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Flat knot 6.273

Min(phi) over symmetries of the knot is: [-4,-1,0,1,1,3,1,3,1,4,4,1,0,1,1,0,1,2,-1,0,2]
Flat knots (up to 7 crossings) with same phi are :['6.273']
Arrow polynomial of the knot is: 4*K1**3 + 4*K1**2*K2 - 14*K1**2 - 6*K1*K2 - 2*K1*K3 + 6*K2 + 2*K3 + 7
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.273', '6.496']
Outer characteristic polynomial of the knot is: t^7+80t^5+56t^3+8t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.273']
2-strand cable arrow polynomial of the knot is: 256*K1**4*K2**3 - 960*K1**4*K2**2 + 3008*K1**4*K2 - 5088*K1**4 - 128*K1**3*K2**2*K3 + 704*K1**3*K2*K3 + 64*K1**3*K3*K4 - 1088*K1**3*K3 - 384*K1**2*K2**4 + 2592*K1**2*K2**3 - 9872*K1**2*K2**2 + 128*K1**2*K2*K3**2 + 64*K1**2*K2*K4**2 - 1088*K1**2*K2*K4 + 12336*K1**2*K2 - 416*K1**2*K3**2 - 160*K1**2*K4**2 - 6600*K1**2 + 736*K1*K2**3*K3 + 32*K1*K2**2*K3*K4 - 2272*K1*K2**2*K3 - 320*K1*K2**2*K5 - 576*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 9848*K1*K2*K3 - 192*K1*K2*K4*K5 + 1824*K1*K3*K4 + 456*K1*K4*K5 + 64*K1*K5*K6 - 32*K2**6 - 64*K2**4*K3**2 - 32*K2**4*K4**2 + 224*K2**4*K4 - 2312*K2**4 + 128*K2**3*K3*K5 + 32*K2**3*K4*K6 - 96*K2**3*K6 - 768*K2**2*K3**2 - 32*K2**2*K3*K7 - 280*K2**2*K4**2 + 3192*K2**2*K4 - 48*K2**2*K5**2 - 8*K2**2*K6**2 - 5832*K2**2 + 1016*K2*K3*K5 + 288*K2*K4*K6 + 16*K2*K5*K7 + 16*K3**2*K6 - 2832*K3**2 - 1368*K4**2 - 368*K5**2 - 72*K6**2 + 6294
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.273']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16550', 'vk6.16641', 'vk6.17512', 'vk6.17567', 'vk6.18871', 'vk6.18949', 'vk6.19202', 'vk6.19495', 'vk6.23070', 'vk6.24107', 'vk6.25501', 'vk6.25576', 'vk6.26008', 'vk6.26392', 'vk6.34943', 'vk6.35061', 'vk6.36299', 'vk6.36366', 'vk6.37598', 'vk6.37687', 'vk6.42515', 'vk6.42624', 'vk6.43476', 'vk6.44589', 'vk6.54781', 'vk6.54870', 'vk6.56438', 'vk6.56550', 'vk6.59297', 'vk6.60183', 'vk6.66095', 'vk6.66137']
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 O1O2O3O4O5U1U4O6U3U6U5U2
R3 orbit {'O1O2O3O4O5U1U4O6U3U6U5U2'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U4U1U6U3O6U2U5
Gauss code of K* O1O2O3O4U5U4U1U6U3O5O6U2
Gauss code of -K* O1O2O3O4U3O5O6U2U5U4U1U6
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 3 1],[ 4 0 4 2 1 3 1],[-1 -4 0 -2 -1 2 1],[ 1 -2 2 0 0 3 1],[ 0 -1 1 0 0 1 0],[-3 -3 -2 -3 -1 0 0],[-1 -1 -1 -1 0 0 0]]
Primitive based matrix [[ 0 3 1 1 0 -1 -4],[-3 0 0 -2 -1 -3 -3],[-1 0 0 -1 0 -1 -1],[-1 2 1 0 -1 -2 -4],[ 0 1 0 1 0 0 -1],[ 1 3 1 2 0 0 -2],[ 4 3 1 4 1 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-1,-1,0,1,4,0,2,1,3,3,1,0,1,1,1,2,4,0,1,2]
Phi over symmetry [-4,-1,0,1,1,3,1,3,1,4,4,1,0,1,1,0,1,2,-1,0,2]
Phi of -K [-4,-1,0,1,1,3,1,3,1,4,4,1,0,1,1,0,1,2,-1,0,2]
Phi of K* [-3,-1,-1,0,1,4,0,2,2,1,4,1,0,0,1,1,1,4,1,3,1]
Phi of -K* [-4,-1,0,1,1,3,2,1,1,4,3,0,1,2,3,0,1,1,-1,0,2]
Symmetry type of based matrix c
u-polynomial t^4-t^3-t
Normalized Jones-Krushkal polynomial 4z^2+25z+35
Enhanced Jones-Krushkal polynomial 4w^3z^2+25w^2z+35w
Inner characteristic polynomial t^6+52t^4+18t^2+1
Outer characteristic polynomial t^7+80t^5+56t^3+8t
Flat arrow polynomial 4*K1**3 + 4*K1**2*K2 - 14*K1**2 - 6*K1*K2 - 2*K1*K3 + 6*K2 + 2*K3 + 7
2-strand cable arrow polynomial 256*K1**4*K2**3 - 960*K1**4*K2**2 + 3008*K1**4*K2 - 5088*K1**4 - 128*K1**3*K2**2*K3 + 704*K1**3*K2*K3 + 64*K1**3*K3*K4 - 1088*K1**3*K3 - 384*K1**2*K2**4 + 2592*K1**2*K2**3 - 9872*K1**2*K2**2 + 128*K1**2*K2*K3**2 + 64*K1**2*K2*K4**2 - 1088*K1**2*K2*K4 + 12336*K1**2*K2 - 416*K1**2*K3**2 - 160*K1**2*K4**2 - 6600*K1**2 + 736*K1*K2**3*K3 + 32*K1*K2**2*K3*K4 - 2272*K1*K2**2*K3 - 320*K1*K2**2*K5 - 576*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 9848*K1*K2*K3 - 192*K1*K2*K4*K5 + 1824*K1*K3*K4 + 456*K1*K4*K5 + 64*K1*K5*K6 - 32*K2**6 - 64*K2**4*K3**2 - 32*K2**4*K4**2 + 224*K2**4*K4 - 2312*K2**4 + 128*K2**3*K3*K5 + 32*K2**3*K4*K6 - 96*K2**3*K6 - 768*K2**2*K3**2 - 32*K2**2*K3*K7 - 280*K2**2*K4**2 + 3192*K2**2*K4 - 48*K2**2*K5**2 - 8*K2**2*K6**2 - 5832*K2**2 + 1016*K2*K3*K5 + 288*K2*K4*K6 + 16*K2*K5*K7 + 16*K3**2*K6 - 2832*K3**2 - 1368*K4**2 - 368*K5**2 - 72*K6**2 + 6294
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {4, 5}, {2, 3}], [{4, 6}, {1, 5}, {2, 3}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {2, 4}, {1, 3}], [{6}, {5}, {2, 4}, {1, 3}]]
If K is slice False
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