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Flat knot 6.877

Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,0,2,1,2,2,1,1,1,1,1,1,1,-1,0,1]
Flat knots (up to 7 crossings) with same phi are :['6.877']
Arrow polynomial of the knot is: 4*K1**3 + 4*K1**2*K2 - 8*K1**2 - 6*K1*K2 - 2*K1*K3 - 2*K2**2 + 3*K2 + 2*K3 + K4 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.877']
Outer characteristic polynomial of the knot is: t^7+52t^5+40t^3+5t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.877']
2-strand cable arrow polynomial of the knot is: -64*K1**6 + 864*K1**4*K2 - 2720*K1**4 + 224*K1**3*K2*K3 + 64*K1**3*K3*K4 - 1568*K1**3*K3 + 32*K1**3*K4*K5 - 128*K1**2*K2**4 + 192*K1**2*K2**3 + 160*K1**2*K2**2*K4 - 2816*K1**2*K2**2 + 128*K1**2*K2*K3**2 + 64*K1**2*K2*K4**2 - 448*K1**2*K2*K4 + 7992*K1**2*K2 - 1344*K1**2*K3**2 - 256*K1**2*K3*K5 - 400*K1**2*K4**2 - 96*K1**2*K4*K6 - 32*K1**2*K5**2 - 6076*K1**2 + 448*K1*K2**3*K3 - 608*K1*K2**2*K3 - 64*K1*K2**2*K5 + 64*K1*K2*K3**3 + 32*K1*K2*K3*K4**2 - 864*K1*K2*K3*K4 + 6768*K1*K2*K3 - 64*K1*K3**2*K5 - 32*K1*K3*K4*K6 + 2456*K1*K3*K4 + 888*K1*K4*K5 + 56*K1*K5*K6 + 8*K1*K6*K7 - 32*K2**6 - 64*K2**4*K3**2 - 32*K2**4*K4**2 + 96*K2**4*K4 - 416*K2**4 + 32*K2**3*K3*K5 + 32*K2**3*K4*K6 + 64*K2**2*K3**2*K4 - 672*K2**2*K3**2 + 32*K2**2*K4**3 - 248*K2**2*K4**2 + 1080*K2**2*K4 - 8*K2**2*K6**2 - 4392*K2**2 - 32*K2*K3*K4*K5 + 888*K2*K3*K5 - 32*K2*K4**2*K6 + 184*K2*K4*K6 + 8*K2*K5*K7 + 8*K2*K6*K8 - 64*K3**4 - 48*K3**2*K4**2 + 56*K3**2*K6 - 2648*K3**2 + 48*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 1146*K4**2 - 412*K5**2 - 56*K6**2 - 16*K7**2 - 2*K8**2 + 4930
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.877']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11531', 'vk6.11862', 'vk6.12881', 'vk6.13188', 'vk6.20346', 'vk6.21689', 'vk6.27650', 'vk6.29196', 'vk6.31310', 'vk6.31705', 'vk6.32468', 'vk6.32883', 'vk6.39084', 'vk6.41340', 'vk6.45840', 'vk6.47507', 'vk6.52318', 'vk6.52578', 'vk6.53162', 'vk6.53462', 'vk6.57205', 'vk6.58424', 'vk6.61819', 'vk6.62948', 'vk6.63823', 'vk6.63955', 'vk6.64269', 'vk6.64465', 'vk6.66818', 'vk6.67688', 'vk6.69458', 'vk6.70182']
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 O1O2O3O4U1U5O6O5U4U2U6U3
R3 orbit {'O1O2O3O4U1U5O6O5U4U2U6U3'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U2U5U3U1O6O5U6U4
Gauss code of K* O1O2O3O4U5U2U4U1O5O6U3U6
Gauss code of -K* O1O2O3O4U5U2O5O6U4U1U3U6
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 -1 2 0 1 1],[ 3 0 2 3 1 3 2],[ 1 -2 0 2 0 1 1],[-2 -3 -2 0 -1 -1 0],[ 0 -1 0 1 0 0 0],[-1 -3 -1 1 0 0 1],[-1 -2 -1 0 0 -1 0]]
Primitive based matrix [[ 0 2 1 1 0 -1 -3],[-2 0 0 -1 -1 -2 -3],[-1 0 0 -1 0 -1 -2],[-1 1 1 0 0 -1 -3],[ 0 1 0 0 0 0 -1],[ 1 2 1 1 0 0 -2],[ 3 3 2 3 1 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,-1,0,1,3,0,1,1,2,3,1,0,1,2,0,1,3,0,1,2]
Phi over symmetry [-3,-1,0,1,1,2,0,2,1,2,2,1,1,1,1,1,1,1,-1,0,1]
Phi of -K [-3,-1,0,1,1,2,0,2,1,2,2,1,1,1,1,1,1,1,-1,0,1]
Phi of K* [-2,-1,-1,0,1,3,0,1,1,1,2,1,1,1,1,1,1,2,1,2,0]
Phi of -K* [-3,-1,0,1,1,2,2,1,2,3,3,0,1,1,2,0,0,1,-1,0,1]
Symmetry type of based matrix c
u-polynomial t^3-t^2-t
Normalized Jones-Krushkal polynomial 3z^2+22z+33
Enhanced Jones-Krushkal polynomial 3w^3z^2+22w^2z+33w
Inner characteristic polynomial t^6+36t^4+19t^2
Outer characteristic polynomial t^7+52t^5+40t^3+5t
Flat arrow polynomial 4*K1**3 + 4*K1**2*K2 - 8*K1**2 - 6*K1*K2 - 2*K1*K3 - 2*K2**2 + 3*K2 + 2*K3 + K4 + 5
2-strand cable arrow polynomial -64*K1**6 + 864*K1**4*K2 - 2720*K1**4 + 224*K1**3*K2*K3 + 64*K1**3*K3*K4 - 1568*K1**3*K3 + 32*K1**3*K4*K5 - 128*K1**2*K2**4 + 192*K1**2*K2**3 + 160*K1**2*K2**2*K4 - 2816*K1**2*K2**2 + 128*K1**2*K2*K3**2 + 64*K1**2*K2*K4**2 - 448*K1**2*K2*K4 + 7992*K1**2*K2 - 1344*K1**2*K3**2 - 256*K1**2*K3*K5 - 400*K1**2*K4**2 - 96*K1**2*K4*K6 - 32*K1**2*K5**2 - 6076*K1**2 + 448*K1*K2**3*K3 - 608*K1*K2**2*K3 - 64*K1*K2**2*K5 + 64*K1*K2*K3**3 + 32*K1*K2*K3*K4**2 - 864*K1*K2*K3*K4 + 6768*K1*K2*K3 - 64*K1*K3**2*K5 - 32*K1*K3*K4*K6 + 2456*K1*K3*K4 + 888*K1*K4*K5 + 56*K1*K5*K6 + 8*K1*K6*K7 - 32*K2**6 - 64*K2**4*K3**2 - 32*K2**4*K4**2 + 96*K2**4*K4 - 416*K2**4 + 32*K2**3*K3*K5 + 32*K2**3*K4*K6 + 64*K2**2*K3**2*K4 - 672*K2**2*K3**2 + 32*K2**2*K4**3 - 248*K2**2*K4**2 + 1080*K2**2*K4 - 8*K2**2*K6**2 - 4392*K2**2 - 32*K2*K3*K4*K5 + 888*K2*K3*K5 - 32*K2*K4**2*K6 + 184*K2*K4*K6 + 8*K2*K5*K7 + 8*K2*K6*K8 - 64*K3**4 - 48*K3**2*K4**2 + 56*K3**2*K6 - 2648*K3**2 + 48*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 1146*K4**2 - 412*K5**2 - 56*K6**2 - 16*K7**2 - 2*K8**2 + 4930
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {2, 5}, {4}, {3}], [{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {2, 5}, {4}, {1}], [{4, 6}, {2, 5}, {1, 3}], [{6}, {2, 5}, {4}, {1, 3}]]
If K is slice False
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