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Flat knot 6.85

Min(phi) over symmetries of the knot is: [-4,-3,0,1,2,4,0,3,3,2,5,2,2,1,3,1,1,3,0,1,0]
Flat knots (up to 7 crossings) with same phi are :['6.85']
Arrow polynomial of the knot is: 8*K1**3 + 4*K1**2*K2 - 10*K1**2 - 6*K1*K2 - 4*K1*K3 - 3*K1 + 5*K2 + K3 + K4 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.85']
Outer characteristic polynomial of the knot is: t^7+125t^5+94t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.28', '6.85']
2-strand cable arrow polynomial of the knot is: 96*K1**4*K2 - 720*K1**4 - 32*K1**3*K3 - 128*K1**2*K2**4 + 576*K1**2*K2**3 - 3376*K1**2*K2**2 - 288*K1**2*K2*K4 + 5976*K1**2*K2 - 192*K1**2*K3**2 - 32*K1**2*K3*K5 - 16*K1**2*K4**2 - 5412*K1**2 + 768*K1*K2**3*K3 + 96*K1*K2**2*K3*K4 - 1664*K1*K2**2*K3 + 32*K1*K2**2*K4*K5 - 256*K1*K2**2*K5 + 32*K1*K2*K3**3 - 448*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 6200*K1*K2*K3 - 32*K1*K2*K4*K5 + 1400*K1*K3*K4 + 296*K1*K4*K5 + 32*K1*K5*K6 + 8*K1*K7*K8 - 64*K2**6 - 128*K2**4*K3**2 - 32*K2**4*K4**2 + 224*K2**4*K4 - 1384*K2**4 + 128*K2**3*K3*K5 + 32*K2**3*K4*K6 - 160*K2**3*K6 + 96*K2**2*K3**2*K4 - 1120*K2**2*K3**2 - 64*K2**2*K3*K7 - 200*K2**2*K4**2 + 2320*K2**2*K4 - 64*K2**2*K5**2 - 16*K2**2*K6**2 - 4382*K2**2 - 64*K2*K3**2*K4 - 64*K2*K3*K4*K5 + 1096*K2*K3*K5 + 232*K2*K4*K6 + 64*K2*K5*K7 + 16*K2*K6*K8 - 48*K3**4 - 32*K3**2*K4**2 + 88*K3**2*K6 - 2456*K3**2 + 40*K3*K4*K7 - 1058*K4**2 - 332*K5**2 - 90*K6**2 - 24*K7**2 - 10*K8**2 + 4570
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.85']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.73356', 'vk6.73387', 'vk6.73518', 'vk6.73564', 'vk6.73715', 'vk6.73832', 'vk6.74268', 'vk6.74891', 'vk6.75325', 'vk6.75524', 'vk6.75832', 'vk6.76441', 'vk6.78240', 'vk6.78305', 'vk6.78486', 'vk6.78627', 'vk6.78820', 'vk6.79312', 'vk6.80060', 'vk6.80089', 'vk6.80209', 'vk6.80257', 'vk6.80393', 'vk6.80773', 'vk6.81961', 'vk6.82688', 'vk6.84752', 'vk6.85048', 'vk6.85141', 'vk6.86514', 'vk6.87339', 'vk6.89433']
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 O1O2O3O4O5O6U2U5U1U4U6U3
R3 orbit {'O1O2O3O4O5O6U2U5U1U4U6U3'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5O6U4U1U3U6U2U5
Gauss code of K* O1O2O3O4O5O6U3U1U6U4U2U5
Gauss code of -K* O1O2O3O4O5O6U2U5U3U1U6U4
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 -4 2 1 0 4],[ 3 0 -1 4 2 1 4],[ 4 1 0 4 2 1 3],[-2 -4 -4 0 -1 -1 2],[-1 -2 -2 1 0 0 2],[ 0 -1 -1 1 0 0 1],[-4 -4 -3 -2 -2 -1 0]]
Primitive based matrix [[ 0 4 2 1 0 -3 -4],[-4 0 -2 -2 -1 -4 -3],[-2 2 0 -1 -1 -4 -4],[-1 2 1 0 0 -2 -2],[ 0 1 1 0 0 -1 -1],[ 3 4 4 2 1 0 -1],[ 4 3 4 2 1 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-4,-2,-1,0,3,4,2,2,1,4,3,1,1,4,4,0,2,2,1,1,1]
Phi over symmetry [-4,-3,0,1,2,4,0,3,3,2,5,2,2,1,3,1,1,3,0,1,0]
Phi of -K [-4,-3,0,1,2,4,0,3,3,2,5,2,2,1,3,1,1,3,0,1,0]
Phi of K* [-4,-2,-1,0,3,4,0,1,3,3,5,0,1,1,2,1,2,3,2,3,0]
Phi of -K* [-4,-3,0,1,2,4,1,1,2,4,3,1,2,4,4,0,1,1,1,2,2]
Symmetry type of based matrix c
u-polynomial t^3-t^2-t
Normalized Jones-Krushkal polynomial 3z^2+20z+29
Enhanced Jones-Krushkal polynomial 3w^3z^2+20w^2z+29w
Inner characteristic polynomial t^6+79t^4+15t^2
Outer characteristic polynomial t^7+125t^5+94t^3+4t
Flat arrow polynomial 8*K1**3 + 4*K1**2*K2 - 10*K1**2 - 6*K1*K2 - 4*K1*K3 - 3*K1 + 5*K2 + K3 + K4 + 5
2-strand cable arrow polynomial 96*K1**4*K2 - 720*K1**4 - 32*K1**3*K3 - 128*K1**2*K2**4 + 576*K1**2*K2**3 - 3376*K1**2*K2**2 - 288*K1**2*K2*K4 + 5976*K1**2*K2 - 192*K1**2*K3**2 - 32*K1**2*K3*K5 - 16*K1**2*K4**2 - 5412*K1**2 + 768*K1*K2**3*K3 + 96*K1*K2**2*K3*K4 - 1664*K1*K2**2*K3 + 32*K1*K2**2*K4*K5 - 256*K1*K2**2*K5 + 32*K1*K2*K3**3 - 448*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 6200*K1*K2*K3 - 32*K1*K2*K4*K5 + 1400*K1*K3*K4 + 296*K1*K4*K5 + 32*K1*K5*K6 + 8*K1*K7*K8 - 64*K2**6 - 128*K2**4*K3**2 - 32*K2**4*K4**2 + 224*K2**4*K4 - 1384*K2**4 + 128*K2**3*K3*K5 + 32*K2**3*K4*K6 - 160*K2**3*K6 + 96*K2**2*K3**2*K4 - 1120*K2**2*K3**2 - 64*K2**2*K3*K7 - 200*K2**2*K4**2 + 2320*K2**2*K4 - 64*K2**2*K5**2 - 16*K2**2*K6**2 - 4382*K2**2 - 64*K2*K3**2*K4 - 64*K2*K3*K4*K5 + 1096*K2*K3*K5 + 232*K2*K4*K6 + 64*K2*K5*K7 + 16*K2*K6*K8 - 48*K3**4 - 32*K3**2*K4**2 + 88*K3**2*K6 - 2456*K3**2 + 40*K3*K4*K7 - 1058*K4**2 - 332*K5**2 - 90*K6**2 - 24*K7**2 - 10*K8**2 + 4570
Genus of based matrix 1
Fillings of based matrix [[{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {5}, {2, 4}, {1}], [{5, 6}, {3, 4}, {1, 2}]]
If K is slice False
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