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Flat knot 6.1230

Min(phi) over symmetries of the knot is: [-2,-1,-1,1,1,2,-1,1,1,2,1,1,1,1,1,1,1,2,0,1,1]
Flat knots (up to 7 crossings) with same phi are :['6.1230']
Arrow polynomial of the knot is: -4*K1*K2 + 2*K1 + 2*K3 + 1
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.540', '6.925', '6.1021', '6.1117', '6.1120', '6.1135', '6.1227', '6.1230', '6.1260', '6.1682', '6.1685', '6.1922']
Outer characteristic polynomial of the knot is: t^7+40t^5+37t^3+8t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1230']
2-strand cable arrow polynomial of the knot is: -784*K1**4 + 224*K1**3*K3*K4 - 64*K1**3*K3 - 800*K1**2*K2**2 + 96*K1**2*K2*K4**2 - 512*K1**2*K2*K4 + 2904*K1**2*K2 - 624*K1**2*K3**2 - 192*K1**2*K3*K5 - 592*K1**2*K4**2 - 3240*K1**2 - 512*K1*K2**2*K3 - 448*K1*K2*K3*K4 + 3096*K1*K2*K3 - 96*K1*K2*K4*K5 + 2472*K1*K3*K4 + 856*K1*K4*K5 + 24*K1*K5*K6 - 112*K2**2*K4**2 + 872*K2**2*K4 - 2500*K2**2 + 344*K2*K3*K5 + 112*K2*K4*K6 - 1820*K3**2 - 1268*K4**2 - 332*K5**2 - 28*K6**2 + 2922
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1230']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.10145', 'vk6.10210', 'vk6.10353', 'vk6.10432', 'vk6.16696', 'vk6.19072', 'vk6.19121', 'vk6.19252', 'vk6.19545', 'vk6.23015', 'vk6.23132', 'vk6.25701', 'vk6.25750', 'vk6.26066', 'vk6.26443', 'vk6.29932', 'vk6.29991', 'vk6.30089', 'vk6.34998', 'vk6.35123', 'vk6.37795', 'vk6.37857', 'vk6.42570', 'vk6.44655', 'vk6.51637', 'vk6.51740', 'vk6.54908', 'vk6.56589', 'vk6.59335', 'vk6.64872', 'vk6.66181', 'vk6.66214']
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 O1O2O3O4U3U2U5O6O5U1U4U6
R3 orbit {'O1O2O3O4U3U2U5O6O5U1U4U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U5U1U4O6O5U6U3U2
Gauss code of K* O1O2O3U4U3O5O6O4U5U2U1U6
Gauss code of -K* O1O2O3U4U1O4O5O6U2U6U5U3
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 2 1 1],[ 2 0 0 0 3 2 1],[ 1 0 0 0 2 1 1],[ 1 0 0 0 1 1 1],[-2 -3 -2 -1 0 -2 0],[-1 -2 -1 -1 2 0 1],[-1 -1 -1 -1 0 -1 0]]
Primitive based matrix [[ 0 2 1 1 -1 -1 -2],[-2 0 0 -2 -1 -2 -3],[-1 0 0 -1 -1 -1 -1],[-1 2 1 0 -1 -1 -2],[ 1 1 1 1 0 0 0],[ 1 2 1 1 0 0 0],[ 2 3 1 2 0 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,-1,1,1,2,0,2,1,2,3,1,1,1,1,1,1,2,0,0,0]
Phi over symmetry [-2,-1,-1,1,1,2,-1,1,1,2,1,1,1,1,1,1,1,2,0,1,1]
Phi of -K [-2,-1,-1,1,1,2,1,1,1,2,1,0,1,1,1,1,1,2,-1,-1,1]
Phi of K* [-2,-1,-1,1,1,2,-1,1,1,2,1,1,1,1,1,1,1,2,0,1,1]
Phi of -K* [-2,-1,-1,1,1,2,0,0,1,2,3,0,1,1,1,1,1,2,-1,0,2]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 3z^2+16z+21
Enhanced Jones-Krushkal polynomial 3w^3z^2-8w^3z+24w^2z+21w
Inner characteristic polynomial t^6+28t^4+13t^2+1
Outer characteristic polynomial t^7+40t^5+37t^3+8t
Flat arrow polynomial -4*K1*K2 + 2*K1 + 2*K3 + 1
2-strand cable arrow polynomial -784*K1**4 + 224*K1**3*K3*K4 - 64*K1**3*K3 - 800*K1**2*K2**2 + 96*K1**2*K2*K4**2 - 512*K1**2*K2*K4 + 2904*K1**2*K2 - 624*K1**2*K3**2 - 192*K1**2*K3*K5 - 592*K1**2*K4**2 - 3240*K1**2 - 512*K1*K2**2*K3 - 448*K1*K2*K3*K4 + 3096*K1*K2*K3 - 96*K1*K2*K4*K5 + 2472*K1*K3*K4 + 856*K1*K4*K5 + 24*K1*K5*K6 - 112*K2**2*K4**2 + 872*K2**2*K4 - 2500*K2**2 + 344*K2*K3*K5 + 112*K2*K4*K6 - 1820*K3**2 - 1268*K4**2 - 332*K5**2 - 28*K6**2 + 2922
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {4, 5}, {1, 3}], [{3, 6}, {2, 5}, {1, 4}]]
If K is slice False
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