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Flat knot 6.741

Min(phi) over symmetries of the knot is: [-3,-2,1,1,1,2,0,1,1,4,3,0,1,2,1,0,0,0,-1,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.741']
Arrow polynomial of the knot is: -12*K1**2 - 6*K1*K2 + 3*K1 + 6*K2 + 3*K3 + 7
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.694', '6.741']
Outer characteristic polynomial of the knot is: t^7+54t^5+105t^3+7t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.741']
2-strand cable arrow polynomial of the knot is: -192*K1**6 - 192*K1**4*K2**2 + 896*K1**4*K2 - 4272*K1**4 + 512*K1**3*K2*K3 + 32*K1**3*K3*K4 - 640*K1**3*K3 - 5712*K1**2*K2**2 - 288*K1**2*K2*K4 + 10376*K1**2*K2 - 1040*K1**2*K3**2 - 80*K1**2*K4**2 - 5748*K1**2 - 800*K1*K2**2*K3 - 32*K1*K2**2*K5 - 224*K1*K2*K3*K4 + 8280*K1*K2*K3 + 1776*K1*K3*K4 + 264*K1*K4*K5 + 32*K1*K5*K6 - 1216*K2**4 - 560*K2**2*K3**2 - 24*K2**2*K4**2 + 1768*K2**2*K4 - 5170*K2**2 + 704*K2*K3*K5 + 56*K2*K4*K6 - 2812*K3**2 - 1044*K4**2 - 280*K5**2 - 38*K6**2 + 5666
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.741']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11259', 'vk6.11339', 'vk6.12524', 'vk6.12637', 'vk6.17611', 'vk6.18927', 'vk6.19005', 'vk6.19340', 'vk6.19635', 'vk6.24064', 'vk6.24158', 'vk6.25521', 'vk6.25622', 'vk6.26116', 'vk6.26536', 'vk6.30941', 'vk6.31066', 'vk6.32121', 'vk6.32242', 'vk6.36408', 'vk6.37668', 'vk6.37717', 'vk6.43506', 'vk6.44777', 'vk6.52025', 'vk6.52115', 'vk6.52939', 'vk6.56492', 'vk6.56656', 'vk6.65388', 'vk6.66122', 'vk6.66158']
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 O1O2O3O4U2O5U1U5U4O6U3U6
R3 orbit {'O1O2O3O4U2O5U1U5U4O6U3U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U5U2O5U1U6U4O6U3
Gauss code of K* O1O2O3U4O5O4U1U6U5U3O6U2
Gauss code of -K* O1O2O3U2O4U1U5U4U3O6O5U6
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 -2 1 2 1 1],[ 3 0 0 4 3 1 1],[ 2 0 0 2 1 0 1],[-1 -4 -2 0 0 0 1],[-2 -3 -1 0 0 0 0],[-1 -1 0 0 0 0 0],[-1 -1 -1 -1 0 0 0]]
Primitive based matrix [[ 0 2 1 1 1 -2 -3],[-2 0 0 0 0 -1 -3],[-1 0 0 1 0 -2 -4],[-1 0 -1 0 0 -1 -1],[-1 0 0 0 0 0 -1],[ 2 1 2 1 0 0 0],[ 3 3 4 1 1 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,-1,-1,2,3,0,0,0,1,3,-1,0,2,4,0,1,1,0,1,0]
Phi over symmetry [-3,-2,1,1,1,2,0,1,1,4,3,0,1,2,1,0,0,0,-1,0,0]
Phi of -K [-3,-2,1,1,1,2,1,0,3,3,2,1,2,3,3,-1,0,1,0,1,1]
Phi of K* [-2,-1,-1,-1,2,3,1,1,1,3,2,-1,0,2,3,0,1,0,3,3,1]
Phi of -K* [-3,-2,1,1,1,2,0,1,1,4,3,0,1,2,1,0,0,0,-1,0,0]
Symmetry type of based matrix c
u-polynomial t^3-3t
Normalized Jones-Krushkal polynomial 2z^2+22z+37
Enhanced Jones-Krushkal polynomial 2w^3z^2+22w^2z+37w
Inner characteristic polynomial t^6+34t^4+29t^2+1
Outer characteristic polynomial t^7+54t^5+105t^3+7t
Flat arrow polynomial -12*K1**2 - 6*K1*K2 + 3*K1 + 6*K2 + 3*K3 + 7
2-strand cable arrow polynomial -192*K1**6 - 192*K1**4*K2**2 + 896*K1**4*K2 - 4272*K1**4 + 512*K1**3*K2*K3 + 32*K1**3*K3*K4 - 640*K1**3*K3 - 5712*K1**2*K2**2 - 288*K1**2*K2*K4 + 10376*K1**2*K2 - 1040*K1**2*K3**2 - 80*K1**2*K4**2 - 5748*K1**2 - 800*K1*K2**2*K3 - 32*K1*K2**2*K5 - 224*K1*K2*K3*K4 + 8280*K1*K2*K3 + 1776*K1*K3*K4 + 264*K1*K4*K5 + 32*K1*K5*K6 - 1216*K2**4 - 560*K2**2*K3**2 - 24*K2**2*K4**2 + 1768*K2**2*K4 - 5170*K2**2 + 704*K2*K3*K5 + 56*K2*K4*K6 - 2812*K3**2 - 1044*K4**2 - 280*K5**2 - 38*K6**2 + 5666
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {3, 5}, {2, 4}], [{3, 6}, {1, 5}, {2, 4}], [{5, 6}, {2, 4}, {1, 3}], [{5, 6}, {4}, {1, 3}, {2}], [{6}, {3, 5}, {2, 4}, {1}]]
If K is slice False
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