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Flat knot 6.842

Min(phi) over symmetries of the knot is: [-3,-1,-1,1,1,3,-1,0,2,3,4,0,0,2,1,1,2,2,0,1,2]
Flat knots (up to 7 crossings) with same phi are :['6.842']
Arrow polynomial of the knot is: 4*K1**2*K2 - 2*K1**2 - 4*K1*K2 - 2*K1*K3 + 2*K1 - 2*K2**2 + 2*K3 + K4 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.110', '6.328', '6.334', '6.842']
Outer characteristic polynomial of the knot is: t^7+63t^5+86t^3+9t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.842']
2-strand cable arrow polynomial of the knot is: -320*K1**4 + 384*K1**3*K2*K3 - 512*K1**3*K3 + 1536*K1**2*K2**3 - 256*K1**2*K2**2*K3**2 - 3824*K1**2*K2**2 + 128*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 + 32*K1**2*K2*K4**2 - 1184*K1**2*K2*K4 + 4720*K1**2*K2 - 416*K1**2*K3**2 - 32*K1**2*K3*K5 - 224*K1**2*K4**2 - 4660*K1**2 - 640*K1*K2**4*K3 + 1088*K1*K2**3*K3 + 864*K1*K2**2*K3*K4 - 1312*K1*K2**2*K3 - 64*K1*K2**2*K5 + 32*K1*K2*K3*K4**2 - 320*K1*K2*K3*K4 + 5552*K1*K2*K3 - 192*K1*K2*K4*K5 + 2256*K1*K3*K4 + 432*K1*K4*K5 + 8*K1*K5*K6 - 32*K2**4*K4**2 + 704*K2**4*K4 - 1880*K2**4 + 32*K2**3*K3*K5 + 32*K2**3*K4*K6 - 1216*K2**2*K3**2 - 32*K2**2*K3*K7 + 32*K2**2*K4**3 - 1136*K2**2*K4**2 - 32*K2**2*K4*K8 + 2152*K2**2*K4 - 16*K2**2*K5**2 - 8*K2**2*K6**2 - 2688*K2**2 - 128*K2*K3**2*K4 - 32*K2*K3*K4*K5 + 736*K2*K3*K5 + 480*K2*K4*K6 + 16*K2*K5*K7 + 8*K2*K6*K8 - 32*K3**4 - 64*K3**2*K4**2 + 24*K3**2*K6 - 2168*K3**2 + 48*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 1360*K4**2 - 272*K5**2 - 32*K6**2 - 4*K7**2 - 2*K8**2 + 3760
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.842']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11504', 'vk6.11823', 'vk6.12840', 'vk6.13161', 'vk6.20277', 'vk6.21604', 'vk6.27547', 'vk6.29113', 'vk6.31265', 'vk6.31632', 'vk6.32405', 'vk6.32838', 'vk6.38952', 'vk6.41193', 'vk6.45727', 'vk6.47426', 'vk6.52263', 'vk6.52510', 'vk6.53086', 'vk6.53420', 'vk6.57118', 'vk6.58306', 'vk6.61710', 'vk6.62854', 'vk6.63786', 'vk6.63904', 'vk6.64220', 'vk6.64430', 'vk6.66745', 'vk6.67627', 'vk6.69403', 'vk6.70127']
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 O1O2O3O4U1U2O5O6U5U4U3U6
R3 orbit {'O1O2O3O4U1U2O5O6U5U4U3U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U5U2U1U6O5O6U3U4
Gauss code of K* O1O2O3O4U5U6U3U2O5O6U1U4
Gauss code of -K* O1O2O3O4U1U4O5O6U3U2U5U6
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 1 -1 3],[ 3 0 1 3 2 0 2],[ 1 -1 0 2 1 0 2],[-1 -3 -2 0 0 0 3],[-1 -2 -1 0 0 0 2],[ 1 0 0 0 0 0 1],[-3 -2 -2 -3 -2 -1 0]]
Primitive based matrix [[ 0 3 1 1 -1 -1 -3],[-3 0 -2 -3 -1 -2 -2],[-1 2 0 0 0 -1 -2],[-1 3 0 0 0 -2 -3],[ 1 1 0 0 0 0 0],[ 1 2 1 2 0 0 -1],[ 3 2 2 3 0 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-1,-1,1,1,3,2,3,1,2,2,0,0,1,2,0,2,3,0,0,1]
Phi over symmetry [-3,-1,-1,1,1,3,-1,0,2,3,4,0,0,2,1,1,2,2,0,1,2]
Phi of -K [-3,-1,-1,1,1,3,1,2,1,2,4,0,0,1,2,2,2,3,0,-1,0]
Phi of K* [-3,-1,-1,1,1,3,-1,0,2,3,4,0,0,2,1,1,2,2,0,1,2]
Phi of -K* [-3,-1,-1,1,1,3,0,1,2,3,2,0,0,0,1,1,2,2,0,2,3]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 6z^2+23z+23
Enhanced Jones-Krushkal polynomial -2w^4z^2+8w^3z^2-2w^3z+25w^2z+23w
Inner characteristic polynomial t^6+41t^4+22t^2+1
Outer characteristic polynomial t^7+63t^5+86t^3+9t
Flat arrow polynomial 4*K1**2*K2 - 2*K1**2 - 4*K1*K2 - 2*K1*K3 + 2*K1 - 2*K2**2 + 2*K3 + K4 + 2
2-strand cable arrow polynomial -320*K1**4 + 384*K1**3*K2*K3 - 512*K1**3*K3 + 1536*K1**2*K2**3 - 256*K1**2*K2**2*K3**2 - 3824*K1**2*K2**2 + 128*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 + 32*K1**2*K2*K4**2 - 1184*K1**2*K2*K4 + 4720*K1**2*K2 - 416*K1**2*K3**2 - 32*K1**2*K3*K5 - 224*K1**2*K4**2 - 4660*K1**2 - 640*K1*K2**4*K3 + 1088*K1*K2**3*K3 + 864*K1*K2**2*K3*K4 - 1312*K1*K2**2*K3 - 64*K1*K2**2*K5 + 32*K1*K2*K3*K4**2 - 320*K1*K2*K3*K4 + 5552*K1*K2*K3 - 192*K1*K2*K4*K5 + 2256*K1*K3*K4 + 432*K1*K4*K5 + 8*K1*K5*K6 - 32*K2**4*K4**2 + 704*K2**4*K4 - 1880*K2**4 + 32*K2**3*K3*K5 + 32*K2**3*K4*K6 - 1216*K2**2*K3**2 - 32*K2**2*K3*K7 + 32*K2**2*K4**3 - 1136*K2**2*K4**2 - 32*K2**2*K4*K8 + 2152*K2**2*K4 - 16*K2**2*K5**2 - 8*K2**2*K6**2 - 2688*K2**2 - 128*K2*K3**2*K4 - 32*K2*K3*K4*K5 + 736*K2*K3*K5 + 480*K2*K4*K6 + 16*K2*K5*K7 + 8*K2*K6*K8 - 32*K3**4 - 64*K3**2*K4**2 + 24*K3**2*K6 - 2168*K3**2 + 48*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 1360*K4**2 - 272*K5**2 - 32*K6**2 - 4*K7**2 - 2*K8**2 + 3760
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {3, 5}, {2, 4}], [{1, 6}, {4, 5}, {2, 3}], [{2, 6}, {4, 5}, {1, 3}], [{3, 6}, {4, 5}, {1, 2}]]
If K is slice False
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