Documentation

TauCeti.KnotTheory.Jimbo.ReducedTrace

The reduced Jimbo trace #

The writhe-normalized Jimbo trace assigns the quantum dimension to the unknot. Over a field, dividing by that dimension gives the reduced sl_N specialization of HOMFLY, with unknot value one whenever the quantum dimension is nonzero. It retains Markov invariance and the skein relation with a = q ^ N and z = q - q⁻¹.

The nonvanishing condition is used explicitly for the unlink and trefoil values. The quotient is defined using total field division; at a zero quantum dimension it is zero and is not an unknot-normalized invariant. No claim of Laurent polynomial integrality or of a two-variable HOMFLY construction is made here.

References #

The construction uses MarkovBraid.jimboTrace, and the calculations use its skein relation and jimboWeight_sum.

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noncomputable def TauCeti.MarkovBraid.jimboReducedTrace {K : Type u_1} [Field K] {N : ℕ} (β : MarkovBraid) (q : Kˣ) :
K

Divide the writhe-normalized Jimbo trace by its unknot value. This gives the reduced sl_N specialization when the quantum dimension is nonzero.

Instances For
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    theorem TauCeti.MarkovBraid.jimboReducedTrace_def {K : Type u_1} [Field K] {N : ℕ} (β : MarkovBraid) (q : Kˣ) :
    β.jimboReducedTrace q = β.jimboTrace q / ∑ a : Fin N, KnotTheory.jimboWeight q a

    The reduced trace is the normalized trace divided by the quantum dimension.

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    @[simp]
    theorem TauCeti.MarkovBraid.jimboReducedTrace_conj {K : Type u_1} [Field K] {N n : ℕ} (q : Kˣ) (b c : BraidGroup (n + 1)) :
    { predStrands := n, braid := c * b * c⁻¹ }.jimboReducedTrace q = { predStrands := n, braid := b }.jimboReducedTrace q

    Conjugation preserves the reduced trace.

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    @[simp]
    theorem TauCeti.MarkovBraid.jimboReducedTrace_stabilize {K : Type u_1} [Field K] {N n : ℕ} (q : Kˣ) (b : BraidGroup (n + 1)) :
    { predStrands := n + 1, braid := BraidGroup.strandIncl b * BraidGroup.sigma (Fin.last n) }.jimboReducedTrace q = { predStrands := n, braid := b }.jimboReducedTrace q

    Positive Markov stabilization preserves the reduced trace.

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    @[simp]
    theorem TauCeti.MarkovBraid.jimboReducedTrace_stabilizeInv {K : Type u_1} [Field K] {N n : ℕ} (q : Kˣ) (b : BraidGroup (n + 1)) :
    { predStrands := n + 1, braid := BraidGroup.strandIncl b * (BraidGroup.sigma (Fin.last n))⁻¹ }.jimboReducedTrace q = { predStrands := n, braid := b }.jimboReducedTrace q

    Negative Markov stabilization preserves the reduced trace.

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    @[simp]
    theorem TauCeti.MarkovBraid.jimboReducedTrace_mul_sum {K : Type u_1} [Field K] {N : ℕ} (β : MarkovBraid) (q : Kˣ) (hδ : ∑ a : Fin N, KnotTheory.jimboWeight q a ≠ 0) :
    β.jimboReducedTrace q * ∑ a : Fin N, KnotTheory.jimboWeight q a = β.jimboTrace q

    Multiplying the reduced trace by the nonzero quantum dimension recovers the writhe-normalized trace.

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    @[simp]
    theorem TauCeti.MarkovBraid.jimboReducedTrace_one {K : Type u_1} [Field K] {N n : ℕ} (q : Kˣ) (hδ : ∑ a : Fin N, KnotTheory.jimboWeight q a ≠ 0) :
    { predStrands := n, braid := 1 }.jimboReducedTrace q = (∑ a : Fin N, KnotTheory.jimboWeight q a) ^ n

    The (n + 1)-component unlink has reduced trace equal to the n-th power of the quantum dimension. In particular the one-strand unknot has value one.

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    theorem TauCeti.MarkovBraid.jimboReducedTrace_skein {K : Type u_1} [Field K] {N n : ℕ} (q : Kˣ) (b c : BraidGroup (n + 1)) (i : Fin n) :
    ↑(q ^ N) * { predStrands := n, braid := b * BraidGroup.sigma i * c }.jimboReducedTrace q - ↑(q⁻¹ ^ N) * { predStrands := n, braid := b * (BraidGroup.sigma i)⁻¹ * c }.jimboReducedTrace q = (↑q - ↑q⁻¹) * { predStrands := n, braid := b * c }.jimboReducedTrace q

    The reduced trace satisfies the oriented HOMFLY skein relation in any braid context.

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    theorem TauCeti.MarkovEquiv.jimboReducedTrace_eq {K : Type u_1} [Field K] {N : ℕ} {β γ : MarkovBraid} (h : MarkovEquiv β γ) (q : Kˣ) :
    β.jimboReducedTrace q = γ.jimboReducedTrace q

    Markov-equivalent braids have the same reduced Jimbo trace.

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    theorem TauCeti.MarkovBraid.jimboReducedTrace_trefoil {K : Type u_1} [Field K] {N : ℕ} (q : Kˣ) (hδ : ∑ a : Fin N, KnotTheory.jimboWeight q a ≠ 0) :
    { predStrands := 1, braid := BraidGroup.sigma 0 ^ 3 }.jimboReducedTrace q = ↑(q⁻¹ ^ N) ^ 2 * ((↑q - ↑q⁻¹) ^ 2 + 2) - ↑(q⁻¹ ^ N) ^ 4

    The closure of the positive two-strand braid σ₀ ^ 3 has the reduced trefoil value a⁻² (z² + 2) - a⁻⁴, where a = q ^ N and z = q - q⁻¹.

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    theorem TauCeti.MarkovBraid.jimboReducedTrace_trefoil_two {K : Type u_1} [Field K] (q : Kˣ) (hδ : ∑ a : Fin 2, KnotTheory.jimboWeight q a ≠ 0) :
    { predStrands := 1, braid := BraidGroup.sigma 0 ^ 3 }.jimboReducedTrace q = ↑q⁻¹ ^ 2 + ↑q⁻¹ ^ 6 - ↑q⁻¹ ^ 8

    With two colours the trefoil value is q⁻² + q⁻⁶ - q⁻⁸, the Jones value t + t³ - t⁴ after the substitution t = q⁻².