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Flat knot 6.870

Min(phi) over symmetries of the knot is: [-3,-2,1,1,1,2,0,1,2,2,2,1,2,2,1,-1,-1,0,0,0,1]
Flat knots (up to 7 crossings) with same phi are :['6.870']
Arrow polynomial of the knot is: 4*K1**3 - 4*K1**2 - 6*K1*K2 - 4*K1*K3 + 4*K2 + 2*K3 + 2*K4 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.870']
Outer characteristic polynomial of the knot is: t^7+66t^5+110t^3+14t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.870']
2-strand cable arrow polynomial of the knot is: -256*K1**4*K2**2 + 96*K1**4*K2 - 928*K1**4 + 128*K1**3*K2**3*K3 + 1376*K1**3*K2*K3 - 544*K1**3*K3 - 320*K1**2*K2**4 + 192*K1**2*K2**3 - 384*K1**2*K2**2*K3**2 - 6640*K1**2*K2**2 + 128*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 736*K1**2*K2*K4 + 8584*K1**2*K2 - 1120*K1**2*K3**2 - 32*K1**2*K3*K5 - 16*K1**2*K4**2 - 32*K1**2*K5**2 - 7164*K1**2 + 1472*K1*K2**3*K3 - 1504*K1*K2**2*K3 - 896*K1*K2**2*K5 + 224*K1*K2*K3**3 - 320*K1*K2*K3*K4 - 96*K1*K2*K3*K6 + 10376*K1*K2*K3 - 96*K1*K2*K4*K5 - 32*K1*K2*K5*K6 + 1672*K1*K3*K4 + 456*K1*K4*K5 + 168*K1*K5*K6 + 8*K1*K6*K7 - 32*K2**6 + 128*K2**4*K4 - 1776*K2**4 + 32*K2**3*K3*K5 - 96*K2**3*K6 - 1472*K2**2*K3**2 - 32*K2**2*K3*K7 - 88*K2**2*K4**2 + 2752*K2**2*K4 - 80*K2**2*K5**2 - 48*K2**2*K6**2 - 5988*K2**2 + 1800*K2*K3*K5 + 264*K2*K4*K6 + 80*K2*K5*K7 + 32*K2*K6*K8 - 32*K3**4 + 40*K3**2*K6 - 3488*K3**2 - 1160*K4**2 - 632*K5**2 - 140*K6**2 - 12*K7**2 - 4*K8**2 + 6154
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.870']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11439', 'vk6.11734', 'vk6.12753', 'vk6.13096', 'vk6.20322', 'vk6.21664', 'vk6.27625', 'vk6.29170', 'vk6.31194', 'vk6.31533', 'vk6.32362', 'vk6.32777', 'vk6.39052', 'vk6.41313', 'vk6.45808', 'vk6.47483', 'vk6.52204', 'vk6.52465', 'vk6.53035', 'vk6.53355', 'vk6.57181', 'vk6.58392', 'vk6.61794', 'vk6.62916', 'vk6.63774', 'vk6.63884', 'vk6.64202', 'vk6.64388', 'vk6.66796', 'vk6.67665', 'vk6.69435', 'vk6.70158']
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 O1O2O3O4U1U5O6O5U2U4U6U3
R3 orbit {'O1O2O3O4U1U5O6O5U2U4U6U3'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U2U5U1U3O6O5U6U4
Gauss code of K* O1O2O3O4U5U1U4U2O5O6U3U6
Gauss code of -K* O1O2O3O4U5U2O5O6U3U1U4U6
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 2 1 1 1],[ 3 0 1 3 2 3 2],[ 2 -1 0 3 1 2 1],[-2 -3 -3 0 -1 -1 0],[-1 -2 -1 1 0 -1 0],[-1 -3 -2 1 1 0 1],[-1 -2 -1 0 0 -1 0]]
Primitive based matrix [[ 0 2 1 1 1 -2 -3],[-2 0 0 -1 -1 -3 -3],[-1 0 0 0 -1 -1 -2],[-1 1 0 0 -1 -1 -2],[-1 1 1 1 0 -2 -3],[ 2 3 1 1 2 0 -1],[ 3 3 2 2 3 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,-1,-1,2,3,0,1,1,3,3,0,1,1,2,1,1,2,2,3,1]
Phi over symmetry [-3,-2,1,1,1,2,0,1,2,2,2,1,2,2,1,-1,-1,0,0,0,1]
Phi of -K [-3,-2,1,1,1,2,0,1,2,2,2,1,2,2,1,-1,-1,0,0,0,1]
Phi of K* [-2,-1,-1,-1,2,3,0,0,1,1,2,-1,0,2,2,1,1,1,2,2,0]
Phi of -K* [-3,-2,1,1,1,2,1,2,2,3,3,1,1,2,3,0,-1,0,-1,1,1]
Symmetry type of based matrix c
u-polynomial t^3-3t
Normalized Jones-Krushkal polynomial 5z^2+26z+33
Enhanced Jones-Krushkal polynomial 5w^3z^2+26w^2z+33w
Inner characteristic polynomial t^6+46t^4+68t^2+4
Outer characteristic polynomial t^7+66t^5+110t^3+14t
Flat arrow polynomial 4*K1**3 - 4*K1**2 - 6*K1*K2 - 4*K1*K3 + 4*K2 + 2*K3 + 2*K4 + 3
2-strand cable arrow polynomial -256*K1**4*K2**2 + 96*K1**4*K2 - 928*K1**4 + 128*K1**3*K2**3*K3 + 1376*K1**3*K2*K3 - 544*K1**3*K3 - 320*K1**2*K2**4 + 192*K1**2*K2**3 - 384*K1**2*K2**2*K3**2 - 6640*K1**2*K2**2 + 128*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 736*K1**2*K2*K4 + 8584*K1**2*K2 - 1120*K1**2*K3**2 - 32*K1**2*K3*K5 - 16*K1**2*K4**2 - 32*K1**2*K5**2 - 7164*K1**2 + 1472*K1*K2**3*K3 - 1504*K1*K2**2*K3 - 896*K1*K2**2*K5 + 224*K1*K2*K3**3 - 320*K1*K2*K3*K4 - 96*K1*K2*K3*K6 + 10376*K1*K2*K3 - 96*K1*K2*K4*K5 - 32*K1*K2*K5*K6 + 1672*K1*K3*K4 + 456*K1*K4*K5 + 168*K1*K5*K6 + 8*K1*K6*K7 - 32*K2**6 + 128*K2**4*K4 - 1776*K2**4 + 32*K2**3*K3*K5 - 96*K2**3*K6 - 1472*K2**2*K3**2 - 32*K2**2*K3*K7 - 88*K2**2*K4**2 + 2752*K2**2*K4 - 80*K2**2*K5**2 - 48*K2**2*K6**2 - 5988*K2**2 + 1800*K2*K3*K5 + 264*K2*K4*K6 + 80*K2*K5*K7 + 32*K2*K6*K8 - 32*K3**4 + 40*K3**2*K6 - 3488*K3**2 - 1160*K4**2 - 632*K5**2 - 140*K6**2 - 12*K7**2 - 4*K8**2 + 6154
Genus of based matrix 1
Fillings of based matrix [[{3, 6}, {2, 5}, {1, 4}], [{5, 6}, {2, 4}, {1, 3}], [{6}, {2, 5}, {1, 4}, {3}]]
If K is slice False
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