Minkowski Units

For every prime p dividing the determinant of a knot K, there is an invariant called the Minkowski unit of K, denoted c_p(K), which is in the multiplicative group formed by {-1, 1}. It is not an additive invariant: that is, it is not always the case that c_p(K # J) = c_p(K)c_p(J).

These invariants were defined by Goeritz [3], based on Minkowski's work on quadratic forms. The fact that they can be used to define knot slicing obstructions was developed by Andrews [1], Andrew-Dristy [2], and Murasugi [5]. An expository account is given in [4].

We will define these invariants in terms of the Legendre symbol: if p is an odd prime and x is relatively prime to p, then the Legendre symbol is $\left(\frac{x}{p}\right)$ is +1 or -1 depending on whether x is a square or not modulo p. The Minkowski symbol of a nonsingular symmetric rational matrix such as the Goeritz matrix can be defined algorithmically as follows.

1. Use a Gauss-Jordan style simultaneous row/column process to convert a Goeritz matrix into a diagonal matrix with square free integer entries.

2. Write the diagonal entries as (a_1, ... , a_r, pb_1, ... , pb_s), where the a_i and b_i are prime to p.

3. For an odd prime p, c_p(G) is defined by

Notation

The notation in the table is illustrated with the example of the knot K = 10_122, in which the value of the Minkowski units is given as: [[3, [1, -1]], [5, [1, 1]], [7, [1, -1]]]. This means that c_3(K) = 1, c_3(-K) = -1; c_5(K) =1; c_5(-K) = 1; c_7(K) = 1, and c_7(-K) = -1.

Notice that for K above, c_3(K) ≠ c_3(K). Thus, K is not amphicheiral (c_p cannot detect reversiblity). This is also detected by c_7, but not by c_5. Minkowski units were among the first invariants that could detect chirality of knots. In his book, Reidemeister [6] included a list of Minkowski units of prime knots through 9 crossings. (Two errors were corrected by Shinohara [7].)

Privacy Notice