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Flat knot 6.959

Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,1,1,3,3,2,1,1,2,1,0,0,0,1,2,0]
Flat knots (up to 7 crossings) with same phi are :['6.959']
Arrow polynomial of the knot is: 4*K1**3 + 4*K1**2*K2 - 4*K1**2 - 6*K1*K2 - 2*K1*K3 - 2*K2**2 + K2 + 2*K3 + K4 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.959', '6.1144']
Outer characteristic polynomial of the knot is: t^7+46t^5+93t^3+5t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.959']
2-strand cable arrow polynomial of the knot is: -208*K1**4 + 608*K1**3*K2*K3 - 224*K1**3*K3 - 128*K1**2*K2**4 + 416*K1**2*K2**3 - 448*K1**2*K2**2*K3**2 + 160*K1**2*K2**2*K4 - 3760*K1**2*K2**2 + 192*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 416*K1**2*K2*K4 + 4288*K1**2*K2 - 512*K1**2*K3**2 - 48*K1**2*K4**2 - 3556*K1**2 + 1536*K1*K2**3*K3 + 512*K1*K2**2*K3*K4 - 1792*K1*K2**2*K3 + 32*K1*K2**2*K4*K5 - 192*K1*K2**2*K5 + 96*K1*K2*K3**3 - 736*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 5240*K1*K2*K3 + 1336*K1*K3*K4 + 200*K1*K4*K5 + 8*K1*K5*K6 - 32*K2**6 - 128*K2**4*K3**2 - 32*K2**4*K4**2 + 160*K2**4*K4 - 1120*K2**4 + 64*K2**3*K3*K5 + 32*K2**3*K4*K6 - 64*K2**3*K6 + 128*K2**2*K3**2*K4 - 1280*K2**2*K3**2 - 32*K2**2*K3*K7 + 32*K2**2*K4**3 - 280*K2**2*K4**2 + 1784*K2**2*K4 - 16*K2**2*K5**2 - 8*K2**2*K6**2 - 2664*K2**2 - 32*K2*K3**2*K4 - 64*K2*K3*K4*K5 + 720*K2*K3*K5 - 32*K2*K4**2*K6 + 96*K2*K4*K6 + 16*K2*K5*K7 + 8*K2*K6*K8 - 16*K3**4 - 32*K3**2*K4**2 + 16*K3**2*K6 - 1820*K3**2 + 24*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 814*K4**2 - 148*K5**2 - 32*K6**2 - 4*K7**2 - 2*K8**2 + 2966
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.959']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4194', 'vk6.4273', 'vk6.5444', 'vk6.5558', 'vk6.7553', 'vk6.7636', 'vk6.9059', 'vk6.9138', 'vk6.18241', 'vk6.18578', 'vk6.24713', 'vk6.25128', 'vk6.36836', 'vk6.37301', 'vk6.44072', 'vk6.44413', 'vk6.48514', 'vk6.48593', 'vk6.49210', 'vk6.49314', 'vk6.50301', 'vk6.50377', 'vk6.51064', 'vk6.51095', 'vk6.56036', 'vk6.56312', 'vk6.60585', 'vk6.60926', 'vk6.65698', 'vk6.65994', 'vk6.68743', 'vk6.68953']
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 O1O2O3O4U5U2O6O5U6U1U3U4
R3 orbit {'O1O2O3O4U5U2O6O5U6U1U3U4'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U1U2U4U5O6O5U3U6
Gauss code of K* O1O2O3O4U2U5U3U4O6O5U1U6
Gauss code of -K* O1O2O3O4U5U4O6O5U1U2U6U3
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 -2 -1 1 3 0 -1],[ 2 0 1 2 3 2 -1],[ 1 -1 0 0 1 1 -1],[-1 -2 0 0 1 0 -1],[-3 -3 -1 -1 0 -2 -1],[ 0 -2 -1 0 2 0 -1],[ 1 1 1 1 1 1 0]]
Primitive based matrix [[ 0 3 1 0 -1 -1 -2],[-3 0 -1 -2 -1 -1 -3],[-1 1 0 0 0 -1 -2],[ 0 2 0 0 -1 -1 -2],[ 1 1 0 1 0 -1 -1],[ 1 1 1 1 1 0 1],[ 2 3 2 2 1 -1 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-1,0,1,1,2,1,2,1,1,3,0,0,1,2,1,1,2,1,1,-1]
Phi over symmetry [-3,-1,0,1,1,2,1,1,3,3,2,1,1,2,1,0,0,0,1,2,0]
Phi of -K [-2,-1,-1,0,1,3,0,2,0,1,2,1,0,2,3,0,1,3,1,1,1]
Phi of K* [-3,-1,0,1,1,2,1,1,3,3,2,1,1,2,1,0,0,0,1,2,0]
Phi of -K* [-2,-1,-1,0,1,3,-1,1,2,2,3,1,1,1,1,1,0,1,0,2,1]
Symmetry type of based matrix c
u-polynomial -t^3+t^2+t
Normalized Jones-Krushkal polynomial 6z^2+23z+23
Enhanced Jones-Krushkal polynomial 6w^3z^2+23w^2z+23w
Inner characteristic polynomial t^6+30t^4+28t^2+1
Outer characteristic polynomial t^7+46t^5+93t^3+5t
Flat arrow polynomial 4*K1**3 + 4*K1**2*K2 - 4*K1**2 - 6*K1*K2 - 2*K1*K3 - 2*K2**2 + K2 + 2*K3 + K4 + 3
2-strand cable arrow polynomial -208*K1**4 + 608*K1**3*K2*K3 - 224*K1**3*K3 - 128*K1**2*K2**4 + 416*K1**2*K2**3 - 448*K1**2*K2**2*K3**2 + 160*K1**2*K2**2*K4 - 3760*K1**2*K2**2 + 192*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 416*K1**2*K2*K4 + 4288*K1**2*K2 - 512*K1**2*K3**2 - 48*K1**2*K4**2 - 3556*K1**2 + 1536*K1*K2**3*K3 + 512*K1*K2**2*K3*K4 - 1792*K1*K2**2*K3 + 32*K1*K2**2*K4*K5 - 192*K1*K2**2*K5 + 96*K1*K2*K3**3 - 736*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 5240*K1*K2*K3 + 1336*K1*K3*K4 + 200*K1*K4*K5 + 8*K1*K5*K6 - 32*K2**6 - 128*K2**4*K3**2 - 32*K2**4*K4**2 + 160*K2**4*K4 - 1120*K2**4 + 64*K2**3*K3*K5 + 32*K2**3*K4*K6 - 64*K2**3*K6 + 128*K2**2*K3**2*K4 - 1280*K2**2*K3**2 - 32*K2**2*K3*K7 + 32*K2**2*K4**3 - 280*K2**2*K4**2 + 1784*K2**2*K4 - 16*K2**2*K5**2 - 8*K2**2*K6**2 - 2664*K2**2 - 32*K2*K3**2*K4 - 64*K2*K3*K4*K5 + 720*K2*K3*K5 - 32*K2*K4**2*K6 + 96*K2*K4*K6 + 16*K2*K5*K7 + 8*K2*K6*K8 - 16*K3**4 - 32*K3**2*K4**2 + 16*K3**2*K6 - 1820*K3**2 + 24*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 814*K4**2 - 148*K5**2 - 32*K6**2 - 4*K7**2 - 2*K8**2 + 2966
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {3, 5}, {2, 4}], [{2, 6}, {1, 5}, {3, 4}]]
If K is slice False
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