can exploit the group structure that we know on braids Alexander's theorem ensures that we can obtain any knot as the closure of a braid but two braids can be very different and still give the same knot For example, they don't even need to have the same number of strands! So the question now is: given two arbitrary braids, do they give the same knot, once they are closed? We introduce a new operation, called conjugation Choose a braid Take another one and its inverse and compose them in this manner: one on the left and the inverse on the right The new braid is called a conjugate of the first Note that the corresponding operation with numbers will not change the starting number: the product is commutative On the contrary, two braids can be different and still be conjugated Here is a simpler example: two generators of the braid group are surely different braids But look: they are conjugated In general understand whether two given braids are conjugate is an intriguing issue Let's go back to our problem: when do two braids close to the same knot? If we conjugate a braid with any other, when we close the new braid we can shift the lateral pieces so that they cancel out since one is the inverse of the other In this way we get the same knot as closing the original braid We can modify our braid in another way: add a strand on the top and link the two top strands together The new braid, when closed, will give the same knot as the old one: we just need to undo the loop Of course we can do vice versa, too: cancel the last strand if it links just once with the second-to-last These operations are called stabilizations A Russian mathematician, Markov, noticed that: Two braids give the same knot if and only if they are related by a sequence of moves of the two kinds we have just seen This is now known as Markov theorem, even if the first proof is probably due to one of his students We didn't show the difficult part of this theorem, namely, that the two kinds of moves are enough We just make an example. We already know that these two braids give the same knot Now we can prove this, without passing through the realm of knots! We have to find a sequence of conjugations and stabilizations that transforms one braid into the other Markov theorem exactly says when two braids give the same knot, but in this form it is of no concrete use: finding a sequence of relations can be very difficult And, as in chapter 2, if we can't find such a sequence, it doesn't mean that none exists! Approaching knots through braids seems not to simplify things But one of the major results on knots was achieved just thanks to the braids! In 1984 Jones, studying braids, proved an outstanding result that revolutionized the theory of knots! It was so important that he won the Fields medal for it, the most important award for mathematicians Jones found a way to associate a formula, a mathematical expression, to each braid The powerful fact is that this permits us to distinguish the knots obtained closing the braids: if two braids have different formulas, then they give different knots On the other hand, if two braids differ by Markov moves, then they are associated to the same formula This means that the formula, called Jones polynomial, only depends on the knot and not on the braid used to get it! As an example, the trefoil and the figure eight knot have different Jones polynomials, so they are surely different knots Later another algorithm to calculate the Jones polynomial was discovered, not involving braids anymore Choose a direction to walk along the knot There are places where we see a crossing The crossings can be of two types depending on the strand that passes behind the other Resolving a crossing means to break the arcs and connect them in the other way, respecting their orientation Introduce a relation between these pieces The symbol V indicates the Jones polynomial Now, associate the polynomial 1 to the unknot Using just these two relations, we can calculate the polynomial on every knot Choose a crossing and apply the first relation to it Simplify... and apply the first relation again to the knot on the right to write a new equation Choose a new crossing and go on like this, writing equations and simplifying Using always the same relations, we can calculate the Jones' polynomial of the simplest knots Then, going back step by step, we can reconstruct the expression for the complex knots in our case the right trefoil We didn't check that this machinery is coherent, that is, making different choices always gives the same expression for a fixed knot This is the difficult part, and the interesting one: the Jones' polynomial is an invariant of knots: calculated on two equivalent knots, it is the same If we calculate the Jones polynomial on the left trefoil, we obtain an expression that is symmetric to the other, in some sense But not equal We have proved that the two trefoils are not equivalent!

Contents

Background

The chirality of certain knots was long suspected, and was proven by Max Dehn in 1914. P. G. Tait conjectured that all amphichiral knots had even crossing number, but a counterexample was found by Morwen Thistlethwaite et al. in 1998.[2] However, Tait's conjecture was proven true for prime, alternating knots.[3]

Number of knots of each type of chirality for each crossing number
Number of crossings 3 4 5 6 7 8 9 10 11 12 13 14 15 16 OEIS sequence
Chiral knots 1 0 2 2 7 16 49 152 552 2118 9988 46698 253292 1387166 N/A
Reversible knots 1 0 2 2 7 16 47 125 365 1015 3069 8813 26712 78717 A051769
Fully chiral knots 0 0 0 0 0 0 2 27 187 1103 6919 37885 226580 1308449 A051766
Amphichiral knots 0 1 0 1 0 5 0 13 0 58 0 274 1 1539 A052401
Positive Amphichiral knots 0 0 0 0 0 0 0 0 0 1 0 6 0 65 A051767
Negative Amphichiral knots 0 0 0 0 0 1 0 6 0 40 0 227 1 1361 A051768
Fully Amphichiral knots 0 1 0 1 0 4 0 7 0 17 0 41 0 113 A052400

The simplest chiral knot is the trefoil knot, which was shown to be chiral by Max Dehn. All torus knots are chiral. The Alexander polynomial cannot detect the chirality of a knot, but the Jones polynomial can in some cases; if Vk(q) ≠ Vk(q−1), then the knot is chiral, however the converse is not true. The HOMFLY polynomial is even better at detecting chirality, but there is no known polynomial knot invariant which can fully detect chirality.[4]

Reversible knot

A chiral knot that is invertible is classified as a reversible knot.[5] Examples include the trefoil knot.

Fully chiral knot

If a knot is not equivalent to its inverse or its mirror image, it is a fully chiral knot, for example the 9 32 knot.[5]

Amphichiral knot

The figure-eight knot is the simplest amphichiral knot.
The figure-eight knot is the simplest amphichiral knot.

An amphichiral knot is one which has an orientation-reversing self-homeomorphism of the 3-sphere, α, fixing the knot set-wise. All amphichiral alternating knots have even crossing number. The first amphichiral knot with odd crossing number is a 15-crossing knot discovered by Hoste et al.[3]

Fully amphichiral

If a knot is isotopic to both its reverse and its mirror image, it is fully amphichiral. The simplest knot with this property is the figure-eight knot.

Positive amphichiral

If the self-homeomorphism, α, preserves the orientation of the knot, it is said to be positive amphichiral. This is equivalent to the knot being isotopic to its mirror. No knots with crossing number smaller than twelve are positive amphichiral.[5]

Negative amphichiral

The first negative amphichiral knot.
The first negative amphichiral knot.

If the self-homeomorphism, α, reverses the orientation of the knot, it is said to be negative amphichiral. This is equivalent to the knot being isotopic to the reverse of its mirror image. The knot with this property that has the fewest crossings is the knot 817.[5]

References

  1. ^ Hoste, Jim; Thistlethwaite, Morwen; Weeks, Jeff (1998), "The first 1,701,936 knots" (PDF), The Mathematical Intelligencer, 20 (4): 33–48, doi:10.1007/BF03025227, MR 1646740, archived from the original (PDF) on 2013-12-15.
  2. ^ Jablan, Slavik & Sazdanovic, Radmila. "History of Knot Theory and Certain Applications of Knots and Links Archived 2011-08-20 at the Wayback Machine.", LinKnot.
  3. ^ a b Weisstein, Eric W. "Amphichiral Knot". MathWorld. Accessed: May 5, 2013.
  4. ^ "Chirality of Knots 942 and 1071 and Chern-Simons Theory" by P. Ramadevi, T. R. Govindarajan, and R. K. Kaul
  5. ^ a b c d "Three Dimensional Invariants", The Knot Atlas.
This page was last edited on 26 January 2018, at 15:21
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