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This work presents formulas for the Kauffman bracket and Jones polynomials of3-bridge knots using the structure of Chebyshev knots and their billiard tablediagrams. In particular, these give far fewer terms than in the Skein relationexpansion. The subject is introduced by considering the easier case of 2-bridgeknots, where some geometric interpretation is provided, as well, viacombinatorial tiling problems.

The Jones polynomials of 3-bridge knots via Chebyshev knots and billiard table diagrams