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The Kazhdan-Lusztig polynomial of a matroid was introduced by Elias,Proudfoot and Wakefield, whose properties need to be further explored. In thispaper we prove that the Kazhdan-Lusztig polynomials of fan matroids coincidewith Motzkin polynomials, which was recently conjectured by Gedeon. As abyproduct, we determine the Kazhdan-Lusztig polynomials of graphic matroids ofsquares of paths. We further obtain explicit formulas of the Kazhdan-Lusztigpolynomials of wheel matroids and whirl matroids. We prove the real-rootednessof the Kazhdan-Lusztig polynomials of these matroids, which provides positiveevidence for a conjecture due to Gedeon, Proudfoot and Young. Based on theresults on the Kazhdan-Lusztig polynomials, we also determine the$Z$-polynomials of fan matroids, wheel matroids and whirl matroids, and provetheir real-rootedness, which provides further evidence in support of aconjecture of Proudfoot, Xu, and Young.

Kazhdan-Lusztig polynomials of fan matroids, wheel matroids and whirl matroids