The Additivity of Crossing Number with Respect to the Composition of Knots

THE ADDITIVITY OF CROSSING NUMBER WITH RESPECT TO THE COMPOSITION OF KNOTS

JASON GRANDY

SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTERS OF SCIENCE IN MATHEMATICS

NIPISSING UNIVERSITY SCHOOL OF GRADUATE STUDIES NORTH BAY, ONTARIO

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Abstract

This paper will investigate the additivity of the crossing number with respect to the composition of knots. The additivity of the crossing number is a long standing conjecture.

The paper presents proofs of this conjecture for alternating knots and torus knots. For alternating knots, the paper uses the Jones Polynomial to show the alternating diagram has minimal degree, and proves the composition of two alternating knots is another alternating knot. For torus knots, the paper’s main ingredient is a closed form equality for the crossing number involving the braid index and genus of the knot. We then show the additivity under composition of these components of the formula to prove the additivity of the crossing number.

Contents

Abstract ...... iii Introduction ...... 1 Knots and Knot Diagrams ...... 1 Crossing Number ...... 2 Knot Diagram Regions ...... 3 Reidemeister Moves ...... 4 Composition of Knots ...... 5 Braids ...... 7 Bridges ...... 8 Writhe Number ...... 10 Torus Knots ...... 10 Alternating Knots ...... 16 Seifert Surfaces ...... 17 A Closed Form Equation for the Crossing Number...... 19 Knot Genus ...... 21 The Bracket Polynomial and Jones Polynomial ...... 23 Span of the Bracket Polynomial and Jones Polynomial ...... 28 Main Conjecture ...... 31 Additivity of Composition Operation for Torus Knots ...... 32 Additivity of Composition Operation for Alternating Knots ...... 33 References ...... 33

Introduction

The goal of this paper is to present the findings concerning the additivity of the crossing number with respect to the composition operation of knots. That is, for two knots A and B, ( ) ( ) ( ) where c(A) is the crossing number for the knot A, and “#” is the composition operation. One direction of the equality requires just an observation, of two knot diagrams, so we can see ( ) ( ) ( ). This conjecture has been solved for alternating knots by Murasugi [4], and a family of knots that includes torus knots by Gruber [1]. As well, bounds have been placed on the other inequality by Lackenby [2]. He proved the inequality ( ) ( ( ) ( )).

The field of knot theory is still very young, with a lot of major progress happening within half a century such as the Jones Polynomial, and the two main conjecture proofs given. Definitions and concepts shown are easy to explain and visualize, and often so are the theorems. Nevertheless the theorems require advanced tools to prove. The main conjecture presented follows this trend with the simplicity that it can be explained to someone with no post secondary mathematics education, yet still remains unsolved.

The findings will be presented in a way that a person with no knowledge of knots will be able to find all information needed to understand the concepts in the paper itself. In short, this paper will be self contained.

Knots and Knot Diagrams

Definition 1. A knot is an embedding of S1 in R3. A link is one or more knots with no intersections.

For the scope of this paper, we will only consider smooth or piece-wise linear embeddings of S1 to R3.

Definition 2. Two links are equivalent if there exists an orientations preserving diffeomorphism or, depending on situations and knots we consider, a piece-wise linear homeomorphism of R3 on itself that maps one link onto the other.

Definition 3. A link diagram is projection of a link onto a plane such that the projection does not cross itself more than twice at any point. Since a link in R3 doesn’t intersect itself, any intersections on a link diagram are labelled with overcrossings and undercrossings to represent which section of the link lies above and which lies below.

Definition 4. Two link diagrams are equivalent if the links they are projections of are equivalent.

For several properties it is needed to give a link an orientation. This will be represented by an arrow on a link diagram for several examples in this paper.

The following terminology will be used throughout the paper.

Definition 5. A string is a connected subset of a knot. A strand is a string on a knot diagram whose endpoints begin and end at crossings.

Definition 6. A splitting point is a crossing that, if deleted, would split the knot into two.

Crossing Number

Definition 7. The crossing number of a link diagram, denoted c(D) for a diagram D of a link K, is the number of times the link crosses over itself. The crossing number of a link, denoted c(K), is the minimum crossing number of all its link diagrams.

Definition 8. Any link with a finite number of crossings is called a tame link, and any link with an infinite number of crossings is called a wild link. Aside from one brief proof, all links considered in this paper will be tame links. Moreover, we will usually assume all links are piece-wise linear or smooth.

The only knot with zero crossings is the trivial knot, which also called the unknot. The only two knots with three crossings are the trefoil knot, and its mirror image. The knot with four crossings is the figure eight knot. These three knots will be used extensively in this paper.

Knot Diagram Regions

Definition 9. A region is a connected component of the complement of a link diagram in the plane.

Each lettered area in the picture below is a region.

If we turn a knot’s crossings into vertices and the knot itself into an edge, we can create a graph where the faces are the regions. By using the Euler Characteristic, we know where V is the number of vertices, E is the number of edges, and F is the number of faces.

We notice that is incident with 2 vertices, and each vertex is incident with 4 edges. This implies that there are twice as many edges as vertices, so , and so . Therefore, including the outside face, there are 2 more regions than crossings in any given knot. Thus, we proved the following proposition.

Proposition 10. The number of regions of a knot diagram equals ( ) .

Reidemeister Moves

The Reidemeister Moves are a set of moves which can transform a link diagram into and only into an equivalent link diagram. These moves are easily described and visualized. The diagrams given also hold for all symmetries possible.

Reidemeister Move 1: To add or remove a loop

Reidemeister Move 2: To pass one string over or under another string, or the reverse

Reidemeister Move 3: To pass a string over or under a crossing

Theorem 11 ([3] Th 2.1). Two link diagrams are equivalent if and only if there exists a series of Reidemeister moves transforming one into the other.

With a series of Reidemeister moves, we may eliminate splitting points by vertically flipping one side. This technique of grouping a part of a knot and moving it is used often to transform diagrams.

Another common technique is to use a series of Reidemeister moves to move one outer strand over the entire knot diagram.

Composition of Knots

Composition is an operation (denoted #) of two oriented knots. The operation is performed by cutting both knots, and attaching them to each other while preserving orientation.

This operation is a special case of the connected sum of two one-manifolds, described using knot diagrams instead of the knots themselves. The only other alteration is the need for an orientation, as two separate knots can be composed by switching it around. The result is still an embedding of the circle in 3-space. The cut point does not matter, for we can “shrink” one knot down, and run it along the second to any other point.

To see that this operation is commutative, merely rotate the plane 180’. The unknot is the identity.

Finally, inverses do not exist. We may prove this by performing a swindle, creating a wild knot. Assume A#B = O, where A and B are not trivial. Then B = B#O#O#O... = B#(A#B)# (A#B)# (A#B)#... = (B#A)#(B#A)#(B#A)#... = (A#B)#(A#B)#(A#B)#... = O#O#O#... = O. But B is nontrivial, a contradiction. The absence of inverses can also be proven with a knot’s genus, which we will do later.

Definition 12. We call a knot a prime knot if it is not the composition of two other nontrivial knots, otherwise it is a composite knot.

We can perform the opposite of the composition operation by factoring off a knot or knot diagram. For knots, this is done by placing a sphere such that it intersects the knot at 2

points, then cutting the knot at those 2 points to create 2 knots. The same can be done for a knot diagram using a circle.

Braids

Definition 13. A braid is a finite number of strings that all begin along the same line, and end on a different line. A braid diagram is a link formed by taking the ends of these strings and connecting them consecutively to arcs, orienting all the arcs in the same direction.

Depending on the braid, this might result in a link, or a knot due to the arcs not being connected when traversing along the link. Every link has a braid diagram representation, that is to say there exists a series of Reidemeister moves that will transform any link diagram into a braid diagram. Note that this representation might not have the minimum number of crossings for the knot. We can see through a series of Reidemeister type 1 moves that the created braid diagram below is actually the unknot. This can also be seen by observing the created diagram only has two crossings, where all nontrivial knots have at least three.

Definition 14. A braid diagram’s braid index is the number of arcs outside the braid. A link’s braid index (denoted b(K)) is the minimum number of braid indices of all possible braid diagrams of the link.

Bridges

Definition 15. A bridge is a strand that starts and ends at an undercrossing, and includes at least one overcrossing and no undercrossings. A links’ bridge number (denoted br(K)) is the minimum number of bridges of every link diagram of the link. The first diagram is a trefoil with one bridge highlighted, and the second diagram is the trefoil with the minimum number of bridges.

The bridge number can also be defined as the minimum number of local maxima with respect to every direction vector of the link, given a smoothing of the link. We may also consider the minimum number of local maxima of all link diagrams of a link. To see this, imagine you are holding a knot in space in front of you. We may project this knot downwards onto the floor while being careful not to overlap the knot more than twice at any one point. We may also project the knot forwards onto the wall. Let the floor diagram have m bridges. Then it is easy to visualize that the wall diagram has a minimum of at most m local maxima. After all, wherever there is a bridge, the knot will be elevated, reach a peak, and then rest back down. The reverse is also true, albeit harder to see: if the wall diagram has n local maxima, the floor diagram will have n bridges after removing unnecessary local maxima.

Since this is true for all embeddings of the original knot, the minimum number of local maxima must equal the bridge number for the knot.

For any braid diagram, the number of circles is the number of local maxima of that diagram. It immediately follows that the minimum number of local maxima, thus the bridge number, is less than or equal to the braid index. Thus we have the following.

Proposition 16. ( ) ( ).

Writhe Number

Definition 17. Consider an oriented link. Assign to each crossing based upon the following rule:

We call these two types of crossings positive and negative respectfully. The writhe number is the sum of all signs of all the crossings.

Note that the writhe number is not a knot invariant. Indeed, we may total the difference in Reidemeister type 2 and 3 moves and see it does not change, but the writhe number changes by depending on the orientation with a Reidemeister type 1 move. We denote the writhe of a knot diagram D by w(D).

Torus Knots

Definition 18. A torus link is a link embedded in a torus.

Consider the representation of the torus as the unit square with opposite sides identified. Embed a link in this representation that winds p times around the latitude of the torus, and q times around the longitude, where p and q are integers. Eliminate any unnecessary passes through the identified sides by deforming the link. The link, obtained after such elimination, is equivalent to the original link. Thus any torus link has a standard representation denoted ( ).

Consider a loop based at any point on a torus. The fundamental group of the torusis isomorphic to . Consider the element ( ) which corresponds to a loop that wraps around the longitude p times, and the meridian q times. The loop ( ) wraps around in the opposite orientation, so the loops ( ) and ( ) represents opposite orientations of the same link. By the same token, ( ) and ( ) are opposite orientations of the mirror image of the link of ( ), seen by the picture below.

We also note that ( ) is equivalent to ( ). To see this, simply rotate the representation above by 90’.

To project ( ) onto the plane, we connect the top and bottom edge points with arcs, and we connect the left and right edges with overcrossing strings. It is easy to see that the diagram created is a braid. The example here is a T(3,2) knot.

We can see through a series of Reidemeister moves that the T(3,2) is also the trefoil.

We can also create the T(2,3) knot, which is another presentation of it:

Because of the way the diagrams are created, we can see that all crossings are positive. In other words, ( ) ( ). Another observation is ( ) ( ) .

It is easy to see that if p and q aren’t relatively prime, more components will exist, resulting in a link instead of a knot. The number of links created is the greatest common divisor of p and q. To see this, let p > q, and consider the leftmost starting point for the braid. The string will end at qth ending point, then run around to the qth starting point. This process will continue until we end back at the 1st starting point again. Treat this as elements of the group of integers under addition modulo p (denoted Zp) and consider the element q. We can see that if q and p aren’t relatively prime, we won’t traverse every starting point of the braid, and instead will traverse a number of strings equal to the order of the element q in

Zp. For example, here is the T(4,2) with one component bolded.

The following theorem regarding the bridge number of a torus knot is extracted from the following paper [5] by Jennifer Schultens. For this, we need to first make a few definitions.

Definition 18. Let be a Morse function with exactly two critical points, such that all level surfaces are two dimensional spheres.

Definition 19. If the minima of occur below all the maxima of , then we say that the knot K is in bridge position. We may put any knot into bridge position by raising all its maxima, and lowering all its minima.

Definition 20. Let K be embedded in a torus T. We say T is taut with respect to br(K) if the number of critical points of is minimal subject to the condition that has br(K) maxima.

Definition 21. Consider a singular foliation of T induced by . Let be a leaf corresponding to a saddle singularity. Then consists of two circles, s1 and s2, wedged at a point s. If either s1 or s2 is nullhomotopic, then the saddle is called an inessential saddle. Otherwise it is an essential saddle.

Theorem 22 [5]. ( ( )) ( )

Proof. Embed a torus in the three dimensional sphere S3, then embed a knot K in the torus T. Consider a Morse function h on S3. Restrict h to T, and consider a particular value of h corresponding to a saddle. Since our torus might be embedded in an “ugly” way, inessential saddles might exist.

Consider any inessential saddle , and let s1 be nullhomotopic. Then we may assume that s1 bounds a disc D contained in T that has only one maximum or minimum of h. For if it had more than one maximum or minimum, it would contain an inessential saddle, which we would consider first. If D intersects K, we may alter K by sliding the knot down below the disc, preserving the number of maximums of K. Then we may assume there is no intersection between D and K.

Consider a level surface L containing . This level surface contains both s1 and s2. We will show that D co-bounds a 3-ball B with a disk D1 contained in L such that B does not contain infinity or negative infinity, and s2-s lies in L-D1.

L-s1 consists of two discs, call these D1 and D2. Let B1 be the 3-ball bounded by D and D1, and B2 be the 3-ball bounded by D and D2. Since the critical point of D is a maximum, then either B1 or B2 contains infinity, and the other contains neither points infinity nor negative

infinity; without loss of generality, let B1 contain infinity. We notice s2 is contained in either

D1 or D2. If s2 is contained in D2, then we just let B=B1 and we’re done. So assume s2 is contained in D1. Let be an arc beginning at the local maximum of D, passing only through local maximums of T, and ending at infinity such that the arc is constantly increasing with respect to h. We may alter K such that does not intersect it. Since T has finite local maximums, let be the points where intersects the T, with being the highest point. Consider a neighbourhood of that does not intersect K. As it is the highest point, we may raise it along above infinity such that we do not alter the isotopy of

T, nor change the number of local maximums of T. We then perform a small tilt to turn into a Morse function. We next raise the neighbourhood of such that it lies between infinity and . Proceeding by induction, we raise all of the neighbourhoods including the neighbourhood of . Since is the local maximum that is contained in D, we have raised

D such that infinity is contained in the ball B2. Since infinity is contained in B2, it is not contained in B1, and we let B=B1. In both cases, the conditions for B have been meet.

Let T be taut with respect to br(K). Then there exist no inessential saddles. To see this, we assume there exists an inessential saddle. Let s1 be nullhomotopic. Then there exists a ball

B bounded by D and D1 such that it does not contain positive or negative infinity, and it does not intersect s2-s. We may assume that D contains a maximum. Shrink ( ) horizontally, and lower it such that it lies below D1. This process does not change the number of critical points of or . Replace D1 with D, and tilt T so is a Morse function. Then the new torus is isotopic to T, but has two fewer critical points, as we removed an inessential saddle and a maximum. But this contradicts T being taut with respect to br(K).

Create a small bicollar N of s1 and s2. Then N has three boundary components c1, c2 and c3 such that c1 is parallel to s1, and c2 is parallel to s2. We may assume that c1 and c2 are contained in the same level surface. Let Di be a disc bounded by ci for i=1,2,3.

Let T be taut with respect to br(K). Let be the highest saddle. We may assume this saddle is essential and as such s1 and s2 are not nullhomotopic. Since is the highest saddle, any

curve in ( ) bounds a disc in T lying above the level surface, for i=1,2. As such, Di is isotopic to a disc whose interior is disjoint from T. This combined with si not being nullhomotopic implies c1 and c2 are both either a latitude or longitude of T, wrapping around exactly once. In either case, ( ) for i=1,2.

Note that points from and are connected by arcs that lie inside the intersection ( ), where D is the disk in the torus T such that its boundary is the curve c3 . Since the endpoints of these arcs are at the same level for h, each arc must contain at least one maximum, and consequently ( ) ( ).

We notice then that the number of disjoint parts between is greater than or equal to ( ). Also, for i=1,2 is a single point. Since and are connected by an arc and occur at the same level for h, each arc must contain at least one maximum, and as such ( ) ( ). Consider ( ). In , the points are connected by arcs for i=1,2. As these points occur on the same level surface and it is the highest saddle, each arc contains a maximum. Therefore ( ) ( ).

Since there exists a standard diagram for T(p,q), that has p bridges, we obtain ( ) ( ).

Consider a T(p,q) knot where . Use the standard torus knot diagram. We note that this knot diagram has p arcing circles. Since the bridge number for our knot is p, and ( ) ( ), we have the following proposition.

Proposition 23. ( ) , where K is a T(p,q) knot, and .

Alternating Knots

Definition 24. An alternating diagram alternates between overcrossings and undercrossings as the knot is traversed.

An alternating knot is a knot with an alternating diagram. The trefoil and its mirror are the simplest alternating knots, with their standard diagram shown in this paper being an alternating diagram. The figure eight knot is also an alternating knot. At one point, it was believed that all knots were alternating knots. In fact, the simplest non-alternating knot has eight crossings, and it is the T(3,4) torus knot. While it seems to be an easy task to tell if a

knot is an alternating knot, it isn’t. It takes complex tools to tell if a knot does not have any alternating diagrams.

The composition of any two alternating knots is an alternating knot. To see this, consider any two alternating diagrams. Choose the cut point of both to be just after an undercrossing, with respect to the orientations. The result is an alternating knot.

Seifert Surfaces

For every oriented link there exists an orientable surface with the link as its boundary. This can be shown via construction, with the resulting surface called a Seifert Surface. First we construct what is called a link diagram’s Seifert Circles. To do this, eliminate each crossing of a link diagram by attaching the outgoing strands to the other incoming strands.

The result is a series of circles with some maybe contained in others. We denote the number of Seifert Circles of the diagram s(D). The number of Seifert Circles in an oriented link, denoted s(L), is the minimum number of s(D) for all diagrams of the link. To create the Seifert Surface, we add twist bands connecting the circles wherever there used to be a crossing. Give each Seifert Circle an orientation, then lift each circle up so it’s sitting on a different plane in 3-space. Circles on top of other circles are given the same orientation,

and circles not contained and joined by twist bands are given opposite orientations. Fill each circle with a disc.

The resulting surface is the Seifert Surface. Different link diagrams belonging to the same link may yield different Seifert Surfaces.

We may calculate the number of Seifert Circles of a composition of two knot diagrams. Let A and B be knot diagrams. Then when we take their composition, one of the Seifert Circles from each diagram combines. The result is ( ) ( ) ( )

The number of Seifert Circles, s(D), is the number of arcs in a braid for a braid diagram D. This is because the orientation always going downward splits each crossing into vertical strings, turning the arcs into nested Seifert Circles as shown by the picture below.

This gives the inequality ( ) ( ) for any diagram D of an oriented link L.

It is possible to show that, given any knot diagram D, there exists an equivalent braid diagram with s(D) arcs. This proves the following theorem by Yamada.

Theorem 25 [6]. ( ) ( ).

Considering the two inequalities, we have

Theorem 26. For any diagram D for a knot K, ( ( )) ( )

Since we have ( ) ( ) ( ) for any two diagrams and since the braid index is the minimum number of Seifert Circles across all diagrams of a knot, an immediate corollary is as follows.

Corollary 27. ( ) ( ) ( ) for any two knots X and Y.

A Closed Form Equation for the Crossing Number

We may express the crossing number of a knot diagram by using its Seifert Surface. We start by gluing a disc along the knot, ie the boundary of the Seifert Surface. We then calculate the Euler Characteristic of a knot diagram’s Seifert Surface, X = V – E + F, where X is the Euler Characteristic, V is the number of vertices, E is the number of edges, and F is the number of faces. Denote X(D) the Euler Characteristic of a Seifert Surface for a diagram D. To calculate X(D), we must triangulate our Seifert Surface. The faces are the discs, bounded

by Seifert Circles, the disc glued along the knot, and the twist bands connecting them. Each circle has a vertex added to create a 0-skeleton. The Seifert Circles and twist bands are joined together with 2 vertices and an edge. Since the Seifert Surface is made of its Seifert Circles s(D) and has a number of twist bands equal to its crossing number c(D), we find F, E and V in terms of those values. According to the diagram below,

we are adding 1 face, 1 edge and 1 vertex for every Seifert Circle, and for every crossing we are adding 1 face, 4 vertices, and 6 edges. We also include the face of the disc that we glued along the boundary of the knot. We have

( ) ( )

Which implies ( ) ( ) ( ) ( )

Therefore ( ) ( ( ) ( ) )

Rearranging for the crossing number, we obtain ( ) ( ) ( )

We may generalize this equation to links, labelling |K| as the number of components:

( ) ( ) ( )

Now we minimize the crossing number of the diagram to find the crossing number of the link. As a result, we obtain the following formula:

Proposition 28. ( ) ( ( ) ( ))

By minimizing and using the fact that the minimum number of Seifert Circles is the braid index, we find the following.

Proposition 29. ( ) ( )+b(K)+|K|-2

Knot Genus

Definition 30. A knot diagram’s genus, denoted g(D), is the genus of the diagram’s Seifert Surface. A knot’s genus, denoted g(K), is the minimum of all of its diagram’s genus.

Theorem 31. The knot genus is additive with respect to composition, that is if A and B are knots, g(A#B)=g(A)+g(B).

Proof. Let A and B be knots, and let sA and sB be Seifert Surfaces associated with A and B respectively with minimum genus. We want to find a Seifert Surface for A#B with a genus equal to the sum of the genii of A and B. This is done by attaching a band from A to B, connecting sA to sB. The result is a Seifert Surface for A#B. The genus of this surface is g(A)+g(B). Therefore, we have ( ) ( ) ( ).

Now we proceed in the other direction.

Consider the knot A#B. Let sA#B be the Seifert Surface of A#B with minimum genus. We need to find Seifert Surfaces for A and B such that the sum of their genii is g(sA#B). There exists some embedding of the sphere such that it divides A#B into A and B, intersecting the knot only at points x and y. The knot and cutting points may be more intricate than the ones in the picture below.

Consider . We may choose the embedding of S2 in such a way that it intersects sA#B at simple closed curves, one of which contains x and y. Selecting neighbourhoods small enough for each closed curve creates a cylinder.

Delete each cylinder, and fill the hole created on each side with a disc. For the hole created on the x-y cylinder, the disc created has an edge that connects where x and y were, reconnecting the knot. The result is two orientable surfaces, one with A as its boundary, and one with B. As such they are Seifert Surfaces for A and B.

Since each operation performed can only decrease the total genus, we have ( ) ( ) ( ).

Therefore ( ) ( ) ( ).

We can see that the genus of the unknot is 0. Now consider any knot K. If the knot has a genus of 0, it has a Seifert Surface of just the sphere, but that implies its edge is the circle. Therefore the only knot with genus 0 is the unknot. Since the genera of knots is additive, we can see that the composition of any two non-trivial knots is not the unknot.

The Bracket Polynomial and Jones Polynomial

The Bracket Polynomial is an attempt at finding a polynomial derived from link diagrams, such that it is invariant under the Reidemeister moves. If a polynomial is invariant under Reidemeister moves, then every diagram of a given link will have the same polynomial associated with it. While the Bracket Polynomial a true invariant as it is not invariant under Reidemeister 1 moves, it paves the way for the Jones Polynomial, which is a knot invariant.

To construct the Bracket Polynomial, we start with an unoriented link diagram D and denote the Bracket Polynomial of L as . We then use the following rules:

- The polynomial of the unknot is equal to 1. = 1 - The polynomial of the unlinked union of the unknot diagram and D is equal to c. = c, and

- where a, b, and c are variables. In the third rule we have strings inside of a circle. It is assumed that the strings are part of the same link diagram outside of the circle, and only vary inside. This notation will be used extensively. Using these rules, we can consider the crossings of any knot, and break it down using rule 3 until we are only considering an unlinked union of unknots. We can then use rule 2 to break these down further to only unknots, then use rule 1 to eliminate it. By requiring the invariance of the polynomial with respect to the Reidemeister moves, we will express variables b and c in terms of a. We proceed by considering a Reidemeister type 2 move. We want

which would imply

Thus for our polynomial to be invariant under type 2 moves, we need to restrict and , thus , and . Rule 2 can now be expanded upon. If we have a link of unlinked unknots, the term is multiplied by ( ) , where n is the number of components. We now see that the polynomial does not change under Reidemeister type 3 moves under the same restriction of variables, using a Reidemeister type 2 move to prove it:

However, we still run into a problem when we consider a Reidemeister type 1 move. We start with

This implies , meaning no restriction of variables can make our Bracket Polynomial invariant under Reidemeister type 1 moves. As such, it needs to be tweaked. Noticing how we have Reidemeister type 2 and 3 moves as invariant, the immediate answer is to use a link’s writhe number, as the writhe number only changes under Reidemeister type 1 moves. We define our new polynomial as

( ) ( ) ( )

Now we perform a change of variables to simplify. The result is the Jones Polynomial. This polynomial is invariant under all three of the Reidemeister moves, although two different knots may have the same Jones Polynomial. While the writhe number needs an orientation, the Jones polynomial will be the same regardless. To illustrate the power of the Jones Polynomial, consider the trefoil and its mirror image. We can show that they are not isotopic by showing that they have different polynomials. Let A be the trefoil diagram, and B be its mirror image:

We calculate w(A)=3, so

( ) ( )( )( )( ) ⁄ ⁄ ⁄

Now we calculate the Bracket Polynomial for B:

And we calculate w(B)=-3, so

( ) ( )( )( )( ) ⁄ ⁄ ⁄

Since the two diagrams have different polynomials, the two knot diagrams must represent different knots, and so the trefoil is distinct from its mirror image. However, the Jones Polynomial isn’t without limits. We have proven that two diagrams of the same knot have the same polynomial, but we haven’t proven two diagrams of different knots cannot have the same polynomial. For example, consider the two knots below (Ex 11.2.5 [4]).

The two knots have the same Jones Polynomial, but have different Alexander Polynomials, which is another knot invariant. Therefore these are two different knots with the same Jones Polynomial.

Consider the figure eight knot pictured below.

The figure eight knot has the same Jones polynomial as its mirror image. However this does not tell us whether or not it is equivalent to its mirror image. To see that they are equivalent, we first rotate the knot, and then move the entire upper strand over the entire knot. Finally we re-organize the crossings.

The following question is open.

Problem 32. Can any nontrivial knot have the same Jones Polynomial as the unknot?

Span of the Bracket Polynomial and Jones Polynomial

Definition 33. The span of a knot polynomial, denoted ( ) for the Bracket Polynomial and ( ( )) for the Jones Polynomial, is defined to be the difference between the smallest and the largest degree of all the terms in the polynomial.

While the Bracket Polynomial is not a knot invariant, its span is, as Reidemeister type 1 moves adjust all terms in the polynomial by , so the difference between the smallest and largest degree remains the same.

Now we note a very important property of the Jones and Bracket Polynomials.

Theorem 34. ( ) ( ), and ( ) ( ( )).

Proof. It is enough to prove this for the Bracket Polynomial, as the Jones Polynomial immediately follows. Let D be the knot diagram of a knot K such that D has minimum crossings c(K).

According to our bracket polynomial rule 3, there are two ways to get delete a crossing: by multiplying by or and joining the strings in a different way. Let X be the number of multiplications by , and Y be the number of multiplications by , defined by the third rule, in a term in the Bracket Polynomial.

Then a particular term for the Bracket Polynomial is of the form ( ) , where ( ), and k is the number of unknots created after deleting all crossings. Let us consider the maximum degree the polynomial has. We claim this is achieved by maximizing ( ) and set . For if we have a deletion of a crossing by increasing Y, then the degree is lowered by 1 for decreasing X. Since the maximum number of unknots that could be added is 1, thus resulting in a 2 more degree increase, the degree of the term overall will not go up. Likewise, for the minimum degree, we minimize ( ) and set . So

( ) ( ( ) ( )) ( ( ) ( )) ( )

Here n is the number of unlinked unknots used in the maximum degree case, and m is the number of unlinked unknots used in the minimum degree case. To simplify, let us consider n and m. An unknot is formed from one or more regions of a knot diagram when deleting crossings. As such, there cannot be a greater number of unknots than total regions. According to Proposition 1, the number of regions is equal to ( ) , so we have ( ) , which implies the following sequence of inequalities.

( ) ( )

( ) ( ) ( )

( ) ( ).

This inequality turns into equality for alternating knots. Consider an alternating diagram with minimum degree. We will create the maximum degree term of the polynomial first by only considering vertical crossing splits found in the Bracket Polynomial rule 3. Start at any crossing, namely the one bolded in the picture below. Since this is an alternating knot, following along the overcrossing, the next crossing must be an undercrossing. Following

along the undercrossing must be an overcrossing. In our maximum degree term, according to the Bracket Polynomial third rule,

Label each point that a region is incident with a crossing, for 4 times per crossing. Label with an “A” if the regions are separate after performing the Bracket Polynomial rule 3, or with a “B” if the regions are joined, as below.

It is easy to see that a region with one A will have only A’s in it, and this region is not joined with any other. This process is reversed for the minimum degree, with the regions with B’s preserved. Below is the figure eight knot divided up for the maximum and minimum degree. The knot outline in the third picture for B that is a region can be considered to be the outside region of the knot.

For the maximum degree term, one set of regions is used, and for the minimum degree term, the other is used. The number of regions used is equal . Since in our case each unknot, formed in the process of obtaining the polynomial, uses exactly one region, we have ( ) , which implies we have an equality instead of an inequality for alternating knots.

Therefore, we have proved the following.

Theorem 35. For an alternating knot K, ( ) ( ) and ( ( )) ( )

Main Conjecture

Composition is a very standard operation on knots. At first glance it appears to operate like the multiplication of natural numbers. Both lack inverses, are commutative, and associative. Also, the definition of prime knots resembles that of prime numbers. Moreover, a knot’s crossing number is a natural way to categorize knots. However one key detail is needed to cement both composition and crossing number. It was conjectured by Peter Tait that a knot’s crossing number is additive with respect to composition.

Conjecture 36. ( ) ( ) ( ).

By inspection of the composition of any two diagrams, we can immediately see that ( ) ( ) ( ), so what is needed is to show the other inequality. This

32 conjecture has been proven for both torus and alternating knots which will be shown in this paper.

Additivity of Composition Operation for Torus Knots

Proposition 5 states ( ) ( )+b(K)+|K|-2. This inequality is an equality for a torus knot. To see this, we consider the standard diagram for a T(p,q) torus knot with p braids, and without loss of generality we let .

To proceed, first we must prove the genus of the torus knot standard diagram is minimal. Since ( ) ( ) , ( ) , and , we have

( ) ( ( ) ( ) ) (( ) ( )) ( )( )

By Murasugi Theorem 7.5.2 [4], this is the genus of the torus knot and not just the knot diagram. Therefore the standard diagram has minimal genus.

Now we consider the standard diagram. Since ( ) and g(D) is minimal for the diagram, we have proved the following.

Theorem 37. For K a (p,q) torus knot, we have ( ) ( ) ( ) ( )

Let A and B be torus knots. We know that ( ) ( ) ( ), ( ) ( ) ( ) , and |A#B|=|A|+|B|-1, so by the above,

( ) ( ) ( )

( ) ( ) ( ) ( )

( ( ) ( ) ) ( ( ) ( ) )

( ) ( )

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Additivity of Composition Operation for Alternating Knots

Previously we showed that the composition of alternating knots is again an alternating knot. In Theorem 35, we also showed that an alternating knot K with c(K) crossings has a Jones Polynomial with span c(K). These two facts are all we need to show the additivity of the crossing number for alternating knots. The additivity follows from the next theorem.

Theorem 38. Any alternating diagram with no splitting points has minimal number of crossings.

Proof. Assume A and B are two knot diagrams for an alternating knot K. Let A be an alternating diagram with n crossings and no splitting points, and let B have less than n crossings. Then span(J(A)) = n (Theorem 35), and span(J(B)) < n (Theorem 34). But A and B are diagrams of the same knot, hence should have the same Jones Polynomial, a contradiction. Hence, K has crossings equal to n, and A is a diagram with minimum crossings.

So given two alternating knots, their composition is an alternating knot, thus in lowest form, and so its crossing number is the sum of the two.

References [1] Gruber, Hermann. : “Estimates for the Minimal Crossing Number”, 2003

[2]Lackenby, Marc. : “The Crossing Number of Composite Knots”, 2009

[3] Monturov, Vassily. : “Knot Theory”, 2004, 1-15

[4] Murasugi, Kunio. : “Knot Theory and its Applications”, 1996, 132-148, 217-241

[5] Schultens, Jennifer. : “Bridge Numbers of torus knots”, 2005

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[6] Yamada, Shuji. : “The minimal number of Seifert Circles equals the braid index of a link”, 1987