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Flat knot 6.107

Min(phi) over symmetries of the knot is: [-4,-1,0,1,1,3,0,1,3,4,3,0,1,1,1,1,1,2,0,2,2]
Flat knots (up to 7 crossings) with same phi are :['6.107']
Arrow polynomial of the knot is: -2*K1*K2 + K1 - 2*K2**2 + K3 + K4 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.95', '6.107', '6.276', '6.292', '6.394', '6.429', '6.463']
Outer characteristic polynomial of the knot is: t^7+80t^5+82t^3+5t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.107']
2-strand cable arrow polynomial of the knot is: 896*K1**4*K2 - 3248*K1**4 + 352*K1**3*K2*K3 + 128*K1**3*K3*K4 - 960*K1**3*K3 + 96*K1**2*K2**2*K4 - 2720*K1**2*K2**2 - 544*K1**2*K2*K4 + 7312*K1**2*K2 - 1360*K1**2*K3**2 - 32*K1**2*K3*K5 - 448*K1**2*K4**2 - 5116*K1**2 + 64*K1*K2**3*K3 + 224*K1*K2**2*K3*K4 - 864*K1*K2**2*K3 - 960*K1*K2*K3*K4 + 6248*K1*K2*K3 - 192*K1*K2*K4*K5 + 64*K1*K3**3*K4 - 32*K1*K3**2*K5 - 32*K1*K3*K4*K6 + 3216*K1*K3*K4 + 888*K1*K4*K5 + 40*K1*K5*K6 - 64*K2**4 - 416*K2**2*K3**2 - 424*K2**2*K4**2 + 1576*K2**2*K4 - 4678*K2**2 - 160*K2*K3**2*K4 + 832*K2*K3*K5 + 408*K2*K4*K6 - 160*K3**4 - 208*K3**2*K4**2 + 208*K3**2*K6 - 2816*K3**2 + 144*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 1720*K4**2 - 424*K5**2 - 114*K6**2 - 20*K7**2 - 2*K8**2 + 5016
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.107']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11493', 'vk6.11802', 'vk6.12817', 'vk6.13148', 'vk6.17067', 'vk6.17309', 'vk6.20908', 'vk6.21068', 'vk6.22318', 'vk6.22494', 'vk6.23788', 'vk6.28384', 'vk6.31254', 'vk6.31609', 'vk6.32825', 'vk6.35577', 'vk6.36029', 'vk6.40030', 'vk6.40313', 'vk6.42082', 'vk6.43272', 'vk6.46560', 'vk6.46770', 'vk6.48025', 'vk6.52248', 'vk6.53403', 'vk6.57714', 'vk6.57722', 'vk6.58898', 'vk6.59950', 'vk6.64417', 'vk6.69754']
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 O1O2O3O4O5O6U2U6U5U3U1U4
R3 orbit {'O1O2O3O4O5O6U2U6U5U3U1U4'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5O6U3U6U4U2U1U5
Gauss code of K* O1O2O3O4O5O6U5U1U4U6U3U2
Gauss code of -K* O1O2O3O4O5O6U5U4U1U3U6U2
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 -4 0 3 1 1],[ 1 0 -3 1 3 1 1],[ 4 3 0 3 4 2 1],[ 0 -1 -3 0 1 0 0],[-3 -3 -4 -1 0 0 0],[-1 -1 -2 0 0 0 0],[-1 -1 -1 0 0 0 0]]
Primitive based matrix [[ 0 3 1 1 0 -1 -4],[-3 0 0 0 -1 -3 -4],[-1 0 0 0 0 -1 -1],[-1 0 0 0 0 -1 -2],[ 0 1 0 0 0 -1 -3],[ 1 3 1 1 1 0 -3],[ 4 4 1 2 3 3 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-1,-1,0,1,4,0,0,1,3,4,0,0,1,1,0,1,2,1,3,3]
Phi over symmetry [-4,-1,0,1,1,3,0,1,3,4,3,0,1,1,1,1,1,2,0,2,2]
Phi of -K [-4,-1,0,1,1,3,0,1,3,4,3,0,1,1,1,1,1,2,0,2,2]
Phi of K* [-3,-1,-1,0,1,4,2,2,2,1,3,0,1,1,3,1,1,4,0,1,0]
Phi of -K* [-4,-1,0,1,1,3,3,3,1,2,4,1,1,1,3,0,0,1,0,0,0]
Symmetry type of based matrix c
u-polynomial t^4-t^3-t
Normalized Jones-Krushkal polynomial 7z^2+28z+29
Enhanced Jones-Krushkal polynomial 7w^3z^2+28w^2z+29w
Inner characteristic polynomial t^6+52t^4+22t^2+1
Outer characteristic polynomial t^7+80t^5+82t^3+5t
Flat arrow polynomial -2*K1*K2 + K1 - 2*K2**2 + K3 + K4 + 2
2-strand cable arrow polynomial 896*K1**4*K2 - 3248*K1**4 + 352*K1**3*K2*K3 + 128*K1**3*K3*K4 - 960*K1**3*K3 + 96*K1**2*K2**2*K4 - 2720*K1**2*K2**2 - 544*K1**2*K2*K4 + 7312*K1**2*K2 - 1360*K1**2*K3**2 - 32*K1**2*K3*K5 - 448*K1**2*K4**2 - 5116*K1**2 + 64*K1*K2**3*K3 + 224*K1*K2**2*K3*K4 - 864*K1*K2**2*K3 - 960*K1*K2*K3*K4 + 6248*K1*K2*K3 - 192*K1*K2*K4*K5 + 64*K1*K3**3*K4 - 32*K1*K3**2*K5 - 32*K1*K3*K4*K6 + 3216*K1*K3*K4 + 888*K1*K4*K5 + 40*K1*K5*K6 - 64*K2**4 - 416*K2**2*K3**2 - 424*K2**2*K4**2 + 1576*K2**2*K4 - 4678*K2**2 - 160*K2*K3**2*K4 + 832*K2*K3*K5 + 408*K2*K4*K6 - 160*K3**4 - 208*K3**2*K4**2 + 208*K3**2*K6 - 2816*K3**2 + 144*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 1720*K4**2 - 424*K5**2 - 114*K6**2 - 20*K7**2 - 2*K8**2 + 5016
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {1, 5}, {3, 4}], [{3, 6}, {1, 5}, {2, 4}], [{4, 6}, {1, 5}, {2, 3}], [{5, 6}, {1, 4}, {2, 3}]]
If K is slice False
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