A Matrix Knot Invariant

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PROC. N. A. S.

THEOREM I. If the condition (2) is satisfied and a. )- 0, then the series (1) uill converge to any arbitrary complex number in an everywhere dense set of points on the circle of convergence. 2. It seems plausible that the following associated theorem is also true, but so far neither Professor Zygmund nor I have succeeded in proving it. THEOREM II. If the condition (2) is satisfied, then the series (1) will converge to any arbitrary complex number in at least one interior point of the circle of convergence. 3. We observe in passing that Theorem I assures us that the function (1) assumes inside the circle values arbitrarily near to any assigned number, so that the set of exceptional values is at any rate nowhere dense. 1 INTERNATIONAL RESEARCH FELLOW and Research Fellow, Harvard University.


Communicated January 11, 1933

Let K be any smooth, closed curve in a Euclidean 3-space R. Then the curve K determines an infinitely sheeted covering space R. spread over the space R and such that as we go around the curve K we traverse the sheets of R,, in cyclical order. The topological invariants of the space R. are, obviously, functions of the knot type of the curve K. We shall fix our attention, more particularly, on the group G formed by the one-cycles of R. that are independent with respect to homologies. Now, the homology group G cannot, in general, be generated by a finite nujmber of cycles, in the restricted sense of the word "generate." Let us, however, introduce an operator x and form the ring 2 consisting

of all finize sums of the form Eaixi,

(i=O, |


where the coefficients as are arbitrary integers. Then, it can be shown that there is a finite set of one-cycles,


l ..(i =1,2,...s)

of the covenng space R. such that every cycle C of R. is homologous to a linear combination of the cycles Ci,

VOL. 19, 1933



with coefficients as in the ring S. Moreover, there is always a finite set of homologies,


aij C, =E

such that every homology among the


Ci's is of the form



where the coefficients ,S are also in the ring S.' The geometrical significance of the operator x is as follows. If C is any curve of the covering 3, . . .), all lie directly 1, 2, space R.. the curves x'C, (i = 0, "'over"y or "under" C but in different sheets of R,.. A fundamental set of homologies (1) can be read off, by inspection, from the diagram of the knot K. It turns out that the coefficients aij are linear =



in a-,

(aij, bij integers),

aij = aij x + bsi,

and that the matrix of the coefficients aij is a square matrix, such that its determinant reduces to unity when we write x = 1. We shall denote the matrix of the aij's by (2) L(x) = M + xN, where M and N are square matrices with integer elements. It can be shown that if the knot K is transformed into a knot K' isotopic to K then the matrix L(x) goes over into a matrix L'(x) e-equivalent to L(x).2 However, necessary and sufficient conditions for the e-equivalence of two matrices are not known, so that we are unable, as yet, to make the fullest possible use of the matrix L(x). We propose, in this note, to translate the problem of the e-equivalence of two knot matrices into a more manageable form. Suppose the determinant of N in (2) is equal to zero. Then, by elementary transformations on the rows of L(x) we can reduce L(x) to such a form that all the elements in the last row of N are zero. Moreover, by elementary transformations on the columns of L(x) we can further reduce L(x) until all the elements in the last row of M are zero, with the exception of the very last one of all which must reduce to ' 1, since the determinant of L(1) is unity. The last row of L(x) will now represent an homology =



in terms of a new set of generators C'i. We may, therefore, strike out the last row and column of L(x), since this is equivalent to suppressing the redundant generator C,s A similar reduction may be carried out if



PROC. N. A. S.

the determinant of M is equal to zero, as we shall, in this case, be led to an homology




whence, again, the generator C" corresponding to the last column of the transformed matrix L(x) will be redundant. We may, thus, keep on reducing the order of the matrix L(x) until we reach the point where the determinants of M and N are both different from zero. The determinant of L(1) will, obviously, be 1. We shall make it +1 by changing the signs of one row, if necessary. Next, let us make the substitution X x - 1 in the right-hand member of (2). The matrix L(x) will then go over into




where J is a matrix of determinant unity. By elementary transformations on the rows of the matrix (3) we shall reduce this matrix to the form


1 + A,

where 1 now represents the unit matrix. It is easy to prove that if K and K' are two isotopic knots then the matrices A and A' corresponding to K and K', respectively, are related in the following manner:





where P is a square matrix with integer elements and of determinant 1. In other words, every invariant of A under the group of all transformations of type (5) must be a knot invariant. Suppose, for example, we form an arbitrary polynomial B in the matrix A, =

B = ao + aiA + a2A2 + ...+ aAl, where the coefficients ai represent diagonal matrices with integer elements ai along their main diagonals. Then the elementary divisors of B will all be invariants of the knot K. For the special case where we write B



l) "




2, 3, 4,



these elementary divisors will reduce to the coefficients of torsion of the n-sheeted covering space R. with the knot as branch curve of order n - 1. The characteristic polynomial (p(X) of A is, obviously, the "x-polynomial" of the knot, except for the change of parameter X = x - 1. Many other invariants of the matrix A suggest themselves, but the arithmetical problem of finding a complete set of invariants of A under all transformations of the group (5) appears to be as yet unsolved. We

VOL. 19, 1933



shall investigate the properties of the matrix A in greater detail in a paper which will appear shortly in the Annals of Mathematics. 1"Topological Invariants of Knots and Links," Trans. Amer. Math. Soc., 30, 275306 (1928). See also Reidemeister's Knotentheorie, Springer, 1932. 2 Loc. cit. 3 Loc. cit.