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Flat knot 6.875

Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,1,1,3,3,2,0,1,2,2,0,0,1,0,1,1]
Flat knots (up to 7 crossings) with same phi are :['6.875']
Arrow polynomial of the knot is: 8*K1**3 + 4*K1**2*K2 - 12*K1**2 - 10*K1*K2 - 2*K1*K3 - K1 - 2*K2**2 + 5*K2 + 3*K3 + K4 + 7
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.875', '6.1163']
Outer characteristic polynomial of the knot is: t^7+52t^5+60t^3+8t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.875']
2-strand cable arrow polynomial of the knot is: -128*K1**6 + 1984*K1**4*K2 - 5024*K1**4 - 256*K1**3*K2**2*K3 - 128*K1**3*K2*K3*K4 + 1088*K1**3*K2*K3 + 224*K1**3*K3*K4 - 1440*K1**3*K3 + 32*K1**3*K4*K5 - 128*K1**2*K2**4 + 640*K1**2*K2**3 + 512*K1**2*K2**2*K4 - 7200*K1**2*K2**2 + 448*K1**2*K2*K3**2 + 128*K1**2*K2*K4**2 - 1024*K1**2*K2*K4 + 12728*K1**2*K2 - 1664*K1**2*K3**2 - 512*K1**2*K4**2 - 64*K1**2*K5**2 - 8056*K1**2 + 704*K1*K2**3*K3 + 160*K1*K2**2*K3*K4 - 2592*K1*K2**2*K3 + 32*K1*K2**2*K4*K5 - 448*K1*K2**2*K5 + 64*K1*K2*K3**3 + 32*K1*K2*K3*K4**2 - 1152*K1*K2*K3*K4 + 11504*K1*K2*K3 + 3144*K1*K3*K4 + 752*K1*K4*K5 + 56*K1*K5*K6 - 64*K2**6 - 64*K2**4*K3**2 - 32*K2**4*K4**2 + 160*K2**4*K4 - 1056*K2**4 + 64*K2**3*K3*K5 + 32*K2**3*K4*K6 - 96*K2**3*K6 + 64*K2**2*K3**2*K4 - 1024*K2**2*K3**2 - 32*K2**2*K3*K7 + 32*K2**2*K4**3 - 376*K2**2*K4**2 + 2960*K2**2*K4 - 32*K2**2*K5**2 - 8*K2**2*K6**2 - 7650*K2**2 - 64*K2*K3**2*K4 - 96*K2*K3*K4*K5 + 1304*K2*K3*K5 - 32*K2*K4**2*K6 + 320*K2*K4*K6 + 48*K2*K5*K7 + 8*K2*K6*K8 - 64*K3**4 - 48*K3**2*K4**2 + 72*K3**2*K6 - 4080*K3**2 + 64*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 1742*K4**2 - 504*K5**2 - 126*K6**2 - 24*K7**2 - 2*K8**2 + 7670
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.875']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4828', 'vk6.5171', 'vk6.6394', 'vk6.6825', 'vk6.8359', 'vk6.8789', 'vk6.9729', 'vk6.10032', 'vk6.11639', 'vk6.11990', 'vk6.12985', 'vk6.20458', 'vk6.20729', 'vk6.21813', 'vk6.27846', 'vk6.29356', 'vk6.31438', 'vk6.32616', 'vk6.39276', 'vk6.39761', 'vk6.41456', 'vk6.46321', 'vk6.47583', 'vk6.47898', 'vk6.49065', 'vk6.49899', 'vk6.51327', 'vk6.51544', 'vk6.53234', 'vk6.57317', 'vk6.62007', 'vk6.64315']
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 O1O2O3O4U1U5O6O5U3U6U4U2
R3 orbit {'O1O2O3O4U1U5O6O5U3U6U4U2'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U3U1U5U2O6O5U6U4
Gauss code of K* O1O2O3O4U5U4U1U3O5O6U2U6
Gauss code of -K* O1O2O3O4U5U3O5O6U2U4U1U6
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 -1 2 1 0],[ 3 0 3 1 2 3 1],[-1 -3 0 -2 1 0 0],[ 1 -1 2 0 2 1 0],[-2 -2 -1 -2 0 -1 -1],[-1 -3 0 -1 1 0 0],[ 0 -1 0 0 1 0 0]]
Primitive based matrix [[ 0 2 1 1 0 -1 -3],[-2 0 -1 -1 -1 -2 -2],[-1 1 0 0 0 -1 -3],[-1 1 0 0 0 -2 -3],[ 0 1 0 0 0 0 -1],[ 1 2 1 2 0 0 -1],[ 3 2 3 3 1 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,-1,0,1,3,1,1,1,2,2,0,0,1,3,0,2,3,0,1,1]
Phi over symmetry [-3,-1,0,1,1,2,1,1,3,3,2,0,1,2,2,0,0,1,0,1,1]
Phi of -K [-3,-1,0,1,1,2,1,2,1,1,3,1,0,1,1,1,1,1,0,0,0]
Phi of K* [-2,-1,-1,0,1,3,0,0,1,1,3,0,1,0,1,1,1,1,1,2,1]
Phi of -K* [-3,-1,0,1,1,2,1,1,3,3,2,0,1,2,2,0,0,1,0,1,1]
Symmetry type of based matrix c
u-polynomial t^3-t^2-t
Normalized Jones-Krushkal polynomial 3z^2+24z+37
Enhanced Jones-Krushkal polynomial 3w^3z^2+24w^2z+37w
Inner characteristic polynomial t^6+36t^4+39t^2+4
Outer characteristic polynomial t^7+52t^5+60t^3+8t
Flat arrow polynomial 8*K1**3 + 4*K1**2*K2 - 12*K1**2 - 10*K1*K2 - 2*K1*K3 - K1 - 2*K2**2 + 5*K2 + 3*K3 + K4 + 7
2-strand cable arrow polynomial -128*K1**6 + 1984*K1**4*K2 - 5024*K1**4 - 256*K1**3*K2**2*K3 - 128*K1**3*K2*K3*K4 + 1088*K1**3*K2*K3 + 224*K1**3*K3*K4 - 1440*K1**3*K3 + 32*K1**3*K4*K5 - 128*K1**2*K2**4 + 640*K1**2*K2**3 + 512*K1**2*K2**2*K4 - 7200*K1**2*K2**2 + 448*K1**2*K2*K3**2 + 128*K1**2*K2*K4**2 - 1024*K1**2*K2*K4 + 12728*K1**2*K2 - 1664*K1**2*K3**2 - 512*K1**2*K4**2 - 64*K1**2*K5**2 - 8056*K1**2 + 704*K1*K2**3*K3 + 160*K1*K2**2*K3*K4 - 2592*K1*K2**2*K3 + 32*K1*K2**2*K4*K5 - 448*K1*K2**2*K5 + 64*K1*K2*K3**3 + 32*K1*K2*K3*K4**2 - 1152*K1*K2*K3*K4 + 11504*K1*K2*K3 + 3144*K1*K3*K4 + 752*K1*K4*K5 + 56*K1*K5*K6 - 64*K2**6 - 64*K2**4*K3**2 - 32*K2**4*K4**2 + 160*K2**4*K4 - 1056*K2**4 + 64*K2**3*K3*K5 + 32*K2**3*K4*K6 - 96*K2**3*K6 + 64*K2**2*K3**2*K4 - 1024*K2**2*K3**2 - 32*K2**2*K3*K7 + 32*K2**2*K4**3 - 376*K2**2*K4**2 + 2960*K2**2*K4 - 32*K2**2*K5**2 - 8*K2**2*K6**2 - 7650*K2**2 - 64*K2*K3**2*K4 - 96*K2*K3*K4*K5 + 1304*K2*K3*K5 - 32*K2*K4**2*K6 + 320*K2*K4*K6 + 48*K2*K5*K7 + 8*K2*K6*K8 - 64*K3**4 - 48*K3**2*K4**2 + 72*K3**2*K6 - 4080*K3**2 + 64*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 1742*K4**2 - 504*K5**2 - 126*K6**2 - 24*K7**2 - 2*K8**2 + 7670
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {4, 5}, {2, 3}], [{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {2, 5}, {1, 3}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {3, 4}, {1, 2}], [{6}, {1, 5}, {4}, {2, 3}], [{6}, {4, 5}, {2, 3}, {1}], [{6}, {5}, {1, 4}, {2, 3}]]
If K is slice False
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