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In my research, I've been playing around with a matrix representation of knots from which various knot invariants are calculable, and I have noticed several families of knots which I don't believe have been named. I believe that I can compute closed formulae for the Jones polynomials of each of these families. With this family in particular, however, because the knots are both visually related and also have other simple relationships (for example, their minimum braid representations are $$\left(\sigma_1^{(n-5)}B_{4_1}\sigma_2^{-1}\right)^{(-1)^\frac{n-2}{2}}$$ where $B_{4_1}$ is the minimum braid representation of the figure-eight knot and $n$ is the crossing number), I would be unsurprised if it already had a name and a closed formula for the Jones polynomial had already been published; however, I haven't the faintest idea what to search to confirm or deny this suspicion.

More generally, what are some resources that enumerate families of knots with closed formulae for their Jones polynomials, so that as I identify more and more families, I might be able to check on whether they've been identified before?

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    $\begingroup$ I doubt that there is a name for the family you describe here. But they seem to be a variation of Twist knots. $\endgroup$
    – N. Owad
    Apr 24, 2018 at 3:06

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I'll agree with Nick in the comments and say that I doubt there is a special name for this family. They are all examples of alternating 3-braids. The Jones polynomial for all 3-braids has been computed in the following paper:

  • Takhashi, Moto-o. Explicit formulas for the Jones polynomials of closed 3-braids. Comment. Math. Univ. St. Paul. 38 (1989), no. 2, 129–167.

The formulas in the above paper are unwieldy, but the set of closed 3-braids contains the knots in your family as a subset. In general, it can be hard to determine which families of knots/links have published formula. Devoted internet searching is my go-to method.

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