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Flat knot 6.394

Min(phi) over symmetries of the knot is: [-4,-1,0,1,1,3,0,3,1,2,5,0,0,0,1,0,0,2,0,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.394']
Arrow polynomial of the knot is: -2*K1*K2 + K1 - 2*K2**2 + K3 + K4 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.95', '6.107', '6.276', '6.292', '6.394', '6.429', '6.463']
Outer characteristic polynomial of the knot is: t^7+72t^5+88t^3+7t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.394']
2-strand cable arrow polynomial of the knot is: -1392*K1**4 + 384*K1**3*K2*K3 + 160*K1**3*K3*K4 - 288*K1**3*K3 - 1696*K1**2*K2**2 - 1120*K1**2*K2*K4 + 3984*K1**2*K2 - 1616*K1**2*K3**2 - 160*K1**2*K3*K5 - 608*K1**2*K4**2 - 3900*K1**2 + 64*K1*K2**3*K3 + 224*K1*K2**2*K3*K4 - 544*K1*K2**2*K3 - 128*K1*K2*K3*K4 + 5304*K1*K2*K3 - 128*K1*K2*K4*K5 + 64*K1*K3**3*K4 + 3776*K1*K3*K4 + 520*K1*K4*K5 - 32*K2**4 - 384*K2**2*K3**2 - 392*K2**2*K4**2 + 1168*K2**2*K4 - 3174*K2**2 - 96*K2*K3**2*K4 + 392*K2*K3*K5 + 272*K2*K4*K6 - 160*K3**4 - 240*K3**2*K4**2 + 96*K3**2*K6 - 2464*K3**2 + 128*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 1512*K4**2 - 160*K5**2 - 26*K6**2 - 12*K7**2 - 2*K8**2 + 3696
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.394']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.10150', 'vk6.10221', 'vk6.10366', 'vk6.10439', 'vk6.16699', 'vk6.19071', 'vk6.19118', 'vk6.19250', 'vk6.19543', 'vk6.23012', 'vk6.23131', 'vk6.25696', 'vk6.25743', 'vk6.26060', 'vk6.26437', 'vk6.29933', 'vk6.29994', 'vk6.30092', 'vk6.34999', 'vk6.35126', 'vk6.37794', 'vk6.37854', 'vk6.42568', 'vk6.44657', 'vk6.51634', 'vk6.51739', 'vk6.54910', 'vk6.56588', 'vk6.59334', 'vk6.64875', 'vk6.66188', 'vk6.66219']
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 O1O2O3O4O5U4U3O6U1U6U2U5
R3 orbit {'O1O2O3O4O5U4U3O6U1U6U2U5'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U1U4U6U5O6U3U2
Gauss code of K* O1O2O3O4U1U3U5U6U4O6O5U2
Gauss code of -K* O1O2O3O4U3O5O6U1U6U5U2U4
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 0 -1 -1 4 1],[ 3 0 2 0 0 5 1],[ 0 -2 0 0 0 3 0],[ 1 0 0 0 0 2 0],[ 1 0 0 0 0 1 0],[-4 -5 -3 -2 -1 0 0],[-1 -1 0 0 0 0 0]]
Primitive based matrix [[ 0 4 1 0 -1 -1 -3],[-4 0 0 -3 -1 -2 -5],[-1 0 0 0 0 0 -1],[ 0 3 0 0 0 0 -2],[ 1 1 0 0 0 0 0],[ 1 2 0 0 0 0 0],[ 3 5 1 2 0 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-4,-1,0,1,1,3,0,3,1,2,5,0,0,0,1,0,0,2,0,0,0]
Phi over symmetry [-4,-1,0,1,1,3,0,3,1,2,5,0,0,0,1,0,0,2,0,0,0]
Phi of -K [-3,-1,-1,0,1,4,2,2,1,3,2,0,1,2,3,1,2,4,1,1,3]
Phi of K* [-4,-1,0,1,1,3,3,1,3,4,2,1,2,2,3,1,1,1,0,2,2]
Phi of -K* [-3,-1,-1,0,1,4,0,0,2,1,5,0,0,0,1,0,0,2,0,3,0]
Symmetry type of based matrix c
u-polynomial -t^4+t^3+t
Normalized Jones-Krushkal polynomial 7z^2+28z+29
Enhanced Jones-Krushkal polynomial 7w^3z^2+28w^2z+29w
Inner characteristic polynomial t^6+44t^4+34t^2
Outer characteristic polynomial t^7+72t^5+88t^3+7t
Flat arrow polynomial -2*K1*K2 + K1 - 2*K2**2 + K3 + K4 + 2
2-strand cable arrow polynomial -1392*K1**4 + 384*K1**3*K2*K3 + 160*K1**3*K3*K4 - 288*K1**3*K3 - 1696*K1**2*K2**2 - 1120*K1**2*K2*K4 + 3984*K1**2*K2 - 1616*K1**2*K3**2 - 160*K1**2*K3*K5 - 608*K1**2*K4**2 - 3900*K1**2 + 64*K1*K2**3*K3 + 224*K1*K2**2*K3*K4 - 544*K1*K2**2*K3 - 128*K1*K2*K3*K4 + 5304*K1*K2*K3 - 128*K1*K2*K4*K5 + 64*K1*K3**3*K4 + 3776*K1*K3*K4 + 520*K1*K4*K5 - 32*K2**4 - 384*K2**2*K3**2 - 392*K2**2*K4**2 + 1168*K2**2*K4 - 3174*K2**2 - 96*K2*K3**2*K4 + 392*K2*K3*K5 + 272*K2*K4*K6 - 160*K3**4 - 240*K3**2*K4**2 + 96*K3**2*K6 - 2464*K3**2 + 128*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 1512*K4**2 - 160*K5**2 - 26*K6**2 - 12*K7**2 - 2*K8**2 + 3696
Genus of based matrix 1
Fillings of based matrix [[{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {2, 5}, {1, 3}], [{4, 6}, {3, 5}, {1, 2}]]
If K is slice False
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