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Flat knot 6.855

Min(phi) over symmetries of the knot is: [-3,-2,0,1,2,2,-1,2,3,2,4,1,3,1,2,1,1,1,1,1,-1]
Flat knots (up to 7 crossings) with same phi are :['6.855']
Arrow polynomial of the knot is: 16*K1**3 - 6*K1**2 - 10*K1*K2 - 7*K1 + 3*K2 + K3 + 4
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.855']
Outer characteristic polynomial of the knot is: t^7+55t^5+65t^3+7t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.855']
2-strand cable arrow polynomial of the knot is: -192*K1**4*K2**2 + 224*K1**4*K2 - 608*K1**4 + 128*K1**3*K2**3*K3 + 224*K1**3*K2*K3 + 640*K1**2*K2**5 - 1856*K1**2*K2**4 + 4384*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 - 8224*K1**2*K2**2 + 64*K1**2*K2*K4**2 - 608*K1**2*K2*K4 + 6504*K1**2*K2 - 160*K1**2*K3**2 - 128*K1**2*K4**2 - 4320*K1**2 - 1152*K1*K2**4*K3 + 2144*K1*K2**3*K3 + 384*K1*K2**2*K3*K4 - 1664*K1*K2**2*K3 - 64*K1*K2**2*K5 + 32*K1*K2*K3**3 - 192*K1*K2*K3*K4 + 5624*K1*K2*K3 - 64*K1*K2*K4*K5 + 856*K1*K3*K4 + 184*K1*K4*K5 + 16*K1*K5*K6 - 768*K2**6 + 992*K2**4*K4 - 3448*K2**4 - 32*K2**3*K6 - 832*K2**2*K3**2 - 376*K2**2*K4**2 + 2432*K2**2*K4 - 1538*K2**2 - 32*K2*K3**2*K4 + 288*K2*K3*K5 + 136*K2*K4*K6 - 1364*K3**2 - 726*K4**2 - 108*K5**2 - 30*K6**2 + 3356
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.855']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.10130', 'vk6.10181', 'vk6.10326', 'vk6.10419', 'vk6.17671', 'vk6.17718', 'vk6.24234', 'vk6.24281', 'vk6.29909', 'vk6.29946', 'vk6.30011', 'vk6.30070', 'vk6.36504', 'vk6.36601', 'vk6.43603', 'vk6.43710', 'vk6.51614', 'vk6.51645', 'vk6.51688', 'vk6.51717', 'vk6.55713', 'vk6.55769', 'vk6.60283', 'vk6.60340', 'vk6.63327', 'vk6.63340', 'vk6.63371', 'vk6.63392', 'vk6.65415', 'vk6.65455', 'vk6.68555', 'vk6.68584']
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 O1O2O3O4U1U4O5O6U2U5U6U3
R3 orbit {'O1O2O3O4U1U4O5O6U2U5U6U3'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U2U5U6U3O5O6U1U4
Gauss code of K* O1O2O3O4U5U1U4U6O5O6U2U3
Gauss code of -K* O1O2O3O4U2U3O5O6U5U1U4U6
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 2 1 0 2],[ 3 0 2 3 1 1 1],[ 2 -2 0 3 0 1 2],[-2 -3 -3 0 0 -1 1],[-1 -1 0 0 0 0 0],[ 0 -1 -1 1 0 0 1],[-2 -1 -2 -1 0 -1 0]]
Primitive based matrix [[ 0 2 2 1 0 -2 -3],[-2 0 1 0 -1 -3 -3],[-2 -1 0 0 -1 -2 -1],[-1 0 0 0 0 0 -1],[ 0 1 1 0 0 -1 -1],[ 2 3 2 0 1 0 -2],[ 3 3 1 1 1 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,-1,0,2,3,-1,0,1,3,3,0,1,2,1,0,0,1,1,1,2]
Phi over symmetry [-3,-2,0,1,2,2,-1,2,3,2,4,1,3,1,2,1,1,1,1,1,-1]
Phi of -K [-3,-2,0,1,2,2,-1,2,3,2,4,1,3,1,2,1,1,1,1,1,-1]
Phi of K* [-2,-2,-1,0,2,3,-1,1,1,2,4,1,1,1,2,1,3,3,1,2,-1]
Phi of -K* [-3,-2,0,1,2,2,2,1,1,1,3,1,0,2,3,0,1,1,0,0,-1]
Symmetry type of based matrix c
u-polynomial t^3-t^2-t
Normalized Jones-Krushkal polynomial 3z^2+16z+21
Enhanced Jones-Krushkal polynomial -2w^4z^2+5w^3z^2-6w^3z+22w^2z+21w
Inner characteristic polynomial t^6+33t^4+24t^2
Outer characteristic polynomial t^7+55t^5+65t^3+7t
Flat arrow polynomial 16*K1**3 - 6*K1**2 - 10*K1*K2 - 7*K1 + 3*K2 + K3 + 4
2-strand cable arrow polynomial -192*K1**4*K2**2 + 224*K1**4*K2 - 608*K1**4 + 128*K1**3*K2**3*K3 + 224*K1**3*K2*K3 + 640*K1**2*K2**5 - 1856*K1**2*K2**4 + 4384*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 - 8224*K1**2*K2**2 + 64*K1**2*K2*K4**2 - 608*K1**2*K2*K4 + 6504*K1**2*K2 - 160*K1**2*K3**2 - 128*K1**2*K4**2 - 4320*K1**2 - 1152*K1*K2**4*K3 + 2144*K1*K2**3*K3 + 384*K1*K2**2*K3*K4 - 1664*K1*K2**2*K3 - 64*K1*K2**2*K5 + 32*K1*K2*K3**3 - 192*K1*K2*K3*K4 + 5624*K1*K2*K3 - 64*K1*K2*K4*K5 + 856*K1*K3*K4 + 184*K1*K4*K5 + 16*K1*K5*K6 - 768*K2**6 + 992*K2**4*K4 - 3448*K2**4 - 32*K2**3*K6 - 832*K2**2*K3**2 - 376*K2**2*K4**2 + 2432*K2**2*K4 - 1538*K2**2 - 32*K2*K3**2*K4 + 288*K2*K3*K5 + 136*K2*K4*K6 - 1364*K3**2 - 726*K4**2 - 108*K5**2 - 30*K6**2 + 3356
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {3, 5}, {2, 4}], [{4, 6}, {2, 5}, {1, 3}]]
If K is slice False
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