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Flat knot 6.1070

Min(phi) over symmetries of the knot is: [-3,0,0,1,1,1,0,1,2,2,3,1,0,0,1,0,0,0,0,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.1070']
Arrow polynomial of the knot is: 8*K1**3 - 8*K1**2 - 6*K1*K2 - 3*K1 + 4*K2 + K3 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.238', '6.431', '6.945', '6.977', '6.981', '6.997', '6.1050', '6.1070', '6.1098', '6.1376']
Outer characteristic polynomial of the knot is: t^7+32t^5+48t^3
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1070']
2-strand cable arrow polynomial of the knot is: -128*K1**4*K2**2 + 448*K1**4*K2 - 768*K1**4 + 320*K1**2*K2**3 - 1792*K1**2*K2**2 + 2256*K1**2*K2 - 64*K1**2*K3**2 - 16*K1**2*K4**2 - 1632*K1**2 + 96*K1*K2**3*K3 + 1656*K1*K2*K3 + 336*K1*K3*K4 + 32*K1*K4*K5 + 8*K1*K5*K6 - 64*K2**6 + 96*K2**4*K4 - 832*K2**4 - 288*K2**2*K3**2 - 72*K2**2*K4**2 + 688*K2**2*K4 - 1026*K2**2 + 208*K2*K3*K5 + 40*K2*K4*K6 - 612*K3**2 - 348*K4**2 - 60*K5**2 - 14*K6**2 + 1530
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1070']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11228', 'vk6.11307', 'vk6.12493', 'vk6.12604', 'vk6.18231', 'vk6.18566', 'vk6.24698', 'vk6.25113', 'vk6.30898', 'vk6.31021', 'vk6.32086', 'vk6.32205', 'vk6.36825', 'vk6.37288', 'vk6.44066', 'vk6.44405', 'vk6.51974', 'vk6.52069', 'vk6.52859', 'vk6.52906', 'vk6.56024', 'vk6.56298', 'vk6.60568', 'vk6.60907', 'vk6.63630', 'vk6.63675', 'vk6.64062', 'vk6.64107', 'vk6.65689', 'vk6.65981', 'vk6.68735', 'vk6.68943', 'vk6.73771', 'vk6.73910', 'vk6.78704', 'vk6.78748', 'vk6.78904', 'vk6.78906', 'vk6.84453', 'vk6.88355']
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 O1O2O3O4U5U3O6U2O5U6U1U4
R3 orbit {'O1O2O3O4U5U3U1O6O5U2U6U4', 'O1O2O3O4U5U3O6U2O5U6U1U4'}
R3 orbit length 2
Gauss code of -K O1O2O3O4U1U4U5O6U3O5U2U6
Gauss code of K* O1O2O3U2U4U5U3O6O5U1O4U6
Gauss code of -K* O1O2O3U4O5U3O6O4U1U6U5U2
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 -1 0 3 -1 0],[ 1 0 0 1 3 0 0],[ 1 0 0 0 2 0 0],[ 0 -1 0 0 0 0 -1],[-3 -3 -2 0 0 -2 -1],[ 1 0 0 0 2 0 0],[ 0 0 0 1 1 0 0]]
Primitive based matrix [[ 0 3 0 0 -1 -1 -1],[-3 0 0 -1 -2 -2 -3],[ 0 0 0 -1 0 0 -1],[ 0 1 1 0 0 0 0],[ 1 2 0 0 0 0 0],[ 1 2 0 0 0 0 0],[ 1 3 1 0 0 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,0,0,1,1,1,0,1,2,2,3,1,0,0,1,0,0,0,0,0,0]
Phi over symmetry [-3,0,0,1,1,1,0,1,2,2,3,1,0,0,1,0,0,0,0,0,0]
Phi of -K [-1,-1,-1,0,0,3,0,0,0,1,1,0,1,1,2,1,1,2,1,3,2]
Phi of K* [-3,0,0,1,1,1,2,3,1,2,2,1,1,1,1,0,1,1,0,0,0]
Phi of -K* [-1,-1,-1,0,0,3,0,0,0,0,2,0,0,0,2,0,1,3,1,1,0]
Symmetry type of based matrix c
u-polynomial -t^3+3t
Normalized Jones-Krushkal polynomial 8z+17
Enhanced Jones-Krushkal polynomial -4w^3z+12w^2z+17w
Inner characteristic polynomial t^6+20t^4+20t^2
Outer characteristic polynomial t^7+32t^5+48t^3
Flat arrow polynomial 8*K1**3 - 8*K1**2 - 6*K1*K2 - 3*K1 + 4*K2 + K3 + 5
2-strand cable arrow polynomial -128*K1**4*K2**2 + 448*K1**4*K2 - 768*K1**4 + 320*K1**2*K2**3 - 1792*K1**2*K2**2 + 2256*K1**2*K2 - 64*K1**2*K3**2 - 16*K1**2*K4**2 - 1632*K1**2 + 96*K1*K2**3*K3 + 1656*K1*K2*K3 + 336*K1*K3*K4 + 32*K1*K4*K5 + 8*K1*K5*K6 - 64*K2**6 + 96*K2**4*K4 - 832*K2**4 - 288*K2**2*K3**2 - 72*K2**2*K4**2 + 688*K2**2*K4 - 1026*K2**2 + 208*K2*K3*K5 + 40*K2*K4*K6 - 612*K3**2 - 348*K4**2 - 60*K5**2 - 14*K6**2 + 1530
Genus of based matrix 1
Fillings of based matrix [[{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {4, 5}, {1, 2}], [{3, 6}, {5}, {1, 4}, {2}], [{4, 6}, {2, 5}, {1, 3}], [{4, 6}, {5}, {1, 3}, {2}]]
If K is slice False
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