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Flat knot 6.1247

Min(phi) over symmetries of the knot is: [-2,-1,0,0,1,2,-1,1,1,1,2,0,1,0,1,1,1,2,0,2,1]
Flat knots (up to 7 crossings) with same phi are :['6.1247']
Arrow polynomial of the knot is: 8*K1**3 + 4*K1**2*K2 - 8*K1**2 - 4*K1*K2 - 4*K1*K3 - 4*K1 + 4*K2 + K4 + 4
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.1247', '6.1262']
Outer characteristic polynomial of the knot is: t^7+31t^5+75t^3
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1247']
2-strand cable arrow polynomial of the knot is: -192*K1**4*K2**2 + 224*K1**4*K2 - 864*K1**4 + 128*K1**3*K2**3*K3 + 320*K1**3*K2*K3 - 512*K1**2*K2**4 + 512*K1**2*K2**3 - 192*K1**2*K2**2*K3**2 - 2832*K1**2*K2**2 + 2552*K1**2*K2 - 320*K1**2*K3**2 - 32*K1**2*K4**2 - 2008*K1**2 + 1408*K1*K2**3*K3 + 96*K1*K2*K3**3 + 3504*K1*K2*K3 + 416*K1*K3*K4 + 64*K1*K4*K5 + 24*K1*K5*K6 - 192*K2**6 - 256*K2**4*K3**2 - 32*K2**4*K4**2 + 192*K2**4*K4 - 1184*K2**4 + 224*K2**3*K3*K5 + 64*K2**3*K4*K6 - 1456*K2**2*K3**2 - 144*K2**2*K4**2 + 568*K2**2*K4 - 64*K2**2*K5**2 - 48*K2**2*K6**2 - 1068*K2**2 + 616*K2*K3*K5 + 120*K2*K4*K6 + 8*K2*K5*K7 + 16*K2*K6*K8 - 32*K3**4 + 48*K3**2*K6 - 1268*K3**2 - 348*K4**2 - 116*K5**2 - 68*K6**2 - 2*K8**2 + 2124
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1247']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.3612', 'vk6.3681', 'vk6.3872', 'vk6.3995', 'vk6.7038', 'vk6.7077', 'vk6.7252', 'vk6.7369', 'vk6.17705', 'vk6.17752', 'vk6.24256', 'vk6.24315', 'vk6.36555', 'vk6.36630', 'vk6.43665', 'vk6.43770', 'vk6.48240', 'vk6.48305', 'vk6.48388', 'vk6.48419', 'vk6.50000', 'vk6.50031', 'vk6.50114', 'vk6.50143', 'vk6.55737', 'vk6.55792', 'vk6.60313', 'vk6.60394', 'vk6.65437', 'vk6.65464', 'vk6.68569', 'vk6.68596']
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 O1O2O3O4U3U5U4O6O5U1U2U6
R3 orbit {'O1O2O3O4U3U5U4O6O5U1U2U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U5U3U4O6O5U1U6U2
Gauss code of K* O1O2O3U4U2O5O6O4U5U6U1U3
Gauss code of -K* O1O2O3U4U1O5O4O6U5U6U2U3
Diagrammatic symmetry type c
Flat genus of the diagram 2
If K is checkerboard colorable False
If K is almost classical False
Based matrix from Gauss code [[ 0 -2 0 -1 2 0 1],[ 2 0 1 -1 2 1 1],[ 0 -1 0 -1 2 -1 0],[ 1 1 1 0 1 0 0],[-2 -2 -2 -1 0 -2 -1],[ 0 -1 1 0 2 0 1],[-1 -1 0 0 1 -1 0]]
Primitive based matrix [[ 0 2 1 0 0 -1 -2],[-2 0 -1 -2 -2 -1 -2],[-1 1 0 0 -1 0 -1],[ 0 2 0 0 -1 -1 -1],[ 0 2 1 1 0 0 -1],[ 1 1 0 1 0 0 1],[ 2 2 1 1 1 -1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,0,0,1,2,1,2,2,1,2,0,1,0,1,1,1,1,0,1,-1]
Phi over symmetry [-2,-1,0,0,1,2,-1,1,1,1,2,0,1,0,1,1,1,2,0,2,1]
Phi of -K [-2,-1,0,0,1,2,2,1,1,2,2,0,1,2,2,1,1,0,0,0,0]
Phi of K* [-2,-1,0,0,1,2,0,0,0,2,2,0,1,2,2,1,1,1,0,1,2]
Phi of -K* [-2,-1,0,0,1,2,-1,1,1,1,2,0,1,0,1,1,1,2,0,2,1]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 11z+23
Enhanced Jones-Krushkal polynomial -6w^3z+17w^2z+23w
Inner characteristic polynomial t^6+21t^4+31t^2
Outer characteristic polynomial t^7+31t^5+75t^3
Flat arrow polynomial 8*K1**3 + 4*K1**2*K2 - 8*K1**2 - 4*K1*K2 - 4*K1*K3 - 4*K1 + 4*K2 + K4 + 4
2-strand cable arrow polynomial -192*K1**4*K2**2 + 224*K1**4*K2 - 864*K1**4 + 128*K1**3*K2**3*K3 + 320*K1**3*K2*K3 - 512*K1**2*K2**4 + 512*K1**2*K2**3 - 192*K1**2*K2**2*K3**2 - 2832*K1**2*K2**2 + 2552*K1**2*K2 - 320*K1**2*K3**2 - 32*K1**2*K4**2 - 2008*K1**2 + 1408*K1*K2**3*K3 + 96*K1*K2*K3**3 + 3504*K1*K2*K3 + 416*K1*K3*K4 + 64*K1*K4*K5 + 24*K1*K5*K6 - 192*K2**6 - 256*K2**4*K3**2 - 32*K2**4*K4**2 + 192*K2**4*K4 - 1184*K2**4 + 224*K2**3*K3*K5 + 64*K2**3*K4*K6 - 1456*K2**2*K3**2 - 144*K2**2*K4**2 + 568*K2**2*K4 - 64*K2**2*K5**2 - 48*K2**2*K6**2 - 1068*K2**2 + 616*K2*K3*K5 + 120*K2*K4*K6 + 8*K2*K5*K7 + 16*K2*K6*K8 - 32*K3**4 + 48*K3**2*K6 - 1268*K3**2 - 348*K4**2 - 116*K5**2 - 68*K6**2 - 2*K8**2 + 2124
Genus of based matrix 1
Fillings of based matrix [[{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {5}, {1, 4}, {2}]]
If K is slice False
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