An elementary proof of the Traczyk-Yokota criteria for periodic knots

Przytycki, Józef H.

doi:10.1090/S0002-9939-1995-1257121-1

An elementary proof of the Traczyk-Yokota criteria for periodic knots
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by Józef H. Przytycki

Proc. Amer. Math. Soc. 123 (1995), 1607-1611

DOI: https://doi.org/10.1090/S0002-9939-1995-1257121-1

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Abstract:

Traczyk used the first coefficient of the skein (Homfly) polynomial to find powerful criteria for r periodic knots. The criteria were extended by Yokota to $(r - 1)/2$ first coefficients of the skein polynomial. We give here a short, elementary proof of the Traczyk-Yokota criteria. The main tool is the Jaeger composition product, the same product which is a base for Turaev’s Hopf algebra structure of links in a handlebody.

References

François Jaeger, Composition products and models for the homfly polynomial, Enseign. Math. (2) 35 (1989), no. 3-4, 323–361 (English, with French summary). MR 1039950
V. F. R. Jones, Hecke algebra representations of braid groups and link polynomials, Ann. of Math. (2) 126 (1987), no. 2, 335–388. MR 908150, DOI 10.2307/1971403
Kunio Murasugi, Jones polynomials of periodic links, Pacific J. Math. 131 (1988), no. 2, 319–329. MR 922222, DOI 10.2140/pjm.1988.131.319
PawełTraczyk, Periodic knots and the skein polynomial, Invent. Math. 106 (1991), no. 1, 73–84. MR 1123374, DOI 10.1007/BF01243905
Yoshiyuki Yokota, The skein polynomial of periodic knots, Math. Ann. 291 (1991), no. 2, 281–291. MR 1129368, DOI 10.1007/BF01445208