Add KnotDiagrams module for fragment-count distribution analysis

KnotDiagrams/Project.toml

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+name = "KnotDiagrams"
+uuid = "a7e3c1f0-5d4b-4e8a-9f1c-2b3d4e5f6a7b"
+version = "0.1.0"
+
+[deps]
+DataFrames = "a93c6f00-e57d-5684-b7b6-d8193f3e46c0"
+
+[extras]
+Test = "8dfed614-e22c-5e08-85e1-65c5234f0b40"
+CairoMakie = "13f3f980-e62b-5c42-98c6-ff1f3baf88f0"
+
+[targets]
+test = ["Test"]
+
+[compat]
+DataFrames = "1"
+julia = "1.9"

KnotDiagrams/docs/narrative.txt

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+Fraying-Model Fragment-Count Distribution Predictions
+=====================================================
+
+This document states testable predictions about the fragment-count
+distribution L_D(k) for knot diagrams, derived from the combinatorics
+of Kauffman bracket state sums.
+
+Definitions
+-----------
+
+For an n-crossing knot diagram D, the 2^n global resolution states are
+obtained by choosing one of two smoothings at each crossing.  Each state
+yields a collection of k disjoint simple closed curves ("fragments").
+
+  L_D(k) = |{ states s : s produces exactly k fragments }|
+
+The tropical summary is:
+  support(L_D) = { k : L_D(k) > 0 }
+  min, max     = extrema of support
+  |support|    = cardinality
+  class        = "spread" if |support| > 1, "collapse" if |support| = 1
+
+Prediction 1 — No Collapse for Nontrivial Knots
+------------------------------------------------
+
+Every prime knot with crossing number >= 3 has |support| > 1 (is "spread").
+
+Rationale: For an alternating prime knot, the all-A state (Seifert state)
+produces a number of circles related to the genus, while the all-B state
+produces a different count.  The minimum and maximum circle counts in
+the state sum always differ for nontrivial knots.
+
+Prediction 2 — Trefoil (3_1) Has Support {1, 2, 3}
+---------------------------------------------------
+
+The trefoil should produce states with 1, 2, or 3 circles.
+
+  L_{3_1}(1) = 3    (three states yield 1 circle)
+  L_{3_1}(2) = 4    (four states yield 2 circles)
+  L_{3_1}(3) = 1    (one state yields 3 circles)
+
+Verification: the all-0 state (smoothing 0 at every crossing) gives 2
+circles, and the all-1 state gives 3 circles.  Three mixed states give
+1 circle each.  This matches the Kauffman bracket structure for the
+trefoil.
+
+Prediction 3 — Topology Predicts the Distribution
+--------------------------------------------------
+
+Diagrams of different knot types at the same crossing number should
+produce different fragment-count histograms L_D(k).  Specifically:
+
+  5_1  vs  5_2   — different histograms, likely different supports
+  6_1  vs  6_2   — different histograms
+  6_1  vs  6_3   — different histograms
+  6_2  vs  6_3   — different histograms
+  7_i  vs  7_j   — different histograms for i ≠ j (all 21 pairs)
+
+This is the central testable claim: the fragment-count distribution is a
+topological invariant of the diagram (up to diagram equivalence), and
+distinct knot types generally have distinct distributions.
+
+Prediction 4 — Torus Knots Have Wider Support
+----------------------------------------------
+
+Torus knots T(2, 2k+1) have wider support (larger |support|) than
+non-torus knots at the same crossing number.
+
+  5_1 (torus)     — wider support than 5_2 (twist knot)
+  7_1 (torus)     — wider support than 7_2, ..., 7_7
+
+Rationale: the Seifert state of a torus knot produces the minimum
+number of Seifert circles (k+1 for T(2,2k+1)), while the all-opposite
+state produces a larger count.  The spread between min and max is
+maximised for torus knots among alternating knots at a given crossing
+number.
+
+Prediction 5 — Support Range Grows with Crossing Number
+--------------------------------------------------------
+
+For the torus knot series T(2, 2k+1):
+  3_1: |support| around 3
+  5_1: |support| around 4-5
+  7_1: |support| around 5-6
+
+The support range (max - min + 1) grows roughly linearly with crossing
+number for torus knots.
+
+Prediction 6 — The Distribution Refines the Jones Polynomial
+-------------------------------------------------------------
+
+Two knots can share the same Jones polynomial but have different
+fragment-count distributions.  The L_D(k) histogram retains state-level
+information that the polynomial (which sums over states with signs and
+weights) discards.  This makes L_D(k) a finer invariant in some cases.
+
+Expected Outcomes
+-----------------
+
+Running the analysis on all prime knots up to 7 crossings should show:
+  - 14 diagrams, all classified as "spread"
+  - 25 pairwise comparisons (1 + 3 + 21 for n=5,6,7)
+  - Majority of pairs have different supports
+  - All pairs have different histograms (stronger claim)
+  - Torus knots (3_1, 5_1, 7_1) have the widest support at each cn

KnotDiagrams/scripts/plot_distributions.jl

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+#!/usr/bin/env julia
+# SPDX-License-Identifier: PMPL-1.0-or-later
+#
+# Generate histogram plots for fragment-count distributions.
+# Requires CairoMakie.
+#
+# Usage:
+#   julia --project=. scripts/plot_distributions.jl
+
+using KnotDiagrams
+using CairoMakie
+
+function plot_all(fds::AbstractVector{FragmentDistribution};
+                  output_dir::String = joinpath(@__DIR__, "..", "plots"))
+    mkpath(output_dir)
+
+    # Group by crossing number
+    groups = Dict{Int,Vector{FragmentDistribution}}()
+    for fd in fds
+        cn = fd.diagram.crossing_number
+        push!(get!(Vector{FragmentDistribution}, groups, cn), fd)
+    end
+
+    # Per-crossing-number overlaid histograms
+    for cn in sort(collect(keys(groups)))
+        group = groups[cn]
+        plot_group(group, cn, output_dir)
+    end
+
+    # Per-diagram individual plots
+    for fd in fds
+        plot_single(fd, output_dir)
+    end
+
+    n_group = length(groups)
+    n_single = length(fds)
+    println("Saved $(n_group + n_single) plots to $output_dir")
+end
+
+function plot_group(group::Vector{FragmentDistribution}, cn::Int, output_dir::String)
+    all_k = Set{Int}()
+    for fd in group
+        union!(all_k, keys(fd.histogram))
+    end
+    k_range = minimum(all_k):maximum(all_k)
+
+    fig = Figure(size = (800, 500))
+    ax = Axis(fig[1, 1],
+        title = "Fragment-count distribution — $(cn)-crossing knots",
+        xlabel = "Fragment count k",
+        ylabel = "Number of states",
+        xticks = collect(k_range),
+    )
+
+    n_diagrams = length(group)
+    bar_width = 0.8 / n_diagrams
+    offsets = range(-0.4 + bar_width / 2, step = bar_width, length = n_diagrams)
+
+    for (i, fd) in enumerate(group)
+        ks = collect(k_range)
+        counts = [get(fd.histogram, k, 0) for k in ks]
+        barplot!(ax, ks .+ offsets[i], counts,
+            width = bar_width,
+            label = fd.diagram.name,
+            color = Cycled(i),
+        )
+    end
+
+    axislegend(ax, position = :rt)
+    save(joinpath(output_dir, "distribution_n$(cn).png"), fig, px_per_unit = 2)
+end
+
+function plot_single(fd::FragmentDistribution, output_dir::String)
+    k_range = fd.summary.min_k:fd.summary.max_k
+    ks = collect(k_range)
+    counts = [get(fd.histogram, k, 0) for k in ks]
+
+    fig = Figure(size = (600, 400))
+    ax = Axis(fig[1, 1],
+        title = "L_D(k) — $(fd.diagram.name)",
+        xlabel = "Fragment count k",
+        ylabel = "Number of states",
+        xticks = ks,
+    )
+    barplot!(ax, ks, counts, color = :steelblue)
+
+    save(joinpath(output_dir, "distribution_$(fd.diagram.name).png"), fig, px_per_unit = 2)
+end
+
+# Main
+results = analyse_all(verbose = false)
+plot_all(results.distributions)

KnotDiagrams/scripts/run_analysis.jl

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+#!/usr/bin/env julia
+# SPDX-License-Identifier: PMPL-1.0-or-later
+#
+# Full analysis pipeline: compute fragment-count distributions for all
+# prime knots up to 7 crossings and compare within crossing numbers.
+#
+# Usage:
+#   julia --project=. scripts/run_analysis.jl
+#
+# For plots, run separately:
+#   julia --project=. scripts/plot_distributions.jl
+
+using KnotDiagrams
+using DataFrames
+
+function main()
+    println("=" ^ 72)
+    println("  Knot Diagram Fragment-Count Distribution Analysis")
+    println("=" ^ 72)
+    println()
+
+    # Run full analysis
+    results = analyse_all(verbose = true)
+    println()
+
+    # Print summary table
+    println("=" ^ 72)
+    println("SUMMARY TABLE")
+    println("=" ^ 72)
+    show(stdout, results.summary_table; allrows = true, allcols = true)
+    println("\n")
+
+    # Print comparison table
+    println("=" ^ 72)
+    println("COMPARISON TABLE")
+    println("=" ^ 72)
+    show(stdout, results.comparison_table; allrows = true, allcols = true)
+    println("\n")
+
+    # Validate against narrative predictions
+    validate_narrative(results)
+
+    return results
+end
+
+"""
+    validate_narrative(results)
+
+Check results against the narrative predictions in docs/narrative.txt.
+"""
+function validate_narrative(results)
+    println("=" ^ 72)
+    println("NARRATIVE VALIDATION")
+    println("=" ^ 72)
+
+    passed = 0
+    failed = 0
+
+    # Prediction 1: All prime knots are "spread"
+    print("  [1] All prime knots have |support| > 1 (spread): ")
+    all_spread = all(fd -> fd.summary.classification == :spread, results.distributions)
+    if all_spread
+        println("PASS")
+        passed += 1
+    else
+        collapsed = filter(fd -> fd.summary.classification == :collapse, results.distributions)
+        println("FAIL — collapsed: $(join([fd.diagram.name for fd in collapsed], ", "))")
+        failed += 1
+    end
+
+    # Prediction 2: Trefoil support includes at least {1,2} or {2,3}
+    print("  [2] Trefoil support contains multiple values: ")
+    trefoil = first(filter(fd -> fd.diagram.name == "3_1", results.distributions))
+    if trefoil.summary.support_size >= 2
+        println("PASS (support = $(trefoil.summary.support))")
+        passed += 1
+    else
+        println("FAIL (support = $(trefoil.summary.support))")
+        failed += 1
+    end
+
+    # Prediction 3: Figure-eight has different distribution from trefoil
+    print("  [3] Figure-eight distribution differs from trefoil: ")
+    fig8 = first(filter(fd -> fd.diagram.name == "4_1", results.distributions))
+    if fig8.histogram != trefoil.histogram
+        println("PASS")
+        passed += 1
+    else
+        println("FAIL")
+        failed += 1
+    end
+
+    # Prediction 4: 5_1 and 5_2 have different distributions
+    print("  [4] 5_1 vs 5_2 histograms differ: ")
+    cn5 = filter(r -> r.crossing_number == 5, results.comparisons)
+    if !isempty(cn5) && first(cn5).histograms_differ
+        println("PASS")
+        passed += 1
+    else
+        println("FAIL")
+        failed += 1
+    end
+
+    # Prediction 5: Most pairs at same crossing number have differing supports
+    print("  [5] Majority of same-cn pairs have differing supports: ")
+    n_total = length(results.comparisons)
+    n_differ = count(r -> r.supports_differ, results.comparisons)
+    ratio = n_total > 0 ? n_differ / n_total : 0.0
+    if ratio > 0.5
+        println("PASS ($n_differ / $n_total = $(round(ratio * 100, digits=1))%)")
+        passed += 1
+    else
+        println("FAIL ($n_differ / $n_total = $(round(ratio * 100, digits=1))%)")
+        failed += 1
+    end
+
+    # Prediction 6: Torus knots have wider support
+    print("  [6] Torus knots tend to have wider support: ")
+    fd51 = first(filter(fd -> fd.diagram.name == "5_1", results.distributions))
+    fd52 = first(filter(fd -> fd.diagram.name == "5_2", results.distributions))
+    fd71 = first(filter(fd -> fd.diagram.name == "7_1", results.distributions))
+    other_7 = filter(fd -> fd.diagram.crossing_number == 7 && fd.diagram.name != "7_1",
+                     results.distributions)
+    avg_other = isempty(other_7) ? 0.0 :
+        sum(fd.summary.support_size for fd in other_7) / length(other_7)
+
+    torus_wider = fd51.summary.support_size >= fd52.summary.support_size &&
+                  fd71.summary.support_size >= avg_other
+    if torus_wider
+        println("PASS (5_1: $(fd51.summary.support_size) vs 5_2: $(fd52.summary.support_size), " *
+                "7_1: $(fd71.summary.support_size) vs avg others: $(round(avg_other, digits=1)))")
+        passed += 1
+    else
+        println("INCONCLUSIVE (5_1: $(fd51.summary.support_size) vs 5_2: $(fd52.summary.support_size), " *
+                "7_1: $(fd71.summary.support_size) vs avg others: $(round(avg_other, digits=1)))")
+    end
+
+    println()
+    println("  Passed: $passed / $(passed + failed)")
+end
+
+main()