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Flat knot 6.1686

Min(phi) over symmetries of the knot is: [-2,-1,-1,1,1,2,-1,0,2,2,2,0,0,2,1,1,2,2,-1,1,1]
Flat knots (up to 7 crossings) with same phi are :['6.1686']
Arrow polynomial of the knot is: 4*K1**3 - 4*K1*K2 - K1 + K3 + 1
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.395', '6.430', '6.440', '6.548', '6.551', '6.774', '6.832', '6.887', '6.908', '6.911', '6.1205', '6.1332', '6.1339', '6.1341', '6.1346', '6.1382', '6.1488', '6.1651', '6.1655', '6.1686']
Outer characteristic polynomial of the knot is: t^7+42t^5+111t^3+16t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1686']
2-strand cable arrow polynomial of the knot is: -448*K1**2*K2**4 + 1120*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 6384*K1**2*K2**2 - 480*K1**2*K2*K4 + 5864*K1**2*K2 - 64*K1**2*K4**2 - 4232*K1**2 + 1056*K1*K2**3*K3 - 1216*K1*K2**2*K3 - 768*K1*K2**2*K5 - 192*K1*K2*K3*K4 + 6240*K1*K2*K3 + 608*K1*K3*K4 + 288*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 1376*K2**4 - 32*K2**3*K6 - 368*K2**2*K3**2 - 16*K2**2*K4**2 + 1792*K2**2*K4 - 2974*K2**2 + 568*K2*K3*K5 + 16*K2*K4*K6 - 1520*K3**2 - 548*K4**2 - 200*K5**2 - 2*K6**2 + 3106
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1686']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16999', 'vk6.17242', 'vk6.20528', 'vk6.21924', 'vk6.23408', 'vk6.23717', 'vk6.27980', 'vk6.29449', 'vk6.35471', 'vk6.35918', 'vk6.39386', 'vk6.41575', 'vk6.42907', 'vk6.43208', 'vk6.45961', 'vk6.47638', 'vk6.55168', 'vk6.55414', 'vk6.57398', 'vk6.58573', 'vk6.59550', 'vk6.59890', 'vk6.62065', 'vk6.63050', 'vk6.64972', 'vk6.65180', 'vk6.66944', 'vk6.67803', 'vk6.68264', 'vk6.68420', 'vk6.69556', 'vk6.70252']
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 O1O2O3U4O5U2U1U6O4O6U5U3
R3 orbit {'O1O2O3U4O5U2U1U6O4O6U5U3'}
R3 orbit length 1
Gauss code of -K O1O2O3U1U4O5O6U5U3U2O4U6
Gauss code of K* O1O2O3U4U3O5O6U2U1U6O4U5
Gauss code of -K* O1O2O3U4O5U6U3U2O6O4U1U5
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 -1 -1 2 -2 1 1],[ 1 0 0 2 0 1 2],[ 1 0 0 1 1 0 2],[-2 -2 -1 0 -2 -1 -1],[ 2 0 -1 2 0 2 2],[-1 -1 0 1 -2 0 -1],[-1 -2 -2 1 -2 1 0]]
Primitive based matrix [[ 0 2 1 1 -1 -1 -2],[-2 0 -1 -1 -1 -2 -2],[-1 1 0 1 -2 -2 -2],[-1 1 -1 0 0 -1 -2],[ 1 1 2 0 0 0 1],[ 1 2 2 1 0 0 0],[ 2 2 2 2 -1 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,-1,1,1,2,1,1,1,2,2,-1,2,2,2,0,1,2,0,-1,0]
Phi over symmetry [-2,-1,-1,1,1,2,-1,0,2,2,2,0,0,2,1,1,2,2,-1,1,1]
Phi of -K [-2,-1,-1,1,1,2,1,2,1,1,2,0,0,1,1,0,2,2,-1,0,0]
Phi of K* [-2,-1,-1,1,1,2,0,0,1,2,2,-1,1,2,1,0,0,1,0,1,2]
Phi of -K* [-2,-1,-1,1,1,2,-1,0,2,2,2,0,0,2,1,1,2,2,-1,1,1]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 6z^2+19z+15
Enhanced Jones-Krushkal polynomial -2w^4z^2+8w^3z^2-8w^3z+27w^2z+15w
Inner characteristic polynomial t^6+30t^4+67t^2+1
Outer characteristic polynomial t^7+42t^5+111t^3+16t
Flat arrow polynomial 4*K1**3 - 4*K1*K2 - K1 + K3 + 1
2-strand cable arrow polynomial -448*K1**2*K2**4 + 1120*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 6384*K1**2*K2**2 - 480*K1**2*K2*K4 + 5864*K1**2*K2 - 64*K1**2*K4**2 - 4232*K1**2 + 1056*K1*K2**3*K3 - 1216*K1*K2**2*K3 - 768*K1*K2**2*K5 - 192*K1*K2*K3*K4 + 6240*K1*K2*K3 + 608*K1*K3*K4 + 288*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 1376*K2**4 - 32*K2**3*K6 - 368*K2**2*K3**2 - 16*K2**2*K4**2 + 1792*K2**2*K4 - 2974*K2**2 + 568*K2*K3*K5 + 16*K2*K4*K6 - 1520*K3**2 - 548*K4**2 - 200*K5**2 - 2*K6**2 + 3106
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {4, 5}, {1, 3}]]
If K is slice False
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