The Tutte polynomial of a morphism of matroids I. Set-pointed matroids and matroid perspectives

Abstract

We study the basic algebraic properties of a 3-variable Tutte polynomial the author has associated with a morphism of matroids, more precisely with a matroid strong map, or matroid perspective in the present paper, or, equivalently by the Factorization Theorem, with a matroid together with a distinguished subset of elements. Most algebraic properties of the usual 2-variable Tutte polynomial of a matroid generalize to the 3-variable polynomial. Among specific properties we show that the 3-variable Tutte polynomial of a matroid <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>M</mml:mi> </mml:math> pointed by a normal subset can be used to abridge the computation of the 2-variable Tutte polynomial of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>M</mml:mi> </mml:math> , and that the 3-variable Tutte polynomial of a matroid perspective <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>M</mml:mi> <mml:mo>→</mml:mo> <mml:msup> <mml:mi>M</mml:mi> <mml:mo>′</mml:mo> </mml:msup> </mml:mrow> </mml:math> is computationally equivalent to the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>r</mml:mi> <mml:mo>(</mml:mo> <mml:mi>M</mml:mi> <mml:mo>)</mml:mo> <mml:mo>-</mml:mo> <mml:mi>r</mml:mi> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>M</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:mo>)</mml:mo> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> two-variable Tutte polynomials of the matroids of its Higgs factorization.