The skein polynomial of freely periodic knots

“…The Tutte polynomial is also related to certain specialization of the HOMFLYPT polynomial as it was proved in [4]. The latter is known to be a good witness of link symmetries, see [1,2,7,11,12,14]. The purpose of this paper is to investigate whether the way the Tutte polynomial interacts with graph symmetries extends to other graph polynomials.…”

Section: Figure 1 a 3-periodic Graph (Left) And Its Quotient Graph (mentioning

confidence: 92%

Graph polynomials and symmetries

Chbili

2019

J. Algebra Appl.

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In a recent paper, we studied the interaction between the automorphism group of a graph and its Tutte polynomial. More precisely, we proved that certain symmetries of graphs are clearly reflected by their Tutte polynomials. The purpose of this paper is to extend this study to other graph polynomials. In particular, we prove that if a graph G has a symmetry of prime order p, then its characteristic polynomial, with coefficients in the finite filed Fp, is determined by the characteristic polynomial of its quotient graph G. Similar results are also proved for some generalization of the Tutte polynomial.

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“…Also note that Chbili in [7] obtains criteria for free periodicity in terms of the HOM-FLYPT polynomial. These criteria are then applied to examples in [16] for which free periods could not be eliminated using Theorem 21 alone.…”

Section: Freely Periodic Knotsmentioning

confidence: 99%

“…These criteria are then applied to examples in [16] for which free periods could not be eliminated using Theorem 21 alone. Free periods for K D 10 62 are not ruled out in [7].…”

Section: Freely Periodic Knotsmentioning

confidence: 99%

Twisted Alexander polynomials of periodic knots

Hillman

Livingston

Naik

2006

Algebr. Geom. Topol.

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Murasugi discovered two criteria that must be satisfied by the Alexander polynomial of a periodic knot. We generalize these to the case of twisted Alexander polynomials. Examples demonstrate the application of these new criteria, including to knots with trivial Alexander polynomial, such as the two polynomial 1 knots with 11 crossings.Hartley found a restrictive condition satisfied by the Alexander polynomial of any freely periodic knot. We generalize this result to the twisted Alexander polynomial and illustrate the applicability of this extension in cases in which Hartley's criterion does not apply. 57M25, 57M27

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The skein polynomial of freely periodic knots

Cited by 9 publications

References 7 publications

A new criterion for knots with free periods

A new criterion for knots with free periods

Graph polynomials and symmetries

Twisted Alexander polynomials of periodic knots

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