Development of polynomial invariants of knots and links
Introduction
The purpose of this chapter is to give a brief outline of the development of polynomial invariants of knots and links. We will also discuss a possible next step in the combinatorial study of the knot problem. For a survey of the knot problem we recommend the articles by Fox [F4], MeA Gordon [M2], Thistlethwaite [T2] and Rolfsen [R2].
Bibliographies in those articles are fairly complete up to the current development of knot theory with the exception of the newly discovered polynomials.
Preliminaries
For arbitrarily small perturbations, the image K contains no triple points, which means that no three points of K are mapped to a single point.
Knot table
This is essentially the method Tait and Little use when constructing their knot tables. However, their tables contain only major nodes of up to 10 crossings and alternating major nodes of 11 crossings.
Alexander polynomials and knot groups
If the intersection is met more than once, Aii would be the sum of the symbols. This polynomial differentiates all nodes up to 8 crossings and all but 6 pairs up to 9 crossings. Until the advent of the HOMFLY polynomial, it remained one of the most powerful knot resolution tools.
In his paper, Alexander showed that his polynomial can also be obtained from the fundamental group of the knot complement, which is always referred to as the knot group.
Fox's algebra
Let n be the number of intersections of D, and r the number of permutations of the process. Otherwise, let v be an intersection and K', L1 and L2 the link types of the switched and the two split diagrams respectively. Let (n, r) denote the complexity of A where r is the number of links of the decoupling process.
Let S be the longest unbroken arc of D, and remove D as in the proof of the previous theorem. Unlike the Conway polynomial, the degree of the H-polynomial bounded above by 1 is less than the crossing number. Thus we can get a lower bound for the crossing number if we know the polynomial.
Conway polynomials and tangles
New polynomials
The real power of the new polynomial is demonstrated by its ability to distinguish the left-hand piano from the right-hand piano, which has the same node group. In fact, if the link has c components, in general there are zc -1 different polynomials for the zc possible orientations of the link. Although some geometric invariants such as the Arf invariants and the signature of the polynomial can be obtained, the geometric nature of the invariants remains to be studied.
The degrees of the new polynomials are much larger than those of the corresponding Conway polynomial.
Computability of the polynomials
Next step towards the knot problem. 10
Although Torres used the link group to define his polynomial, our definition is combinatorial and simpler.
Definition of Alexander polynomials
In fact, we believe that our polynomial is the same as Torres' Alexander polynomial. Let L1 be an oriented link with more than one component, and let L2 be the link obtained by changing the orientation of the ith component of L1. In A2, for the rows corresponding to those intersections, divide by ti and let A' be the resulting matrix.
So, if L1 and L2 are equivalent to a link with reversed orientation, then they have the same reduced Alexander polynomial.
Proof of Theorem 2.2.1
Starting from the first component, cross the i component from the starting point along the orientation direction. This completes the proof of the existence and uniqueness of the polynomial F for any type of connection. Let K be a diagram with complexity ( n, r ) and a switch intersection v that minimizes the complexity of switch and split diagrams.
Alternatively, let G be a disk in 3-space spanned by C' such that G does not intersect the rest of Li's diagram.
Torres' conditions
Conway polynomials
Definition of Conway polynomials
Denote the diagram D in Axiom 2 as the distribution of the link and L as its link type. Order the complexities lexicographically; we can, by induction of the complexity, calculate the polynomial of a link L as follows. Suppose K' is a link obtained by a cross-link of a diagram of K, and L is the division of the link.
The relationship is a combination of the first and second coefficients of all its sublinks. However, the following definition results in a polynomial invariant for nodes and links that is indifferent to the orientation of the link. Next, we show that H(D) is independent of the choice of starting point on Ci, the i-th component.
We have the following charts. fig. 4.3) and equations, and find that H(D) is independent of the motion. In addition to the degree of the polynomial, we would like to examine the leading coefficient of the polynomial. For if so, let G be a disc spanned by C in R3 such that G does not intersect with the rest of the link.
Therefore, the cyclothon of the connection is defined up to permutations of words, turns and cyclical permutations of words. Since we are only interested in the degree of the polynomial in this chapter, we can assume that L has no non-intersecting non-knotted component. A cycloton is loopless if {vv-1} does not appear in any word of the cycloton.
The general case is more complicated, because the degree of the polynomial does not reflect much information.
A new polynomial invariant for knots and links
Existence of the invariant polynomial
Let n be the number of transitions of D and R the number of switches required to break the connection D. If for some choice of order of components and starting points r equals 0, Dis is descending. First, we define a polynomial function on the diagrams with a certain choice of order of components, direction and origin.
To show F is indeed a link invariant, we need to verify that it is independent of the choice of component order, starting points and directions, and diagrams of the same link type. We will also show that the polynomials for trivial decouplings (link type of descending diagrams) correspond to equation (4) of section 1. Since both polynomials are independent of the link order and links required for both choices of starting points are the same, H(D ) = G(D).
Intersection v will be toggled for one choice of starting point and not the other. Without loss, let E be the diagram with starting point b, H(E) a polynomial from E, and E' the diagram just before the change of vis. Let x and y be the orthoimages of v on the component C, S the segment on C connecting x and y by crossing along the given direction, and T the remaining segment.
Since any two diagrams of a connection are n-equivalent for some n, we can verify that F is an invariant of the connection type, up to the choice of component ordering and directions. Since descending diagrams take polynomials that are independent of the order and directions of the components, a simple induction argument on the complexity of a diagram will conclude that the polynomial of any diagram is independent of the order and directions of the components.
Basic properties of the polynomial
Use induction on the complexity pair (n, r) and consider the unknot and the trivial unlinks as the base cases. In both cases, the result follows from the base case when the number of crossings is 0. In addition to the ten properties listed above, there are nice features relating the polynomial and some geometric and algebraic properties of the link.
Although the proof is elementary, I cannot find a proof that does not use a clamped connection surface.
Arf invariant
If m is the number of circuits, since the resulting connection is a separated union of N and M, according to Theorem 4.2.3.
Polynomials of tangles
Select a starting point for each component, except for two strings; the starting points will always be a and c or b depending on whether a and c are on the same string (fig. 4.8).
Degree of the polynomial
If K is a disjoint union of knots, then the leading coefficient is obviously (-1)c-l, where c is the number of components of K. If C is a diagram of a disjoint disjoint component in D, then C cannot have any overpass intersections. This means that C does not have any self-transition and lies below all other components.
If C' does not include the two arcs at v, then it is also a separate, unknotted component in K, so C' does not intersect D. Connect the two preimages of v in K by a vertical line segment; G is also a non-intersecting disk in K (Fig. 4.10). A non-intersecting unknotted component, a component that has no intersection with itself or with other components, gets a null word.
Introducing a non-intersecting unknotted component will increase the degree of H(L) by one, and removing such a component decreases the degree by one. If S is alternating and loop-free and has no sub-cyclone, then deg(H(S)) = n- 1 where n is the number of crossings of L. But in Section 5.6 both L1 and L2 have the same number of non- intersecting unknotted components and thus their conducting coefficients are of the same sign.
It is not true that any alternative link reaches the maximum degree, because a link composed of two alternative links does not have the maximum degree.
Alternating Knots and Links
Degree of polynomials of Alternating diagrams
If L is alternating, the words in its cyclotone alternate in symbols and their inverses. Therefore we must have Y = PIQ!I, where PI and QI are the suffixes of P and Q respectively; or the prefixes are P and Q. Then Rp1 ~ T1, T2 and Rp2• Choose T1 from all possible subcyclotons of S1 containing P 1 and no other symbol from P, with Q 1 having the shortest length.
Choose T1 among all subcyclotons of S 1 containing Q 1 but no other symbol of Q with P1 having the shortest length. So some symbol in Q1 or words in Rq1 has its inverse in P1 and thus in P2. If there is a simple closed curve in the diagram that intersects the connection at two points and the segments inside and outside are knotted, then the connection has subcyclotons, namely cyclotons corresponding to the connection factors.
An alternating diagram with maximum degree is minimal, since any diagram with a lower number of intersections has a smaller degree. Since it is an open problem whether the intersection numbers are additive, in the case of alternate and simple connections, when we connect two such connections, the best we can do is to reduce one intersection to many of the given connections.
Remarks. 76
The first coefficient is the constant term for a node and is that of x-1 if it is a 2-component connection. Conway, An enumeration of knots and links, and some of their algebraic properties, Computational Problems in Abstract Algebra, 329-358. Fox, A Quick Trip Through Knot Theory, The Topology of 3-Manifolds and Related Topics, Prentice-Hall, Englewood Cliffs, New Jersey.
