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Flat knot 6.1162

Min(phi) over symmetries of the knot is: [-3,-1,1,1,1,1,0,1,1,2,3,1,1,1,2,-1,-1,-1,0,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.1162']
Arrow polynomial of the knot is: 4*K1**3 + 8*K1**2*K2 - 12*K1**2 - 2*K1*K2 - 4*K1*K3 - 2*K1 + 4*K2 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.1162']
Outer characteristic polynomial of the knot is: t^7+47t^5+67t^3+5t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1162']
2-strand cable arrow polynomial of the knot is: -128*K1**6 - 384*K1**4*K2**2 + 832*K1**4*K2 - 1472*K1**4 + 384*K1**3*K2*K3 - 128*K1**3*K3 - 256*K1**2*K2**4 + 608*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 + 64*K1**2*K2**2*K4 - 4544*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 416*K1**2*K2*K4 + 5896*K1**2*K2 - 128*K1**2*K3**2 - 80*K1**2*K4**2 - 3856*K1**2 + 1344*K1*K2**3*K3 + 448*K1*K2**2*K3*K4 - 1312*K1*K2**2*K3 + 32*K1*K2**2*K4*K5 - 192*K1*K2**2*K5 - 320*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 5056*K1*K2*K3 + 792*K1*K3*K4 + 104*K1*K4*K5 - 32*K2**6 - 64*K2**4*K3**2 - 64*K2**4*K4**2 + 224*K2**4*K4 - 1504*K2**4 + 96*K2**3*K3*K5 + 128*K2**3*K4*K6 - 96*K2**3*K6 - 1104*K2**2*K3**2 - 488*K2**2*K4**2 + 1776*K2**2*K4 - 48*K2**2*K5**2 - 48*K2**2*K6**2 - 2752*K2**2 - 32*K2*K3**2*K4 + 432*K2*K3*K5 + 224*K2*K4*K6 + 8*K3**2*K6 - 1488*K3**2 - 596*K4**2 - 48*K5**2 - 40*K6**2 + 3266
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1162']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.10917', 'vk6.10926', 'vk6.10932', 'vk6.12081', 'vk6.12083', 'vk6.12096', 'vk6.12098', 'vk6.14481', 'vk6.14494', 'vk6.15704', 'vk6.15715', 'vk6.16144', 'vk6.16149', 'vk6.30515', 'vk6.30527', 'vk6.30543', 'vk6.30557', 'vk6.31798', 'vk6.34068', 'vk6.34161', 'vk6.34181', 'vk6.34506', 'vk6.51748', 'vk6.51769', 'vk6.52626', 'vk6.54137', 'vk6.54144', 'vk6.54326', 'vk6.54531', 'vk6.63464', 'vk6.63473', 'vk6.63487']
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 O1O2O3O4U1U4U5O6O5U2U3U6
R3 orbit {'O1O2O3O4U1U4U5O6O5U2U3U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U5U2U3O6O5U6U1U4
Gauss code of K* O1O2O3U4U3O5O6O4U1U5U6U2
Gauss code of -K* O1O2O3U4U1O4O5O6U5U2U3U6
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 -1 1 1 1 1],[ 3 0 2 3 1 3 2],[ 1 -2 0 1 0 1 1],[-1 -3 -1 0 0 -1 0],[-1 -1 0 0 0 -1 0],[-1 -3 -1 1 1 0 1],[-1 -2 -1 0 0 -1 0]]
Primitive based matrix [[ 0 1 1 1 1 -1 -3],[-1 0 1 1 1 -1 -3],[-1 -1 0 0 0 0 -1],[-1 -1 0 0 0 -1 -2],[-1 -1 0 0 0 -1 -3],[ 1 1 0 1 1 0 -2],[ 3 3 1 2 3 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-1,-1,-1,-1,1,3,-1,-1,-1,1,3,0,0,0,1,0,1,2,1,3,2]
Phi over symmetry [-3,-1,1,1,1,1,0,1,1,2,3,1,1,1,2,-1,-1,-1,0,0,0]
Phi of -K [-3,-1,1,1,1,1,0,1,1,2,3,1,1,1,2,-1,-1,-1,0,0,0]
Phi of K* [-1,-1,-1,-1,1,3,-1,0,0,1,1,1,1,1,1,0,1,2,2,3,0]
Phi of -K* [-3,-1,1,1,1,1,2,1,2,3,3,0,1,1,1,0,-1,0,-1,0,1]
Symmetry type of based matrix c
u-polynomial t^3-3t
Normalized Jones-Krushkal polynomial 4z^2+21z+27
Enhanced Jones-Krushkal polynomial 4w^3z^2+21w^2z+27w
Inner characteristic polynomial t^6+33t^4+25t^2+1
Outer characteristic polynomial t^7+47t^5+67t^3+5t
Flat arrow polynomial 4*K1**3 + 8*K1**2*K2 - 12*K1**2 - 2*K1*K2 - 4*K1*K3 - 2*K1 + 4*K2 + 5
2-strand cable arrow polynomial -128*K1**6 - 384*K1**4*K2**2 + 832*K1**4*K2 - 1472*K1**4 + 384*K1**3*K2*K3 - 128*K1**3*K3 - 256*K1**2*K2**4 + 608*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 + 64*K1**2*K2**2*K4 - 4544*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 416*K1**2*K2*K4 + 5896*K1**2*K2 - 128*K1**2*K3**2 - 80*K1**2*K4**2 - 3856*K1**2 + 1344*K1*K2**3*K3 + 448*K1*K2**2*K3*K4 - 1312*K1*K2**2*K3 + 32*K1*K2**2*K4*K5 - 192*K1*K2**2*K5 - 320*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 5056*K1*K2*K3 + 792*K1*K3*K4 + 104*K1*K4*K5 - 32*K2**6 - 64*K2**4*K3**2 - 64*K2**4*K4**2 + 224*K2**4*K4 - 1504*K2**4 + 96*K2**3*K3*K5 + 128*K2**3*K4*K6 - 96*K2**3*K6 - 1104*K2**2*K3**2 - 488*K2**2*K4**2 + 1776*K2**2*K4 - 48*K2**2*K5**2 - 48*K2**2*K6**2 - 2752*K2**2 - 32*K2*K3**2*K4 + 432*K2*K3*K5 + 224*K2*K4*K6 + 8*K3**2*K6 - 1488*K3**2 - 596*K4**2 - 48*K5**2 - 40*K6**2 + 3266
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{3, 6}, {2, 5}, {1, 4}], [{4, 6}, {2, 5}, {1, 3}], [{4, 6}, {3, 5}, {1, 2}]]
If K is slice False
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