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Flat knot 6.1523

Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,-1,1,1,2,2,1,0,1,1,0,0,0,-1,0,1]
Flat knots (up to 7 crossings) with same phi are :['6.1523']
Arrow polynomial of the knot is: 4*K1**3 - 10*K1**2 - 8*K1*K2 + K1 + 5*K2 + 3*K3 + 6
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.374', '6.446', '6.527', '6.1218', '6.1237', '6.1276', '6.1498', '6.1523', '6.1595', '6.1703', '6.1751', '6.1766', '6.1849', '6.1926']
Outer characteristic polynomial of the knot is: t^7+24t^5+28t^3+3t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1523']
2-strand cable arrow polynomial of the knot is: -640*K1**6 - 448*K1**4*K2**2 + 3904*K1**4*K2 - 8368*K1**4 + 1408*K1**3*K2*K3 - 1792*K1**3*K3 - 192*K1**2*K2**4 + 1472*K1**2*K2**3 + 192*K1**2*K2**2*K4 - 9936*K1**2*K2**2 - 1184*K1**2*K2*K4 + 13856*K1**2*K2 - 1232*K1**2*K3**2 - 48*K1**2*K4**2 - 4432*K1**2 + 608*K1*K2**3*K3 - 1280*K1*K2**2*K3 - 256*K1*K2**2*K5 - 416*K1*K2*K3*K4 + 9344*K1*K2*K3 + 1592*K1*K3*K4 + 184*K1*K4*K5 - 32*K2**6 + 96*K2**4*K4 - 1208*K2**4 - 32*K2**3*K6 - 400*K2**2*K3**2 - 128*K2**2*K4**2 + 1544*K2**2*K4 - 4866*K2**2 + 384*K2*K3*K5 + 104*K2*K4*K6 - 2228*K3**2 - 674*K4**2 - 92*K5**2 - 22*K6**2 + 5216
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1523']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4464', 'vk6.4559', 'vk6.5850', 'vk6.5977', 'vk6.7904', 'vk6.8022', 'vk6.9337', 'vk6.9456', 'vk6.13407', 'vk6.13502', 'vk6.13695', 'vk6.14062', 'vk6.15039', 'vk6.15159', 'vk6.17797', 'vk6.17828', 'vk6.18839', 'vk6.19418', 'vk6.19713', 'vk6.24344', 'vk6.25438', 'vk6.25469', 'vk6.26592', 'vk6.33261', 'vk6.33320', 'vk6.37558', 'vk6.44867', 'vk6.48651', 'vk6.50551', 'vk6.53645', 'vk6.55812', 'vk6.65484']
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 O1O2O3U2O4U5U1O5O6U4U6U3
R3 orbit {'O1O2O3U2O4U5U1O5O6U4U6U3'}
R3 orbit length 1
Gauss code of -K O1O2O3U1U4U5O4O6U3U6O5U2
Gauss code of K* O1O2O3U4U5U3O5U1O6O4U6U2
Gauss code of -K* O1O2O3U2U4O5O4U3O6U1U6U5
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 2 0 -1 1],[ 1 0 0 2 0 1 1],[ 1 0 0 1 0 1 0],[-2 -2 -1 0 -1 -2 1],[ 0 0 0 1 0 0 1],[ 1 -1 -1 2 0 0 1],[-1 -1 0 -1 -1 -1 0]]
Primitive based matrix [[ 0 2 1 0 -1 -1 -1],[-2 0 1 -1 -1 -2 -2],[-1 -1 0 -1 0 -1 -1],[ 0 1 1 0 0 0 0],[ 1 1 0 0 0 1 0],[ 1 2 1 0 -1 0 -1],[ 1 2 1 0 0 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,0,1,1,1,-1,1,1,2,2,1,0,1,1,0,0,0,-1,0,1]
Phi over symmetry [-2,-1,0,1,1,1,-1,1,1,2,2,1,0,1,1,0,0,0,-1,0,1]
Phi of -K [-1,-1,-1,0,1,2,-1,0,1,1,1,1,1,1,1,1,2,2,0,1,2]
Phi of K* [-2,-1,0,1,1,1,2,1,1,1,2,0,1,1,2,1,1,1,-1,-1,0]
Phi of -K* [-1,-1,-1,0,1,2,-1,-1,0,1,2,0,0,0,1,0,1,2,1,1,-1]
Symmetry type of based matrix c
u-polynomial -t^2+2t
Normalized Jones-Krushkal polynomial 2z^2+23z+39
Enhanced Jones-Krushkal polynomial 2w^3z^2+23w^2z+39w
Inner characteristic polynomial t^6+16t^4+11t^2
Outer characteristic polynomial t^7+24t^5+28t^3+3t
Flat arrow polynomial 4*K1**3 - 10*K1**2 - 8*K1*K2 + K1 + 5*K2 + 3*K3 + 6
2-strand cable arrow polynomial -640*K1**6 - 448*K1**4*K2**2 + 3904*K1**4*K2 - 8368*K1**4 + 1408*K1**3*K2*K3 - 1792*K1**3*K3 - 192*K1**2*K2**4 + 1472*K1**2*K2**3 + 192*K1**2*K2**2*K4 - 9936*K1**2*K2**2 - 1184*K1**2*K2*K4 + 13856*K1**2*K2 - 1232*K1**2*K3**2 - 48*K1**2*K4**2 - 4432*K1**2 + 608*K1*K2**3*K3 - 1280*K1*K2**2*K3 - 256*K1*K2**2*K5 - 416*K1*K2*K3*K4 + 9344*K1*K2*K3 + 1592*K1*K3*K4 + 184*K1*K4*K5 - 32*K2**6 + 96*K2**4*K4 - 1208*K2**4 - 32*K2**3*K6 - 400*K2**2*K3**2 - 128*K2**2*K4**2 + 1544*K2**2*K4 - 4866*K2**2 + 384*K2*K3*K5 + 104*K2*K4*K6 - 2228*K3**2 - 674*K4**2 - 92*K5**2 - 22*K6**2 + 5216
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {3, 5}, {2, 4}], [{1, 6}, {4, 5}, {2, 3}], [{3, 6}, {4, 5}, {1, 2}], [{6}, {3, 5}, {2, 4}, {1}]]
If K is slice False
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