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Flat knot 6.75

Min(phi) over symmetries of the knot is: [-4,-2,-1,1,2,4,0,1,4,3,4,0,2,1,2,2,2,3,1,3,1]
Flat knots (up to 7 crossings) with same phi are :['6.75']
Arrow polynomial of the knot is: 12*K1**3 + 4*K1**2*K2 - 4*K1**2 - 8*K1*K2 - 4*K1*K3 - 5*K1 + 2*K2 + K3 + K4 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.75']
Outer characteristic polynomial of the knot is: t^7+121t^5+115t^3+7t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.75']
2-strand cable arrow polynomial of the knot is: -512*K1**4*K2**2 + 736*K1**4*K2 - 1136*K1**4 + 128*K1**3*K2**3*K3 + 1120*K1**3*K2*K3 - 544*K1**3*K3 - 832*K1**2*K2**4 + 2816*K1**2*K2**3 - 384*K1**2*K2**2*K3**2 + 128*K1**2*K2**2*K4 - 9424*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 608*K1**2*K2*K4 + 7864*K1**2*K2 - 1040*K1**2*K3**2 - 32*K1**2*K3*K5 - 4808*K1**2 + 3232*K1*K2**3*K3 + 352*K1*K2**2*K3*K4 - 2848*K1*K2**2*K3 - 640*K1*K2**2*K5 + 288*K1*K2*K3**3 - 608*K1*K2*K3*K4 - 160*K1*K2*K3*K6 + 9560*K1*K2*K3 - 32*K1*K3**2*K5 + 1296*K1*K3*K4 + 80*K1*K4*K5 + 8*K1*K5*K6 - 224*K2**6 - 384*K2**4*K3**2 - 32*K2**4*K4**2 + 832*K2**4*K4 - 4208*K2**4 + 640*K2**3*K3*K5 + 32*K2**3*K4*K6 - 160*K2**3*K6 - 128*K2**2*K3**4 + 32*K2**2*K3**2*K4 + 128*K2**2*K3**2*K6 - 3056*K2**2*K3**2 - 64*K2**2*K3*K7 - 528*K2**2*K4**2 + 3200*K2**2*K4 - 208*K2**2*K5**2 - 48*K2**2*K6**2 - 2570*K2**2 - 128*K2*K3**2*K4 - 32*K2*K3*K4*K5 + 1848*K2*K3*K5 + 208*K2*K4*K6 + 32*K2*K5*K7 + 8*K2*K6*K8 - 160*K3**4 + 176*K3**2*K6 - 2484*K3**2 + 8*K3*K4*K7 - 664*K4**2 - 280*K5**2 - 54*K6**2 - 4*K7**2 - 2*K8**2 + 4208
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.75']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16911', 'vk6.17153', 'vk6.20523', 'vk6.21911', 'vk6.23299', 'vk6.23598', 'vk6.27968', 'vk6.29441', 'vk6.35317', 'vk6.35753', 'vk6.39372', 'vk6.41554', 'vk6.42818', 'vk6.43100', 'vk6.45941', 'vk6.47624', 'vk6.55058', 'vk6.55303', 'vk6.57384', 'vk6.58550', 'vk6.59450', 'vk6.59739', 'vk6.62039', 'vk6.63035', 'vk6.64899', 'vk6.65112', 'vk6.66929', 'vk6.67780', 'vk6.68204', 'vk6.68348', 'vk6.69533', 'vk6.70237']
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 O1O2O3O4O5O6U2U3U6U4U1U5
R3 orbit {'O1O2O3O4O5O6U2U3U6U4U1U5'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5O6U2U6U3U1U4U5
Gauss code of K* O1O2O3O4O5O6U5U1U2U4U6U3
Gauss code of -K* O1O2O3O4O5O6U4U1U3U5U6U2
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 -4 -2 1 4 2],[ 1 0 -3 -1 2 4 2],[ 4 3 0 1 3 4 2],[ 2 1 -1 0 2 3 1],[-1 -2 -3 -2 0 1 0],[-4 -4 -4 -3 -1 0 0],[-2 -2 -2 -1 0 0 0]]
Primitive based matrix [[ 0 4 2 1 -1 -2 -4],[-4 0 0 -1 -4 -3 -4],[-2 0 0 0 -2 -1 -2],[-1 1 0 0 -2 -2 -3],[ 1 4 2 2 0 -1 -3],[ 2 3 1 2 1 0 -1],[ 4 4 2 3 3 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-4,-2,-1,1,2,4,0,1,4,3,4,0,2,1,2,2,2,3,1,3,1]
Phi over symmetry [-4,-2,-1,1,2,4,0,1,4,3,4,0,2,1,2,2,2,3,1,3,1]
Phi of -K [-4,-2,-1,1,2,4,1,0,2,4,4,0,1,3,3,0,1,1,1,2,2]
Phi of K* [-4,-2,-1,1,2,4,2,2,1,3,4,1,1,3,4,0,1,2,0,0,1]
Phi of -K* [-4,-2,-1,1,2,4,1,3,3,2,4,1,2,1,3,2,2,4,0,1,0]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 8z^2+29z+27
Enhanced Jones-Krushkal polynomial 8w^3z^2+29w^2z+27w
Inner characteristic polynomial t^6+79t^4+31t^2+1
Outer characteristic polynomial t^7+121t^5+115t^3+7t
Flat arrow polynomial 12*K1**3 + 4*K1**2*K2 - 4*K1**2 - 8*K1*K2 - 4*K1*K3 - 5*K1 + 2*K2 + K3 + K4 + 2
2-strand cable arrow polynomial -512*K1**4*K2**2 + 736*K1**4*K2 - 1136*K1**4 + 128*K1**3*K2**3*K3 + 1120*K1**3*K2*K3 - 544*K1**3*K3 - 832*K1**2*K2**4 + 2816*K1**2*K2**3 - 384*K1**2*K2**2*K3**2 + 128*K1**2*K2**2*K4 - 9424*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 608*K1**2*K2*K4 + 7864*K1**2*K2 - 1040*K1**2*K3**2 - 32*K1**2*K3*K5 - 4808*K1**2 + 3232*K1*K2**3*K3 + 352*K1*K2**2*K3*K4 - 2848*K1*K2**2*K3 - 640*K1*K2**2*K5 + 288*K1*K2*K3**3 - 608*K1*K2*K3*K4 - 160*K1*K2*K3*K6 + 9560*K1*K2*K3 - 32*K1*K3**2*K5 + 1296*K1*K3*K4 + 80*K1*K4*K5 + 8*K1*K5*K6 - 224*K2**6 - 384*K2**4*K3**2 - 32*K2**4*K4**2 + 832*K2**4*K4 - 4208*K2**4 + 640*K2**3*K3*K5 + 32*K2**3*K4*K6 - 160*K2**3*K6 - 128*K2**2*K3**4 + 32*K2**2*K3**2*K4 + 128*K2**2*K3**2*K6 - 3056*K2**2*K3**2 - 64*K2**2*K3*K7 - 528*K2**2*K4**2 + 3200*K2**2*K4 - 208*K2**2*K5**2 - 48*K2**2*K6**2 - 2570*K2**2 - 128*K2*K3**2*K4 - 32*K2*K3*K4*K5 + 1848*K2*K3*K5 + 208*K2*K4*K6 + 32*K2*K5*K7 + 8*K2*K6*K8 - 160*K3**4 + 176*K3**2*K6 - 2484*K3**2 + 8*K3*K4*K7 - 664*K4**2 - 280*K5**2 - 54*K6**2 - 4*K7**2 - 2*K8**2 + 4208
Genus of based matrix 1
Fillings of based matrix [[{3, 6}, {2, 5}, {1, 4}]]
If K is slice False
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