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Flat knot 6.448

Min(phi) over symmetries of the knot is: [-3,-3,1,1,1,3,0,0,2,3,3,1,3,3,4,1,-1,1,0,1,2]
Flat knots (up to 7 crossings) with same phi are :['6.448']
Arrow polynomial of the knot is: 4*K1**2*K2 - 4*K1**2 - 2*K1*K2 - 4*K1*K3 + K1 + 2*K2 + K3 + K4 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.115', '6.407', '6.413', '6.448', '6.844', '6.879', '6.888', '6.926', '6.934', '6.1140', '6.1143', '6.1161', '6.1177']
Outer characteristic polynomial of the knot is: t^7+95t^5+341t^3+12t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.448']
2-strand cable arrow polynomial of the knot is: -256*K1**4*K2**2 + 288*K1**4*K2 - 480*K1**4 + 384*K1**3*K2*K3 - 384*K1**3*K3 + 928*K1**2*K2**3 - 3184*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 480*K1**2*K2*K4 + 4888*K1**2*K2 - 288*K1**2*K3**2 - 4388*K1**2 + 192*K1*K2**3*K3 + 416*K1*K2**2*K3*K4 - 1824*K1*K2**2*K3 + 160*K1*K2**2*K4*K5 - 160*K1*K2**2*K5 - 384*K1*K2*K3*K4 + 5568*K1*K2*K3 - 128*K1*K2*K4*K5 + 1432*K1*K3*K4 + 192*K1*K4*K5 + 32*K1*K5*K6 - 32*K2**4*K4**2 + 480*K2**4*K4 - 1792*K2**4 + 224*K2**3*K3*K5 + 64*K2**3*K4*K6 - 224*K2**3*K6 - 1088*K2**2*K3**2 - 64*K2**2*K3*K7 - 648*K2**2*K4**2 - 32*K2**2*K4*K8 + 2336*K2**2*K4 - 208*K2**2*K5**2 - 16*K2**2*K6**2 - 3134*K2**2 - 128*K2*K3**2*K4 + 1008*K2*K3*K5 + 360*K2*K4*K6 + 48*K2*K5*K7 + 16*K2*K6*K8 + 24*K3**2*K6 - 2128*K3**2 - 952*K4**2 - 272*K5**2 - 58*K6**2 - 4*K7**2 - 2*K8**2 + 3632
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.448']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11506', 'vk6.11827', 'vk6.12844', 'vk6.13163', 'vk6.20272', 'vk6.21595', 'vk6.27540', 'vk6.29110', 'vk6.31271', 'vk6.31643', 'vk6.32417', 'vk6.32844', 'vk6.38943', 'vk6.41176', 'vk6.45712', 'vk6.47419', 'vk6.52273', 'vk6.52530', 'vk6.53106', 'vk6.53430', 'vk6.57109', 'vk6.58289', 'vk6.61696', 'vk6.62847', 'vk6.63792', 'vk6.63916', 'vk6.64232', 'vk6.64436', 'vk6.66740', 'vk6.67618', 'vk6.69396', 'vk6.70124']
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 O1O2O3O4O5U6U3O6U1U2U5U4
R3 orbit {'O1O2O3O4O5U6U3O6U1U2U5U4'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U2U1U4U5O6U3U6
Gauss code of K* O1O2O3O4U1U2U5U4U3O6O5U6
Gauss code of -K* O1O2O3O4U5O6O5U2U1U6U3U4
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 -1 -1 3 3 -1],[ 3 0 1 1 4 3 2],[ 1 -1 0 1 3 2 0],[ 1 -1 -1 0 1 0 1],[-3 -4 -3 -1 0 0 -3],[-3 -3 -2 0 0 0 -3],[ 1 -2 0 -1 3 3 0]]
Primitive based matrix [[ 0 3 3 -1 -1 -1 -3],[-3 0 0 0 -2 -3 -3],[-3 0 0 -1 -3 -3 -4],[ 1 0 1 0 -1 1 -1],[ 1 2 3 1 0 0 -1],[ 1 3 3 -1 0 0 -2],[ 3 3 4 1 1 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-3,1,1,1,3,0,0,2,3,3,1,3,3,4,1,-1,1,0,1,2]
Phi over symmetry [-3,-3,1,1,1,3,0,0,2,3,3,1,3,3,4,1,-1,1,0,1,2]
Phi of -K [-3,-1,-1,-1,3,3,0,1,1,2,3,0,1,1,1,-1,1,2,3,4,0]
Phi of K* [-3,-3,1,1,1,3,0,1,1,3,2,1,2,4,3,0,-1,0,1,1,1]
Phi of -K* [-3,-1,-1,-1,3,3,1,1,2,3,4,-1,1,0,1,0,2,3,3,3,0]
Symmetry type of based matrix c
u-polynomial -t^3+3t
Normalized Jones-Krushkal polynomial 6z^2+23z+23
Enhanced Jones-Krushkal polynomial -2w^4z^2+8w^3z^2-2w^3z+25w^2z+23w
Inner characteristic polynomial t^6+65t^4+187t^2+4
Outer characteristic polynomial t^7+95t^5+341t^3+12t
Flat arrow polynomial 4*K1**2*K2 - 4*K1**2 - 2*K1*K2 - 4*K1*K3 + K1 + 2*K2 + K3 + K4 + 2
2-strand cable arrow polynomial -256*K1**4*K2**2 + 288*K1**4*K2 - 480*K1**4 + 384*K1**3*K2*K3 - 384*K1**3*K3 + 928*K1**2*K2**3 - 3184*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 480*K1**2*K2*K4 + 4888*K1**2*K2 - 288*K1**2*K3**2 - 4388*K1**2 + 192*K1*K2**3*K3 + 416*K1*K2**2*K3*K4 - 1824*K1*K2**2*K3 + 160*K1*K2**2*K4*K5 - 160*K1*K2**2*K5 - 384*K1*K2*K3*K4 + 5568*K1*K2*K3 - 128*K1*K2*K4*K5 + 1432*K1*K3*K4 + 192*K1*K4*K5 + 32*K1*K5*K6 - 32*K2**4*K4**2 + 480*K2**4*K4 - 1792*K2**4 + 224*K2**3*K3*K5 + 64*K2**3*K4*K6 - 224*K2**3*K6 - 1088*K2**2*K3**2 - 64*K2**2*K3*K7 - 648*K2**2*K4**2 - 32*K2**2*K4*K8 + 2336*K2**2*K4 - 208*K2**2*K5**2 - 16*K2**2*K6**2 - 3134*K2**2 - 128*K2*K3**2*K4 + 1008*K2*K3*K5 + 360*K2*K4*K6 + 48*K2*K5*K7 + 16*K2*K6*K8 + 24*K3**2*K6 - 2128*K3**2 - 952*K4**2 - 272*K5**2 - 58*K6**2 - 4*K7**2 - 2*K8**2 + 3632
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {3, 5}, {1, 4}], [{3, 6}, {2, 5}, {1, 4}], [{4, 6}, {3, 5}, {1, 2}], [{5, 6}, {1, 4}, {2, 3}]]
If K is slice False
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