[1] smoothing formulas of Section 2 against the half-edge model
    12063 smoothings compared: the formulas agree with the half-edge model
[2] Theorem A: K1, K2
    K1 endpoints: a 0,2 b 1,5 e 3,10 c 4,12 d 6,8 f 7,13 g 9,11
    K1 chords linked with: a: b; b: ace; c: bef; d: f; e: bcfg; f: cde; g: e
    Gaussian parity: e is the only even chord of K1 and of K2
    (i) the only triangle of K1 has the edges ec, ge, gc at positions [(3, 4), (9, 10), (11, 12)]; the third move gives a b a c e b d f d e g c g f = K2
    (ii) arcs at the unique even chord e: K1 (6, 6), K2 (4, 8); canonical forms differ: K1 and K2 are not isomorphic
    K1 = e A e B with A = c b d f d g, B = g c f a b a
    D^1 = A.rev(B) = c b d f d g a b a f c g -> R2-normal form c b d f d g a b a f c g (6 vertices, 1 components, free loop: False; all reduction orders give 1 normal form)
    D^2 = {A, B} = c b d f d g | g c f a b a -> R2-normal form b d f d | f a b a (4 vertices, 2 components, free loop: False; all reduction orders give 1 normal form)
    (iii) {K1} = [T6] + [T4], T6 = c b d f d g a b a f c g, T4 = b d f d | f a b a: two different basis vectors; {K2} = {K1}
    K2 also has exactly one even vertex and a smoothing isomorphic to T6 (Proposition 4.1)
[3] Theorem B: D1, D2
    mixed parity: e is the only even vertex of D1 and of D2; 1, ..., 6 are odd
    (i) the only triangle of D1 is (3, 4, e); the third move gives 4 1 3 e 5 2 6 e | 1 2 4 3 5 6, isomorphic to D2
    (ii) arcs of the first word between the occurrences of e: D1 (1, 5), D2 (3, 3); D1, D2 not isomorphic
    D^1 = G6 = 1 4 6 2 5 3 | 1 2 3 4 5 6 -> R2-normal form 1 4 6 2 5 3 | 1 2 3 4 5 6 (6 vertices, 2 components, free loop: False; all reduction orders give 1 normal form)
    D^2 = 1 | 3 5 2 6 4 | 1 2 3 4 5 6 -> R2-normal form 1 | 6 | 1 6 (2 vertices, 3 components, free loop: False; all reduction orders give 1 normal form)
    (iii) {D1} = [G6] + [H2] = {D2}; G6 has 2 components and 6 vertices, H2 = 1 | 6 | 1 6 has 3 components
[4] local tables of the invariance proof
    R1, v even
      T   state (0,)      0-1                                        
      T   state (1,)      0-1                                        circle -> 0
      T'  state ()        0-1                                        
      mod-2 sums agree (1 term(s))
    R2 parallel, u, v even
      T   state (0, 0)    0-2 1-3                                    circle -> 0
      T   state (0, 1)    0-2 1-3                                    
      T   state (1, 0)    0-2 1-3                                    
      T   state (1, 1)    0-1 2-3                                    
      T'  state ()        0-1 2-3                                    
      mod-2 sums agree (1 term(s))
    R2 antiparallel, u, v even
      T   state (0, 0)    0-1 2-3                                    
      T   state (0, 1)    0-3 1-2                                    
      T   state (1, 0)    0-3 1-2                                    
      T   state (1, 1)    0-3 1-2                                    circle -> 0
      T'  state ()        0-1 2-3                                    
      mod-2 sums agree (1 term(s))
    R2 parallel, u, v odd
      T   state ()        0-u-v-1 2-u-v-3                            -> after R2: 0-1 2-3
      T'  state ()        0-1 2-3                                    
      mod-2 sums agree (1 term(s))
    R2 antiparallel, u, v odd
      T   state ()        0-u-v-1 2-v-u-3                            -> after R2: 0-1 2-3
      T'  state ()        0-1 2-3                                    
      mod-2 sums agree (1 term(s))
    R3, a, b, c even
      T   state (0, 0, 0) 0-3 1-4 2-5                                
      T   state (0, 0, 1) 0-1 2-5 3-4                                
      T   state (0, 1, 0) 0-3 1-2 4-5                                
      T   state (0, 1, 1) 0-5 1-2 3-4                                
      T   state (1, 0, 0) 0-5 1-4 2-3                                
      T   state (1, 0, 1) 0-5 1-2 3-4                                
      T   state (1, 1, 0) 0-5 1-2 3-4                                
      T   state (1, 1, 1) 0-5 1-2 3-4                                circle -> 0
      T'  state (0, 0, 0) 0-5 1-2 3-4                                
      T'  state (0, 0, 1) 0-1 2-5 3-4                                
      T'  state (0, 1, 0) 0-3 1-2 4-5                                
      T'  state (0, 1, 1) 0-3 1-4 2-5                                
      T'  state (1, 0, 0) 0-5 1-4 2-3                                
      T'  state (1, 0, 1) 0-3 1-4 2-5                                
      T'  state (1, 1, 0) 0-3 1-4 2-5                                
      T'  state (1, 1, 1) 0-3 1-4 2-5                                circle -> 0
      mod-2 sums agree (5 term(s))
    R3, only a even
      T   state (0,)      0-c-4 1-b-5 2-b-c-3                        
      T   state (1,)      0-5 1-b-c-4 2-b-c-3                        -> after R2: 0-5 1-4 2-3
      T'  state (0,)      0-b-4 1-c-5 2-c-b-3                        
      T'  state (1,)      0-b-c-5 1-4 2-c-b-3                        -> after R2: 0-5 1-4 2-3
      mod-2 sums agree (2 term(s))
    R3, only b even
      T   state (0,)      0-a-2 1-c-3 4-c-a-5                        
      T   state (1,)      0-a-c-3 1-2 4-c-a-5                        -> after R2: 0-3 1-2 4-5
      T'  state (0,)      0-c-2 1-a-3 4-a-c-5                        
      T'  state (1,)      0-3 1-a-c-2 4-a-c-5                        -> after R2: 0-3 1-2 4-5
      mod-2 sums agree (2 term(s))
    R3, only c even
      T   state (0,)      0-a-b-1 2-b-4 3-a-5                        
      T   state (1,)      0-a-b-1 2-b-a-5 3-4                        -> after R2: 0-1 2-5 3-4
      T'  state (0,)      0-b-a-1 2-a-4 3-b-5                        
      T'  state (1,)      0-b-a-1 2-5 3-b-a-4                        -> after R2: 0-1 2-5 3-4
      mod-2 sums agree (2 term(s))
    controls: for the parities excluded by axiom (P3) (two even, or three odd vertices) the sums differ
[5] Proposition 4.2: un-smoothings of T6 and of G6 (cross-check of part (iv))
    knot: 312 candidates, 14 isomorphism classes, 2 with the bracket of K1: K1 and K2
    link: 168 candidates, 8 isomorphism classes, 2 with the bracket of D1: D1 and D2
[6] the unoriented states K_us (criterion of Kauffman and Manturov) and the example X1, X2
    K1: K_us has 0 bigons and 6 quadrilaterals; girth 4
    K2: K_us has 0 bigons and 5 quadrilaterals; girth 4
    X1: K_us has 0 bigons and 0 quadrilaterals; girth 6
    X2: K_us has 0 bigons and 0 quadrilaterals; girth 6
    X1 has exactly one triangle, X2 is isomorphic to its image under the third move, X1 and X2 are not
    isomorphic; X1 has 5 even chords and {X1} = {X2} = 0 (the parity bracket does not see them)
[7] the unused figure non_uniq.eps of the source archive of the problem list
    left diagram: A D G E G A C B C F E D F B; linked pairs AD AE BC BD BE DF EF EG; bigons []; bracket terms 0
    right diagram: A F E D F A C B C D G E G B; linked pairs AD AE BC BD BE DF EF EG; bigons ['AF']; bracket terms 0
    same labelled intersection graph, not isomorphic, both brackets zero, a second move applies to the right one
ALL CHECKS PASSED (12124 assertions)
