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Flat knot 6.1852

Min(phi) over symmetries of the knot is: [-1,-1,-1,1,1,1,-1,-1,1,1,2,0,0,1,1,1,0,1,0,0,1]
Flat knots (up to 7 crossings) with same phi are :['6.1852']
Arrow polynomial of the knot is: 4*K1**3 - 12*K1**2 - 8*K1*K2 + K1 + 6*K2 + 3*K3 + 7
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.906', '6.1223', '6.1338', '6.1351', '6.1571', '6.1670', '6.1718', '6.1743', '6.1765', '6.1793', '6.1852', '6.2070']
Outer characteristic polynomial of the knot is: t^7+19t^5+33t^3+11t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1852']
2-strand cable arrow polynomial of the knot is: -384*K1**6 - 256*K1**4*K2**2 + 2176*K1**4*K2 - 4288*K1**4 + 704*K1**3*K2*K3 - 1664*K1**3*K3 + 896*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 6560*K1**2*K2**2 + 192*K1**2*K2*K3**2 - 704*K1**2*K2*K4 + 12104*K1**2*K2 - 736*K1**2*K3**2 - 32*K1**2*K4**2 - 7488*K1**2 + 224*K1*K2**3*K3 - 1792*K1*K2**2*K3 - 256*K1*K2**2*K5 - 608*K1*K2*K3*K4 + 9520*K1*K2*K3 + 1488*K1*K3*K4 + 256*K1*K4*K5 - 32*K2**6 + 224*K2**4*K4 - 1248*K2**4 - 96*K2**3*K6 - 416*K2**2*K3**2 - 128*K2**2*K4**2 + 2064*K2**2*K4 - 6194*K2**2 + 632*K2*K3*K5 + 72*K2*K4*K6 - 2952*K3**2 - 844*K4**2 - 184*K5**2 - 6*K6**2 + 6186
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1852']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.10947', 'vk6.10959', 'vk6.10978', 'vk6.10990', 'vk6.12117', 'vk6.12129', 'vk6.12148', 'vk6.12160', 'vk6.13791', 'vk6.13805', 'vk6.14224', 'vk6.14237', 'vk6.14673', 'vk6.14684', 'vk6.14866', 'vk6.14876', 'vk6.15831', 'vk6.15844', 'vk6.31823', 'vk6.31827', 'vk6.33623', 'vk6.33645', 'vk6.33654', 'vk6.33676', 'vk6.51775', 'vk6.51803', 'vk6.52642', 'vk6.52670', 'vk6.53801', 'vk6.53815', 'vk6.54226', 'vk6.54239']
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 O1O2O3U2U4O5U3O6O4U6U1U5
R3 orbit {'O1O2O3U2U4O5U3O6O4U6U1U5'}
R3 orbit length 1
Gauss code of -K O1O2O3U4U3U5O6O5U1O4U6U2
Gauss code of K* O1O2O3U2U4U5O4O6U3O5U1U6
Gauss code of -K* O1O2O3U4U3O5U1O4O6U5U6U2
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 -1 -1 1 1 1 -1],[ 1 0 -1 2 1 1 -1],[ 1 1 0 1 0 1 0],[-1 -2 -1 0 -1 0 -1],[-1 -1 0 1 0 0 -1],[-1 -1 -1 0 0 0 0],[ 1 1 0 1 1 0 0]]
Primitive based matrix [[ 0 1 1 1 -1 -1 -1],[-1 0 1 0 0 -1 -1],[-1 -1 0 0 -1 -1 -2],[-1 0 0 0 -1 0 -1],[ 1 0 1 1 0 0 1],[ 1 1 1 0 0 0 1],[ 1 1 2 1 -1 -1 0]]
If based matrix primitive True
Phi of primitive based matrix [-1,-1,-1,1,1,1,-1,0,0,1,1,0,1,1,2,1,0,1,0,-1,-1]
Phi over symmetry [-1,-1,-1,1,1,1,-1,-1,1,1,2,0,0,1,1,1,0,1,0,0,1]
Phi of -K [-1,-1,-1,1,1,1,-1,0,1,1,2,1,0,1,1,1,2,1,0,1,0]
Phi of K* [-1,-1,-1,1,1,1,-1,0,0,1,1,0,1,1,2,1,2,1,-1,-1,0]
Phi of -K* [-1,-1,-1,1,1,1,-1,-1,1,1,2,0,0,1,1,1,0,1,0,0,1]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial z^2+22z+41
Enhanced Jones-Krushkal polynomial w^3z^2+22w^2z+41w
Inner characteristic polynomial t^6+13t^4+15t^2+1
Outer characteristic polynomial t^7+19t^5+33t^3+11t
Flat arrow polynomial 4*K1**3 - 12*K1**2 - 8*K1*K2 + K1 + 6*K2 + 3*K3 + 7
2-strand cable arrow polynomial -384*K1**6 - 256*K1**4*K2**2 + 2176*K1**4*K2 - 4288*K1**4 + 704*K1**3*K2*K3 - 1664*K1**3*K3 + 896*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 6560*K1**2*K2**2 + 192*K1**2*K2*K3**2 - 704*K1**2*K2*K4 + 12104*K1**2*K2 - 736*K1**2*K3**2 - 32*K1**2*K4**2 - 7488*K1**2 + 224*K1*K2**3*K3 - 1792*K1*K2**2*K3 - 256*K1*K2**2*K5 - 608*K1*K2*K3*K4 + 9520*K1*K2*K3 + 1488*K1*K3*K4 + 256*K1*K4*K5 - 32*K2**6 + 224*K2**4*K4 - 1248*K2**4 - 96*K2**3*K6 - 416*K2**2*K3**2 - 128*K2**2*K4**2 + 2064*K2**2*K4 - 6194*K2**2 + 632*K2*K3*K5 + 72*K2*K4*K6 - 2952*K3**2 - 844*K4**2 - 184*K5**2 - 6*K6**2 + 6186
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {1, 5}, {3, 4}], [{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {2, 5}, {1, 4}], [{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {2, 5}, {1, 3}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {2, 4}, {1, 3}]]
If K is slice False
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