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The -Turk's head link is presented by the alternating diagram which is obtained from the standard diagram of the -torus link by crossing changes. In this paper, we show that for any integers and , the coefficients of in the Conway polynomials of the - and -Turk's head links coincide for and differ by sign for . We conjecture that this property holds for any . [ABSTRACT FROM AUTHOR]

For any knot K represented as an n-braid (n ≥ 3), we construct an infinite sequence of pairwise non-conjugate (n + 1)-braids {bm, m ∈ ℕ} representing K. [ABSTRACT FROM AUTHOR]

We define a measure of "complexity" of a braid which is natural with respect to both an algebraic and a geometric point of view. Algebraically, we modify the standard notion of the length of a braid by introducing generators Δij, which are Garside-like half-twists involving strings i through j, and by counting powered generators Δkij as log(ǀkǀ + 1) instead of simply ǀkǀ. The geometrical complexity is some natural measure of the amount of distortion of the n times punctured disk caused by a homeomorphism. Our main result is that the two notions of complexity are comparable. This gives rise to a new combinatorial model for the Teichmüller space of an n + 1 times punctured sphere. We also show how to recover a braid from its curve diagram in polynomial time. The key role in the proofs is played by a technique introduced by Agol, Hass, and Thurston. [ABSTRACT FROM AUTHOR]

We review the q-deformed spin network approach to Topological Quantum Field Theory and apply these methods to produce unitary representations of the braid groups that are dense in the unitary groups. Our methods are rooted in the bracket state sum model for the Jones polynomial. We give our results for a large class of representations based on values for the bracket polynomial that are roots of unity. We make a separate and self-contained study of the quantum universal Fibonacci model in this framework. We apply our results to give quantum algorithms for the computation of the colored Jones polynomials for knots and links, and the Witten–Reshetikhin–Turaev invariant of three manifolds. [ABSTRACT FROM AUTHOR]

We show, using Mellit's recent results, that Kálmán's full twist formula for the HOMFLY polynomial can be generalized to a formula for superpolynomials in the case of positive toric braids. [ABSTRACT FROM AUTHOR]