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Flat knot 6.88

Min(phi) over symmetries of the knot is: [-4,-1,-1,0,3,3,0,1,3,3,4,0,1,1,1,1,2,2,2,2,-1]
Flat knots (up to 7 crossings) with same phi are :['6.88']
Arrow polynomial of the knot is: -4*K1*K2 + 2*K1 - 2*K2**2 + 2*K3 + K4 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.87', '6.88', '6.184', '6.302', '6.459', '6.467', '6.506']
Outer characteristic polynomial of the knot is: t^7+104t^5+20t^3
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.88']
2-strand cable arrow polynomial of the knot is: -112*K1**2*K3**2 - 48*K1**2*K4**2 - 432*K1**2 + 392*K1*K2*K3 + 544*K1*K3*K4 + 104*K1*K4*K5 - 16*K2**2*K4**2 + 128*K2**2*K4 - 320*K2**2 + 48*K2*K3*K5 + 16*K2*K4*K6 + 8*K2*K5*K7 - 16*K3**4 - 32*K3**2*K4**2 + 16*K3**2*K6 - 420*K3**2 + 24*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 332*K4**2 - 68*K5**2 - 8*K6**2 - 8*K7**2 - 2*K8**2 + 532
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.88']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11437', 'vk6.11732', 'vk6.12747', 'vk6.13090', 'vk6.20339', 'vk6.21680', 'vk6.27639', 'vk6.29183', 'vk6.31184', 'vk6.31525', 'vk6.32348', 'vk6.32765', 'vk6.39067', 'vk6.41325', 'vk6.45819', 'vk6.47490', 'vk6.52190', 'vk6.52447', 'vk6.53017', 'vk6.53333', 'vk6.57198', 'vk6.58413', 'vk6.61808', 'vk6.62933', 'vk6.63756', 'vk6.63866', 'vk6.64180', 'vk6.64366', 'vk6.66807', 'vk6.67675', 'vk6.69443', 'vk6.70165', 'vk6.73319', 'vk6.75211', 'vk6.78200', 'vk6.79576', 'vk6.80019', 'vk6.80997', 'vk6.82361', 'vk6.82736', 'vk6.84306', 'vk6.84321', 'vk6.84365', 'vk6.85219', 'vk6.86758', 'vk6.87582', 'vk6.88279', 'vk6.88714']
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 O1O2O3O4O5O6U2U5U3U6U1U4
R3 orbit {'O1O2O3O4O5U1O6U5U3U2U6U4', 'O1O2O3O4O5U1U4O6U3U2U5U6', 'O1O2O3O4O5O6U2U5U3U6U1U4'}
R3 orbit length 3
Gauss code of -K O1O2O3O4O5O6U3U6U1U4U2U5
Gauss code of K* O1O2O3O4O5O6U5U1U3U6U2U4
Gauss code of -K* O1O2O3O4O5O6U3U5U1U4U6U2
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 -1 -4 -1 3 0 3],[ 1 0 -3 0 3 0 3],[ 4 3 0 2 4 1 3],[ 1 0 -2 0 2 0 2],[-3 -3 -4 -2 0 -1 1],[ 0 0 -1 0 1 0 1],[-3 -3 -3 -2 -1 -1 0]]
Primitive based matrix [[ 0 3 3 0 -1 -1 -4],[-3 0 1 -1 -2 -3 -4],[-3 -1 0 -1 -2 -3 -3],[ 0 1 1 0 0 0 -1],[ 1 2 2 0 0 0 -2],[ 1 3 3 0 0 0 -3],[ 4 4 3 1 2 3 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-3,0,1,1,4,-1,1,2,3,4,1,2,3,3,0,0,1,0,2,3]
Phi over symmetry [-4,-1,-1,0,3,3,0,1,3,3,4,0,1,1,1,1,2,2,2,2,-1]
Phi of -K [-4,-1,-1,0,3,3,0,1,3,3,4,0,1,1,1,1,2,2,2,2,-1]
Phi of K* [-3,-3,0,1,1,4,-1,2,1,2,4,2,1,2,3,1,1,3,0,0,1]
Phi of -K* [-4,-1,-1,0,3,3,2,3,1,3,4,0,0,2,2,0,3,3,1,1,-1]
Symmetry type of based matrix c
u-polynomial t^4-2t^3+2t
Normalized Jones-Krushkal polynomial 5z+11
Enhanced Jones-Krushkal polynomial -4w^3z+9w^2z+11w
Inner characteristic polynomial t^6+68t^4
Outer characteristic polynomial t^7+104t^5+20t^3
Flat arrow polynomial -4*K1*K2 + 2*K1 - 2*K2**2 + 2*K3 + K4 + 2
2-strand cable arrow polynomial -112*K1**2*K3**2 - 48*K1**2*K4**2 - 432*K1**2 + 392*K1*K2*K3 + 544*K1*K3*K4 + 104*K1*K4*K5 - 16*K2**2*K4**2 + 128*K2**2*K4 - 320*K2**2 + 48*K2*K3*K5 + 16*K2*K4*K6 + 8*K2*K5*K7 - 16*K3**4 - 32*K3**2*K4**2 + 16*K3**2*K6 - 420*K3**2 + 24*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 332*K4**2 - 68*K5**2 - 8*K6**2 - 8*K7**2 - 2*K8**2 + 532
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {3, 5}, {4}, {1}], [{3, 6}, {1, 5}, {2, 4}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {4}, {2, 3}, {1}]]
If K is slice False
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