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Flat knot 6.565

Min(phi) over symmetries of the knot is: [-2,-1,-1,1,1,2,-1,-1,1,2,2,0,0,1,1,0,1,2,0,0,2]
Flat knots (up to 7 crossings) with same phi are :['6.565']
Arrow polynomial of the knot is: 4*K1**3 + 4*K1**2*K2 - 4*K1*K2 - 4*K1*K3 - K1 + K3 + K4
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.565', '6.1229', '6.1243', '6.1920']
Outer characteristic polynomial of the knot is: t^7+34t^5+77t^3+8t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.548', '6.565']
2-strand cable arrow polynomial of the knot is: 1536*K1**4*K2 - 3360*K1**4 - 384*K1**3*K2**2*K3 + 1792*K1**3*K2*K3 - 576*K1**3*K3 - 128*K1**2*K2**4 + 1664*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 + 128*K1**2*K2**2*K4 - 7664*K1**2*K2**2 + 768*K1**2*K2*K3**2 - 1088*K1**2*K2*K4 + 6832*K1**2*K2 - 2976*K1**2*K3**2 - 192*K1**2*K3*K5 - 32*K1**2*K4**2 - 2648*K1**2 + 1472*K1*K2**3*K3 + 256*K1*K2**2*K3*K4 - 2752*K1*K2**2*K3 - 576*K1*K2**2*K5 + 384*K1*K2*K3**3 - 672*K1*K2*K3*K4 - 384*K1*K2*K3*K6 + 8816*K1*K2*K3 + 2688*K1*K3*K4 + 192*K1*K4*K5 + 48*K1*K5*K6 - 32*K2**6 - 32*K2**4*K4**2 + 192*K2**4*K4 - 1872*K2**4 + 64*K2**3*K3*K5 + 64*K2**3*K4*K6 - 96*K2**3*K6 - 2352*K2**2*K3**2 - 32*K2**2*K3*K7 - 272*K2**2*K4**2 + 2272*K2**2*K4 - 64*K2**2*K5**2 - 48*K2**2*K6**2 - 2934*K2**2 - 64*K2*K3**2*K4 + 1648*K2*K3*K5 + 272*K2*K4*K6 + 48*K2*K5*K7 + 16*K2*K6*K8 - 320*K3**4 + 208*K3**2*K6 - 1924*K3**2 - 724*K4**2 - 220*K5**2 - 74*K6**2 - 8*K7**2 - 2*K8**2 + 3268
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.565']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.3252', 'vk6.3278', 'vk6.3300', 'vk6.3378', 'vk6.3411', 'vk6.3435', 'vk6.3479', 'vk6.3511', 'vk6.4609', 'vk6.5896', 'vk6.6023', 'vk6.7949', 'vk6.8072', 'vk6.9383', 'vk6.17837', 'vk6.17852', 'vk6.19066', 'vk6.19886', 'vk6.24350', 'vk6.25680', 'vk6.25693', 'vk6.26326', 'vk6.26771', 'vk6.37786', 'vk6.43779', 'vk6.43794', 'vk6.45063', 'vk6.48108', 'vk6.48121', 'vk6.48146', 'vk6.48197', 'vk6.50661']
The R3 orbit of minmal crossing diagrams contains:
The diagrammatic symmetry type of this knot is c.
The reverse -K is
The mirror image K* is
The reversed mirror image -K* is
The fillings (up to the first 10) associated to the algebraic genus:
Or click here to check the fillings

invariant value
Gauss code O1O2O3O4O5U4U3U5O6U1U6U2
R3 orbit {'O1O2O3O4O5U4U3U5O6U1U6U2'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U4U6U5O6U1U3U2
Gauss code of K* O1O2O3U4O5O4O6U5U6U2U1U3
Gauss code of -K* O1O2O3U2O4O5O6U4U6U5U1U3
Diagrammatic symmetry type c
Flat genus of the diagram 3
If K is checkerboard colorable False
If K is almost classical False
Based matrix from Gauss code [[ 0 -2 1 -1 -1 2 1],[ 2 0 2 -1 -1 2 1],[-1 -2 0 -1 -1 2 0],[ 1 1 1 0 0 2 0],[ 1 1 1 0 0 1 0],[-2 -2 -2 -2 -1 0 0],[-1 -1 0 0 0 0 0]]
Primitive based matrix [[ 0 2 1 1 -1 -1 -2],[-2 0 0 -2 -1 -2 -2],[-1 0 0 0 0 0 -1],[-1 2 0 0 -1 -1 -2],[ 1 1 0 1 0 0 1],[ 1 2 0 1 0 0 1],[ 2 2 1 2 -1 -1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,-1,1,1,2,0,2,1,2,2,0,0,0,1,1,1,2,0,-1,-1]
Phi over symmetry [-2,-1,-1,1,1,2,-1,-1,1,2,2,0,0,1,1,0,1,2,0,0,2]
Phi of -K [-2,-1,-1,1,1,2,2,2,1,2,2,0,1,2,1,1,2,2,0,-1,1]
Phi of K* [-2,-1,-1,1,1,2,-1,1,1,2,2,0,1,1,1,2,2,2,0,2,2]
Phi of -K* [-2,-1,-1,1,1,2,-1,-1,1,2,2,0,0,1,1,0,1,2,0,0,2]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 9z^2+30z+25
Enhanced Jones-Krushkal polynomial 9w^3z^2+30w^2z+25w
Inner characteristic polynomial t^6+22t^4+33t^2+1
Outer characteristic polynomial t^7+34t^5+77t^3+8t
Flat arrow polynomial 4*K1**3 + 4*K1**2*K2 - 4*K1*K2 - 4*K1*K3 - K1 + K3 + K4
2-strand cable arrow polynomial 1536*K1**4*K2 - 3360*K1**4 - 384*K1**3*K2**2*K3 + 1792*K1**3*K2*K3 - 576*K1**3*K3 - 128*K1**2*K2**4 + 1664*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 + 128*K1**2*K2**2*K4 - 7664*K1**2*K2**2 + 768*K1**2*K2*K3**2 - 1088*K1**2*K2*K4 + 6832*K1**2*K2 - 2976*K1**2*K3**2 - 192*K1**2*K3*K5 - 32*K1**2*K4**2 - 2648*K1**2 + 1472*K1*K2**3*K3 + 256*K1*K2**2*K3*K4 - 2752*K1*K2**2*K3 - 576*K1*K2**2*K5 + 384*K1*K2*K3**3 - 672*K1*K2*K3*K4 - 384*K1*K2*K3*K6 + 8816*K1*K2*K3 + 2688*K1*K3*K4 + 192*K1*K4*K5 + 48*K1*K5*K6 - 32*K2**6 - 32*K2**4*K4**2 + 192*K2**4*K4 - 1872*K2**4 + 64*K2**3*K3*K5 + 64*K2**3*K4*K6 - 96*K2**3*K6 - 2352*K2**2*K3**2 - 32*K2**2*K3*K7 - 272*K2**2*K4**2 + 2272*K2**2*K4 - 64*K2**2*K5**2 - 48*K2**2*K6**2 - 2934*K2**2 - 64*K2*K3**2*K4 + 1648*K2*K3*K5 + 272*K2*K4*K6 + 48*K2*K5*K7 + 16*K2*K6*K8 - 320*K3**4 + 208*K3**2*K6 - 1924*K3**2 - 724*K4**2 - 220*K5**2 - 74*K6**2 - 8*K7**2 - 2*K8**2 + 3268
Genus of based matrix 1
Fillings of based matrix [[{3, 6}, {1, 5}, {2, 4}], [{4, 6}, {1, 5}, {2, 3}]]
If K is slice False
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