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Flat knot 6.443

Min(phi) over symmetries of the knot is: [-4,-1,0,1,2,2,1,3,1,2,4,1,1,1,2,1,1,2,0,-1,0]
Flat knots (up to 7 crossings) with same phi are :['6.443']
Arrow polynomial of the knot is: 4*K1**3 + 4*K1**2*K2 - 12*K1**2 - 8*K1*K2 + K1 - 2*K2**2 + 4*K2 + 3*K3 + 7
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.443']
Outer characteristic polynomial of the knot is: t^7+79t^5+132t^3+8t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.443']
2-strand cable arrow polynomial of the knot is: -64*K1**6 + 1184*K1**4*K2 - 3856*K1**4 + 608*K1**3*K2*K3 + 32*K1**3*K3*K4 - 1216*K1**3*K3 - 128*K1**2*K2**4 + 576*K1**2*K2**3 + 416*K1**2*K2**2*K4 - 5696*K1**2*K2**2 + 224*K1**2*K2*K3**2 + 64*K1**2*K2*K4**2 - 928*K1**2*K2*K4 + 11312*K1**2*K2 - 1680*K1**2*K3**2 - 32*K1**2*K3*K5 - 288*K1**2*K4**2 - 7832*K1**2 + 416*K1*K2**3*K3 - 1568*K1*K2**2*K3 - 320*K1*K2**2*K5 + 192*K1*K2*K3**3 + 64*K1*K2*K3*K4**2 - 736*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 9968*K1*K2*K3 - 32*K1*K2*K4*K5 + 2776*K1*K3*K4 + 392*K1*K4*K5 + 16*K1*K5*K6 - 32*K2**6 - 64*K2**4*K3**2 - 32*K2**4*K4**2 + 288*K2**4*K4 - 1232*K2**4 + 32*K2**3*K3*K5 - 96*K2**3*K6 + 64*K2**2*K3**2*K4 - 944*K2**2*K3**2 - 32*K2**2*K3*K7 + 32*K2**2*K4**3 - 288*K2**2*K4**2 + 2176*K2**2*K4 - 6402*K2**2 - 96*K2*K3**2*K4 - 32*K2*K3*K4*K5 + 1008*K2*K3*K5 + 184*K2*K4*K6 - 160*K3**4 - 80*K3**2*K4**2 + 128*K3**2*K6 - 3396*K3**2 + 40*K3*K4*K7 - 8*K4**4 - 1236*K4**2 - 268*K5**2 - 54*K6**2 + 6698
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.443']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16944', 'vk6.17186', 'vk6.20533', 'vk6.21934', 'vk6.23343', 'vk6.23637', 'vk6.27991', 'vk6.29458', 'vk6.35390', 'vk6.35811', 'vk6.39395', 'vk6.41588', 'vk6.42867', 'vk6.43146', 'vk6.45975', 'vk6.47651', 'vk6.55107', 'vk6.55366', 'vk6.57409', 'vk6.58584', 'vk6.59508', 'vk6.59807', 'vk6.62080', 'vk6.63062', 'vk6.64958', 'vk6.65165', 'vk6.66953', 'vk6.67814', 'vk6.68250', 'vk6.68392', 'vk6.69568', 'vk6.70265']
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 O1O2O3O4O5U6U2O6U4U1U3U5
R3 orbit {'O1O2O3O4O5U6U2O6U4U1U3U5'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U1U3U5U2O6U4U6
Gauss code of K* O1O2O3O4U2U5U3U1U4O6O5U6
Gauss code of -K* O1O2O3O4U5O6O5U1U4U2U6U3
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 -2 -2 1 0 4 -1],[ 2 0 0 2 1 4 1],[ 2 0 0 1 0 2 2],[-1 -2 -1 0 0 2 -1],[ 0 -1 0 0 0 1 0],[-4 -4 -2 -2 -1 0 -4],[ 1 -1 -2 1 0 4 0]]
Primitive based matrix [[ 0 4 1 0 -1 -2 -2],[-4 0 -2 -1 -4 -2 -4],[-1 2 0 0 -1 -1 -2],[ 0 1 0 0 0 0 -1],[ 1 4 1 0 0 -2 -1],[ 2 2 1 0 2 0 0],[ 2 4 2 1 1 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-4,-1,0,1,2,2,2,1,4,2,4,0,1,1,2,0,0,1,2,1,0]
Phi over symmetry [-4,-1,0,1,2,2,1,3,1,2,4,1,1,1,2,1,1,2,0,-1,0]
Phi of -K [-2,-2,-1,0,1,4,0,-1,2,2,4,0,1,1,2,1,1,1,1,3,1]
Phi of K* [-4,-1,0,1,2,2,1,3,1,2,4,1,1,1,2,1,1,2,0,-1,0]
Phi of -K* [-2,-2,-1,0,1,4,0,1,1,2,4,2,0,1,2,0,1,4,0,1,2]
Symmetry type of based matrix c
u-polynomial -t^4+2t^2
Normalized Jones-Krushkal polynomial 3z^2+24z+37
Enhanced Jones-Krushkal polynomial 3w^3z^2+24w^2z+37w
Inner characteristic polynomial t^6+53t^4+78t^2+1
Outer characteristic polynomial t^7+79t^5+132t^3+8t
Flat arrow polynomial 4*K1**3 + 4*K1**2*K2 - 12*K1**2 - 8*K1*K2 + K1 - 2*K2**2 + 4*K2 + 3*K3 + 7
2-strand cable arrow polynomial -64*K1**6 + 1184*K1**4*K2 - 3856*K1**4 + 608*K1**3*K2*K3 + 32*K1**3*K3*K4 - 1216*K1**3*K3 - 128*K1**2*K2**4 + 576*K1**2*K2**3 + 416*K1**2*K2**2*K4 - 5696*K1**2*K2**2 + 224*K1**2*K2*K3**2 + 64*K1**2*K2*K4**2 - 928*K1**2*K2*K4 + 11312*K1**2*K2 - 1680*K1**2*K3**2 - 32*K1**2*K3*K5 - 288*K1**2*K4**2 - 7832*K1**2 + 416*K1*K2**3*K3 - 1568*K1*K2**2*K3 - 320*K1*K2**2*K5 + 192*K1*K2*K3**3 + 64*K1*K2*K3*K4**2 - 736*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 9968*K1*K2*K3 - 32*K1*K2*K4*K5 + 2776*K1*K3*K4 + 392*K1*K4*K5 + 16*K1*K5*K6 - 32*K2**6 - 64*K2**4*K3**2 - 32*K2**4*K4**2 + 288*K2**4*K4 - 1232*K2**4 + 32*K2**3*K3*K5 - 96*K2**3*K6 + 64*K2**2*K3**2*K4 - 944*K2**2*K3**2 - 32*K2**2*K3*K7 + 32*K2**2*K4**3 - 288*K2**2*K4**2 + 2176*K2**2*K4 - 6402*K2**2 - 96*K2*K3**2*K4 - 32*K2*K3*K4*K5 + 1008*K2*K3*K5 + 184*K2*K4*K6 - 160*K3**4 - 80*K3**2*K4**2 + 128*K3**2*K6 - 3396*K3**2 + 40*K3*K4*K7 - 8*K4**4 - 1236*K4**2 - 268*K5**2 - 54*K6**2 + 6698
Genus of based matrix 1
Fillings of based matrix [[{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {1, 5}, {4}, {2}], [{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {2, 5}, {4}, {1}], [{3, 6}, {4, 5}, {1, 2}], [{3, 6}, {5}, {4}, {1, 2}]]
If K is slice False
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