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Flat knot 6.384

Min(phi) over symmetries of the knot is: [-4,-1,-1,1,2,3,0,2,1,3,5,0,0,0,1,0,1,3,0,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.384']
Arrow polynomial of the knot is: 4*K1**3 + 4*K1**2*K2 - 16*K1**2 - 6*K1*K2 - 2*K1*K3 + 7*K2 + 2*K3 + 8
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.384']
Outer characteristic polynomial of the knot is: t^7+82t^5+110t^3+6t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.384']
2-strand cable arrow polynomial of the knot is: -256*K1**6 - 384*K1**4*K2**2 + 1152*K1**4*K2 - 3600*K1**4 + 576*K1**3*K2*K3 - 448*K1**3*K3 - 256*K1**2*K2**4 + 736*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 + 64*K1**2*K2**2*K4 - 7312*K1**2*K2**2 + 192*K1**2*K2*K3**2 - 832*K1**2*K2*K4 + 10008*K1**2*K2 - 816*K1**2*K3**2 - 96*K1**2*K4**2 - 5252*K1**2 + 896*K1*K2**3*K3 + 256*K1*K2**2*K3*K4 - 1600*K1*K2**2*K3 - 448*K1*K2**2*K5 + 64*K1*K2*K3**3 - 320*K1*K2*K3*K4 + 9112*K1*K2*K3 - 32*K1*K2*K4*K5 + 1672*K1*K3*K4 + 152*K1*K4*K5 - 32*K2**6 - 64*K2**4*K3**2 - 32*K2**4*K4**2 + 160*K2**4*K4 - 1776*K2**4 + 96*K2**3*K3*K5 + 32*K2**3*K4*K6 - 64*K2**3*K6 - 1328*K2**2*K3**2 - 376*K2**2*K4**2 + 2512*K2**2*K4 - 48*K2**2*K5**2 - 8*K2**2*K6**2 - 4728*K2**2 - 32*K2*K3**2*K4 + 952*K2*K3*K5 + 216*K2*K4*K6 + 8*K2*K5*K7 - 64*K3**4 - 32*K3**2*K4**2 + 32*K3**2*K6 - 2488*K3**2 + 16*K3*K4*K7 - 922*K4**2 - 148*K5**2 - 32*K6**2 + 5048
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.384']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11054', 'vk6.11134', 'vk6.12220', 'vk6.12329', 'vk6.16432', 'vk6.19226', 'vk6.19319', 'vk6.19519', 'vk6.19612', 'vk6.22736', 'vk6.22837', 'vk6.26034', 'vk6.26083', 'vk6.26418', 'vk6.26505', 'vk6.30623', 'vk6.30720', 'vk6.31931', 'vk6.34779', 'vk6.38105', 'vk6.38125', 'vk6.42393', 'vk6.44625', 'vk6.44745', 'vk6.51847', 'vk6.52711', 'vk6.52813', 'vk6.56571', 'vk6.56629', 'vk6.64724', 'vk6.66271', 'vk6.66301']
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 O1O2O3O4O5U4U2O6U1U6U3U5
R3 orbit {'O1O2O3O4O5U4U2O6U1U6U3U5'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U1U3U6U5O6U4U2
Gauss code of K* O1O2O3O4U1U5U3U6U4O6O5U2
Gauss code of -K* O1O2O3O4U3O5O6U1U6U2U5U4
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 -3 -2 1 -1 4 1],[ 3 0 0 3 0 5 1],[ 2 0 0 1 0 3 0],[-1 -3 -1 0 0 2 0],[ 1 0 0 0 0 1 0],[-4 -5 -3 -2 -1 0 0],[-1 -1 0 0 0 0 0]]
Primitive based matrix [[ 0 4 1 1 -1 -2 -3],[-4 0 0 -2 -1 -3 -5],[-1 0 0 0 0 0 -1],[-1 2 0 0 0 -1 -3],[ 1 1 0 0 0 0 0],[ 2 3 0 1 0 0 0],[ 3 5 1 3 0 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-4,-1,-1,1,2,3,0,2,1,3,5,0,0,0,1,0,1,3,0,0,0]
Phi over symmetry [-4,-1,-1,1,2,3,0,2,1,3,5,0,0,0,1,0,1,3,0,0,0]
Phi of -K [-3,-2,-1,1,1,4,1,2,1,3,2,1,2,3,3,2,2,4,0,1,3]
Phi of K* [-4,-1,-1,1,2,3,1,3,4,3,2,0,2,2,1,2,3,3,1,2,1]
Phi of -K* [-3,-2,-1,1,1,4,0,0,1,3,5,0,0,1,3,0,0,1,0,0,2]
Symmetry type of based matrix c
u-polynomial -t^4+t^3+t^2-t
Normalized Jones-Krushkal polynomial 3z^2+22z+33
Enhanced Jones-Krushkal polynomial 3w^3z^2+22w^2z+33w
Inner characteristic polynomial t^6+50t^4+41t^2+1
Outer characteristic polynomial t^7+82t^5+110t^3+6t
Flat arrow polynomial 4*K1**3 + 4*K1**2*K2 - 16*K1**2 - 6*K1*K2 - 2*K1*K3 + 7*K2 + 2*K3 + 8
2-strand cable arrow polynomial -256*K1**6 - 384*K1**4*K2**2 + 1152*K1**4*K2 - 3600*K1**4 + 576*K1**3*K2*K3 - 448*K1**3*K3 - 256*K1**2*K2**4 + 736*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 + 64*K1**2*K2**2*K4 - 7312*K1**2*K2**2 + 192*K1**2*K2*K3**2 - 832*K1**2*K2*K4 + 10008*K1**2*K2 - 816*K1**2*K3**2 - 96*K1**2*K4**2 - 5252*K1**2 + 896*K1*K2**3*K3 + 256*K1*K2**2*K3*K4 - 1600*K1*K2**2*K3 - 448*K1*K2**2*K5 + 64*K1*K2*K3**3 - 320*K1*K2*K3*K4 + 9112*K1*K2*K3 - 32*K1*K2*K4*K5 + 1672*K1*K3*K4 + 152*K1*K4*K5 - 32*K2**6 - 64*K2**4*K3**2 - 32*K2**4*K4**2 + 160*K2**4*K4 - 1776*K2**4 + 96*K2**3*K3*K5 + 32*K2**3*K4*K6 - 64*K2**3*K6 - 1328*K2**2*K3**2 - 376*K2**2*K4**2 + 2512*K2**2*K4 - 48*K2**2*K5**2 - 8*K2**2*K6**2 - 4728*K2**2 - 32*K2*K3**2*K4 + 952*K2*K3*K5 + 216*K2*K4*K6 + 8*K2*K5*K7 - 64*K3**4 - 32*K3**2*K4**2 + 32*K3**2*K6 - 2488*K3**2 + 16*K3*K4*K7 - 922*K4**2 - 148*K5**2 - 32*K6**2 + 5048
Genus of based matrix 1
Fillings of based matrix [[{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {1, 5}, {3}, {2}], [{4, 6}, {2, 5}, {1, 3}], [{4, 6}, {3, 5}, {1, 2}]]
If K is slice False
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