Journal of Knot Theory and Its Ramifications
c World Scientific Publishing Company
°
THE RANK OF THE TRIP MATRIX
OF A POSITIVE KNOT DIAGRAM
LOUIS ZULLI
Union College
Department of Mathematics
Schenectady, NY 12308-3166
[email protected]
ABSTRACT
In this note we show that the rank of the trip matrix of a positive knot diagram is
exactly twice the genus of the associated positive knot. From this, we give a quick proof
of the following result of Murasugi: The term of lowest degree in the Jones polynomial
of a positive knot is 1 · tg , where g is the genus of the knot.
Keywords: trip matrix, positive knot, genus, Jones polynomial
1. Introduction
The trip matrix of an unoriented knot diagram was introduced in [1], where an
algorithm was presented for computing the Jones polynomial of a knot from the
trip matrix of any diagram for the knot. We begin by recalling the definition of
the trip matrix. Given an unoriented knot diagram K, enumerate the crossings
1, 2, . . . , n in any manner whatsoever. At each crossing i, place an arrow (i.e. a
local orientation) on the overcrossing strand at crossing i. There are two choices at
each crossing for the direction of this arrow; you may choose either. In particular,
no global consistency of these local orientations is required. Let i+ denote the
+
overcrossing arrow at crossing i, and let i+
+ denote the vertex of i represented by
the arrow-head. At each crossing i, place an arrow on the undercrossing strand
so that the crossing sign at i is +1. (The crossing sign of an oriented crossing is
defined as follows: Imagine placing the palm of the right hand upon the overcrossing
strand at the crossing, with the fingers pointing in the direction of the arrow. If
the thumb of that hand points in the direction of the arrow on the undercrossing
strand, then the crossing sign is +1. Otherwise the crossing sign is −1.) Let i −
−
denote the undercrossing arrow at crossing i, and let i−
− denote the tail of i . See
Figure 1.
From this adorned knot diagram, define an n×n matrix T over Z2 as follows: Tij
is defined to be the number of times (mod 2) that a traveler passes through crossing
j while making the following trip—the traveler begins at the vertex i+
+ , and proceeds
along the knot diagram in the direction of the arrow i+ until he reaches the vertex
i−
− . The matrix T so constructed is called the (mod 2) trip matrix of the adorned
diagram. One may check that this matrix is symmetric, and that it doesn’t depend
on which directions are chosen for the overcrossing strands in the process of adorning
the diagram.
i ++
i
i_ _
Figure 1: An adorned crossing
2. Results
Suppose now that K is a positive knot diagram, meaning that, with respect to
either global orientation of K, all the induced crossing signs are positive. (In this
case, it is easy to check that each element on the diagonal of the trip matrix is zero.)
Then
Theorem. The rank (over Z2 ) of the trip matrix T is twice the genus of the
positive knot represented by the diagram K.
Proof. Let us use the terminology diagrammatic genus, and the notation g K ,
for the genus of the Seifert surface for K produced by Seifert’s algorithm. For a
positive knot diagram K, it is a result of Murasugi that the diagrammatic genus
gK equals the genus g of the associated positive knot. (See Prop. 4.1. in [2].) That
is, for positive knot diagrams, Seifert’s algorithm yields a minimal genus Seifert
surface for the associated positive knot. Thus, it suffices to restrict our attention to
the diagrammatic genus of K. By a simple Euler characteristic computation, one
obtains the formula 2gK = n − s + 1, where n is the number of crossings in the
diagram K and s is the number of Seifert circles produced when Seifert’s algorithm
is applied to K. But for a positive diagram, the Seifert circles are precisely the
circles produced when the diagram is split open according to the state AA . . . A.
(See [1] or [3] for an explanation of this terminology.) By the main result in [1], the
number of such circles is precisely 1 + nullity(T ). Thus s = 1 + n − rank(T ), so
2gK = n − s + 1 = n − (1 + n − rank(T )) + 1 = rank(T ) and the result is proved.
From this theorem, we obtain a quick proof of the following result of Murasugi
(see Thm. C in [2]).
Corollary. The term of lowest degree in the Jones polynomial of a positive
knot K is 1 · tg , where g is the genus of K.
Proof. Using the notation in [1], let fK (A) denote the Laurent polynomial
knot invariant obtained from Kauffman’s bracket polynomial < K >. Since no
Seifert circle is self-approaching, the term of highest degree in fK (A) is the term of
highest degree produced by the state AA . . . A. This state contributes
An (−A2 − A−2 )n−rank(T ) (−A−3 )w(K)
to fK (A), where w(K) denotes the writhe of K. Since K is a positive diagram with
n crossings, w(K) = n. Thus the term of highest degree in fK (A) is
An (−1)n−rank(T ) A2(n−rank(T )) (−1)n A−3n
= (−1)2n−rank(T ) A−2rank(T )
= (−1)2n−2g A−4g
= 1 · A−4g .
Since the Jones polynomial is produced from fK (A) by substituting A = t−1/4 , the
term of lowest degree in the Jones polynomial of K is 1 · tg .
Remark: Suppose K is any knot diagram, not necessarily a positive one. Let
+
K denote the positification of K, the diagram obtained from K by changing any
negative crossings in K to positive crossings. Since the genus of the Seifert surface
produced by Seifert’s algorithm is independent of the crossing signs in a diagram,
we have proved gK = gK + = 21 rank(T + ), where T + is the trip matrix of K + . (T +
is the positification of the trip matrix T for K—it is T with each entry on the main
diagonal replaced by zero.) Thus we have given an upper bound for the genus of
the knot represented by K, and this upper bound is sharp if and only if Seifert’s
algorithm yields a minimal genus Seifert surface for the knot. In particular, the
genus of an alternating knot is half the rank of the positified trip matrix of any
alternating diagram for the knot.
References
[1] L. Zulli, A matrix for computing the Jones polynomial of a knot, Topology 34 (1995)
717–729.
[2] K. Murasugi, Jones polynomials of alternating links, Trans. Amer. Math. Soc. 295
(1986). 147-174.
[3] L. H. Kauffman, State models and the Jones polynomial, Topology 26 (1987) 395-407.