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Flat knot 6.290

Min(phi) over symmetries of the knot is: [-4,-1,0,1,1,3,0,2,1,4,4,1,1,2,1,1,1,2,0,0,2]
Flat knots (up to 7 crossings) with same phi are :['6.290']
Arrow polynomial of the knot is: 4*K1**3 + 4*K1**2*K2 - 10*K1**2 - 2*K1*K2 - 2*K1*K3 - 2*K1 + 4*K2 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.290', '6.427']
Outer characteristic polynomial of the knot is: t^7+82t^5+110t^3+13t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.290']
2-strand cable arrow polynomial of the knot is: -128*K1**6 - 256*K1**4*K2**2 + 896*K1**4*K2 - 2448*K1**4 + 320*K1**3*K2*K3 - 192*K1**3*K3 - 768*K1**2*K2**4 + 2336*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 + 64*K1**2*K2**2*K4 - 8048*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 512*K1**2*K2*K4 + 8856*K1**2*K2 - 144*K1**2*K3**2 - 32*K1**2*K4**2 - 4896*K1**2 - 256*K1*K2**4*K3 + 2048*K1*K2**3*K3 + 448*K1*K2**2*K3*K4 - 2048*K1*K2**2*K3 - 320*K1*K2**2*K5 - 224*K1*K2*K3*K4 + 7264*K1*K2*K3 + 728*K1*K3*K4 + 88*K1*K4*K5 - 32*K2**6 - 192*K2**4*K3**2 - 288*K2**4*K4**2 + 1152*K2**4*K4 - 3496*K2**4 + 128*K2**3*K3*K5 + 160*K2**3*K4*K6 - 160*K2**3*K6 - 1184*K2**2*K3**2 - 872*K2**2*K4**2 + 3080*K2**2*K4 - 16*K2**2*K5**2 - 8*K2**2*K6**2 - 3364*K2**2 + 456*K2*K3*K5 + 208*K2*K4*K6 - 1832*K3**2 - 816*K4**2 - 72*K5**2 - 4*K6**2 + 4438
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.290']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.17012', 'vk6.17255', 'vk6.20240', 'vk6.21542', 'vk6.23422', 'vk6.23726', 'vk6.27457', 'vk6.29057', 'vk6.35495', 'vk6.35943', 'vk6.38870', 'vk6.41066', 'vk6.42920', 'vk6.43218', 'vk6.45629', 'vk6.47374', 'vk6.55191', 'vk6.55432', 'vk6.57073', 'vk6.58224', 'vk6.59570', 'vk6.59903', 'vk6.61605', 'vk6.62781', 'vk6.64986', 'vk6.65196', 'vk6.66698', 'vk6.67554', 'vk6.68272', 'vk6.68427', 'vk6.69350', 'vk6.70095']
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 O1O2O3O4O5U1U5O6U4U2U3U6
R3 orbit {'O1O2O3O4O5U1U5O6U4U2U3U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U6U3U4U2O6U1U5
Gauss code of K* O1O2O3O4U5U2U3U1U6O5O6U4
Gauss code of -K* O1O2O3O4U1O5O6U5U4U2U3U6
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 -4 -1 1 0 1 3],[ 4 0 3 4 2 1 3],[ 1 -3 0 1 0 0 3],[-1 -4 -1 0 0 0 2],[ 0 -2 0 0 0 0 1],[-1 -1 0 0 0 0 0],[-3 -3 -3 -2 -1 0 0]]
Primitive based matrix [[ 0 3 1 1 0 -1 -4],[-3 0 0 -2 -1 -3 -3],[-1 0 0 0 0 0 -1],[-1 2 0 0 0 -1 -4],[ 0 1 0 0 0 0 -2],[ 1 3 0 1 0 0 -3],[ 4 3 1 4 2 3 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-1,-1,0,1,4,0,2,1,3,3,0,0,0,1,0,1,4,0,2,3]
Phi over symmetry [-4,-1,0,1,1,3,0,2,1,4,4,1,1,2,1,1,1,2,0,0,2]
Phi of -K [-4,-1,0,1,1,3,0,2,1,4,4,1,1,2,1,1,1,2,0,0,2]
Phi of K* [-3,-1,-1,0,1,4,0,2,2,1,4,0,1,1,1,1,2,4,1,2,0]
Phi of -K* [-4,-1,0,1,1,3,3,2,1,4,3,0,0,1,3,0,0,1,0,0,2]
Symmetry type of based matrix c
u-polynomial t^4-t^3-t
Normalized Jones-Krushkal polynomial 4z^2+21z+27
Enhanced Jones-Krushkal polynomial -2w^4z^2+6w^3z^2-4w^3z+25w^2z+27w
Inner characteristic polynomial t^6+54t^4+38t^2+1
Outer characteristic polynomial t^7+82t^5+110t^3+13t
Flat arrow polynomial 4*K1**3 + 4*K1**2*K2 - 10*K1**2 - 2*K1*K2 - 2*K1*K3 - 2*K1 + 4*K2 + 5
2-strand cable arrow polynomial -128*K1**6 - 256*K1**4*K2**2 + 896*K1**4*K2 - 2448*K1**4 + 320*K1**3*K2*K3 - 192*K1**3*K3 - 768*K1**2*K2**4 + 2336*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 + 64*K1**2*K2**2*K4 - 8048*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 512*K1**2*K2*K4 + 8856*K1**2*K2 - 144*K1**2*K3**2 - 32*K1**2*K4**2 - 4896*K1**2 - 256*K1*K2**4*K3 + 2048*K1*K2**3*K3 + 448*K1*K2**2*K3*K4 - 2048*K1*K2**2*K3 - 320*K1*K2**2*K5 - 224*K1*K2*K3*K4 + 7264*K1*K2*K3 + 728*K1*K3*K4 + 88*K1*K4*K5 - 32*K2**6 - 192*K2**4*K3**2 - 288*K2**4*K4**2 + 1152*K2**4*K4 - 3496*K2**4 + 128*K2**3*K3*K5 + 160*K2**3*K4*K6 - 160*K2**3*K6 - 1184*K2**2*K3**2 - 872*K2**2*K4**2 + 3080*K2**2*K4 - 16*K2**2*K5**2 - 8*K2**2*K6**2 - 3364*K2**2 + 456*K2*K3*K5 + 208*K2*K4*K6 - 1832*K3**2 - 816*K4**2 - 72*K5**2 - 4*K6**2 + 4438
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{2, 6}, {1, 5}, {3, 4}], [{3, 6}, {2, 5}, {1, 4}], [{4, 6}, {2, 5}, {1, 3}], [{6}, {2, 5}, {1, 4}, {3}]]
If K is slice False
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