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Flat knot 6.895

Min(phi) over symmetries of the knot is: [-3,-2,1,1,1,2,-1,1,1,3,3,0,1,2,1,0,0,0,0,0,1]
Flat knots (up to 7 crossings) with same phi are :['6.895']
Arrow polynomial of the knot is: 4*K1**3 + 4*K1**2*K2 - 8*K1**2 - 6*K1*K2 - 4*K1*K3 + 4*K2 + 2*K3 + K4 + 4
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.364', '6.895', '6.907']
Outer characteristic polynomial of the knot is: t^7+48t^5+46t^3+9t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.895']
2-strand cable arrow polynomial of the knot is: -192*K1**6 + 256*K1**4*K2**3 - 640*K1**4*K2**2 + 1408*K1**4*K2 - 2832*K1**4 - 512*K1**3*K2**2*K3 + 992*K1**3*K2*K3 + 32*K1**3*K3*K4 - 1568*K1**3*K3 - 384*K1**2*K2**4 + 1920*K1**2*K2**3 - 5280*K1**2*K2**2 + 352*K1**2*K2*K3**2 - 928*K1**2*K2*K4 + 8208*K1**2*K2 - 1072*K1**2*K3**2 - 32*K1**2*K3*K5 - 80*K1**2*K4**2 - 5996*K1**2 + 1216*K1*K2**3*K3 - 1088*K1*K2**2*K3 - 64*K1*K2**2*K5 + 32*K1*K2*K3**3 - 288*K1*K2*K3*K4 + 7832*K1*K2*K3 - 64*K1*K3**2*K5 + 1832*K1*K3*K4 + 224*K1*K4*K5 + 72*K1*K5*K6 - 32*K2**6 - 64*K2**4*K3**2 - 32*K2**4*K4**2 + 160*K2**4*K4 - 1360*K2**4 + 128*K2**3*K3*K5 + 64*K2**3*K4*K6 - 1104*K2**2*K3**2 - 248*K2**2*K4**2 + 1296*K2**2*K4 - 112*K2**2*K5**2 - 48*K2**2*K6**2 - 4132*K2**2 + 912*K2*K3*K5 + 208*K2*K4*K6 + 64*K2*K5*K7 + 16*K2*K6*K8 - 32*K3**4 + 80*K3**2*K6 - 2852*K3**2 - 948*K4**2 - 360*K5**2 - 116*K6**2 - 16*K7**2 - 2*K8**2 + 5172
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.895']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4208', 'vk6.4289', 'vk6.5469', 'vk6.5582', 'vk6.7569', 'vk6.7659', 'vk6.9073', 'vk6.9154', 'vk6.11190', 'vk6.12276', 'vk6.12385', 'vk6.19379', 'vk6.19672', 'vk6.19766', 'vk6.26159', 'vk6.26205', 'vk6.26575', 'vk6.26648', 'vk6.30776', 'vk6.31981', 'vk6.38167', 'vk6.38189', 'vk6.44824', 'vk6.44934', 'vk6.48518', 'vk6.49215', 'vk6.49324', 'vk6.50305', 'vk6.52758', 'vk6.63596', 'vk6.66331', 'vk6.66337']
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 O1O2O3O4U2U4O5O6U1U6U3U5
R3 orbit {'O1O2O3O4U2U4O5O6U1U6U3U5'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U5U2U6U4O6O5U1U3
Gauss code of K* O1O2O3O4U1U5U3U6O5O6U4U2
Gauss code of -K* O1O2O3O4U3U1O5O6U5U2U6U4
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 1 2 1],[ 3 0 -1 3 1 3 1],[ 2 1 0 2 1 1 0],[-1 -3 -2 0 0 1 0],[-1 -1 -1 0 0 0 0],[-2 -3 -1 -1 0 0 0],[-1 -1 0 0 0 0 0]]
Primitive based matrix [[ 0 2 1 1 1 -2 -3],[-2 0 0 0 -1 -1 -3],[-1 0 0 0 0 0 -1],[-1 0 0 0 0 -1 -1],[-1 1 0 0 0 -2 -3],[ 2 1 0 1 2 0 1],[ 3 3 1 1 3 -1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,-1,-1,2,3,0,0,1,1,3,0,0,0,1,0,1,1,2,3,-1]
Phi over symmetry [-3,-2,1,1,1,2,-1,1,1,3,3,0,1,2,1,0,0,0,0,0,1]
Phi of -K [-3,-2,1,1,1,2,2,1,3,3,2,1,2,3,3,0,0,0,0,1,1]
Phi of K* [-2,-1,-1,-1,2,3,0,1,1,3,2,0,0,1,1,0,2,3,3,3,2]
Phi of -K* [-3,-2,1,1,1,2,-1,1,1,3,3,0,1,2,1,0,0,0,0,0,1]
Symmetry type of based matrix c
u-polynomial t^3-3t
Normalized Jones-Krushkal polynomial z^2+18z+33
Enhanced Jones-Krushkal polynomial w^3z^2-4w^3z+22w^2z+33w
Inner characteristic polynomial t^6+28t^4+18t^2+1
Outer characteristic polynomial t^7+48t^5+46t^3+9t
Flat arrow polynomial 4*K1**3 + 4*K1**2*K2 - 8*K1**2 - 6*K1*K2 - 4*K1*K3 + 4*K2 + 2*K3 + K4 + 4
2-strand cable arrow polynomial -192*K1**6 + 256*K1**4*K2**3 - 640*K1**4*K2**2 + 1408*K1**4*K2 - 2832*K1**4 - 512*K1**3*K2**2*K3 + 992*K1**3*K2*K3 + 32*K1**3*K3*K4 - 1568*K1**3*K3 - 384*K1**2*K2**4 + 1920*K1**2*K2**3 - 5280*K1**2*K2**2 + 352*K1**2*K2*K3**2 - 928*K1**2*K2*K4 + 8208*K1**2*K2 - 1072*K1**2*K3**2 - 32*K1**2*K3*K5 - 80*K1**2*K4**2 - 5996*K1**2 + 1216*K1*K2**3*K3 - 1088*K1*K2**2*K3 - 64*K1*K2**2*K5 + 32*K1*K2*K3**3 - 288*K1*K2*K3*K4 + 7832*K1*K2*K3 - 64*K1*K3**2*K5 + 1832*K1*K3*K4 + 224*K1*K4*K5 + 72*K1*K5*K6 - 32*K2**6 - 64*K2**4*K3**2 - 32*K2**4*K4**2 + 160*K2**4*K4 - 1360*K2**4 + 128*K2**3*K3*K5 + 64*K2**3*K4*K6 - 1104*K2**2*K3**2 - 248*K2**2*K4**2 + 1296*K2**2*K4 - 112*K2**2*K5**2 - 48*K2**2*K6**2 - 4132*K2**2 + 912*K2*K3*K5 + 208*K2*K4*K6 + 64*K2*K5*K7 + 16*K2*K6*K8 - 32*K3**4 + 80*K3**2*K6 - 2852*K3**2 - 948*K4**2 - 360*K5**2 - 116*K6**2 - 16*K7**2 - 2*K8**2 + 5172
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {3, 5}, {2, 4}], [{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {5}, {1, 4}, {2}], [{4, 6}, {2, 5}, {1, 3}]]
If K is slice False
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