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Flat knot 6.735

Min(phi) over symmetries of the knot is: [-3,-2,1,1,1,2,-1,1,3,4,3,0,1,2,1,0,-1,0,0,1,2]
Flat knots (up to 7 crossings) with same phi are :['6.735']
Arrow polynomial of the knot is: 4*K1**3 - 8*K1**2 - 2*K1*K2 - 2*K1 + 4*K2 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.635', '6.639', '6.659', '6.727', '6.732', '6.735', '6.799', '6.1088', '6.1090', '6.1095', '6.1383']
Outer characteristic polynomial of the knot is: t^7+64t^5+52t^3+5t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.735']
2-strand cable arrow polynomial of the knot is: -128*K1**6 - 384*K1**4*K2**2 + 3904*K1**4*K2 - 7584*K1**4 + 352*K1**3*K2*K3 - 2144*K1**3*K3 - 192*K1**2*K2**4 + 1248*K1**2*K2**3 + 192*K1**2*K2**2*K4 - 6208*K1**2*K2**2 - 480*K1**2*K2*K4 + 11968*K1**2*K2 - 480*K1**2*K3**2 - 32*K1**2*K3*K5 - 48*K1**2*K4**2 - 4600*K1**2 + 128*K1*K2**3*K3 - 576*K1*K2**2*K3 - 64*K1*K2**2*K5 - 32*K1*K2*K3*K4 + 5816*K1*K2*K3 + 664*K1*K3*K4 + 56*K1*K4*K5 - 32*K2**6 + 32*K2**4*K4 - 608*K2**4 - 32*K2**2*K3**2 - 8*K2**2*K4**2 + 448*K2**2*K4 - 3976*K2**2 + 8*K2*K3*K5 - 1344*K3**2 - 200*K4**2 - 8*K5**2 + 4342
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.735']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4837', 'vk6.5182', 'vk6.6403', 'vk6.6836', 'vk6.8364', 'vk6.8794', 'vk6.9730', 'vk6.10035', 'vk6.11640', 'vk6.11993', 'vk6.12982', 'vk6.20457', 'vk6.20726', 'vk6.21810', 'vk6.27841', 'vk6.29349', 'vk6.31439', 'vk6.32613', 'vk6.39267', 'vk6.39766', 'vk6.41445', 'vk6.46326', 'vk6.47572', 'vk6.47901', 'vk6.49064', 'vk6.49894', 'vk6.51316', 'vk6.51535', 'vk6.53231', 'vk6.57328', 'vk6.62014', 'vk6.64312']
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 O1O2O3O4U1O5U6U5U3O6U4U2
R3 orbit {'O1O2O3O4U1O5U6U5U3O6U4U2'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U3U1O5U2U6U5O6U4
Gauss code of K* O1O2O3U1O4O5U6U5U3U4O6U2
Gauss code of -K* O1O2O3U2O4U5U1U6U4O6O5U3
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 2 1 -2],[ 3 0 3 1 2 0 2],[-1 -3 0 0 1 1 -3],[-1 -1 0 0 0 0 -2],[-2 -2 -1 0 0 1 -3],[-1 0 -1 0 -1 0 -1],[ 2 -2 3 2 3 1 0]]
Primitive based matrix [[ 0 2 1 1 1 -2 -3],[-2 0 1 0 -1 -3 -2],[-1 -1 0 0 -1 -1 0],[-1 0 0 0 0 -2 -1],[-1 1 1 0 0 -3 -3],[ 2 3 1 2 3 0 -2],[ 3 2 0 1 3 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,-1,-1,2,3,-1,0,1,3,2,0,1,1,0,0,2,1,3,3,2]
Phi over symmetry [-3,-2,1,1,1,2,-1,1,3,4,3,0,1,2,1,0,-1,0,0,1,2]
Phi of -K [-3,-2,1,1,1,2,-1,1,3,4,3,0,1,2,1,0,-1,0,0,1,2]
Phi of K* [-2,-1,-1,-1,2,3,0,1,2,1,3,0,1,0,1,0,1,3,2,4,-1]
Phi of -K* [-3,-2,1,1,1,2,2,0,1,3,2,1,2,3,3,0,-1,-1,0,0,1]
Symmetry type of based matrix c
u-polynomial t^3-3t
Normalized Jones-Krushkal polynomial 3z^2+24z+37
Enhanced Jones-Krushkal polynomial 3w^3z^2+24w^2z+37w
Inner characteristic polynomial t^6+44t^4+30t^2+1
Outer characteristic polynomial t^7+64t^5+52t^3+5t
Flat arrow polynomial 4*K1**3 - 8*K1**2 - 2*K1*K2 - 2*K1 + 4*K2 + 5
2-strand cable arrow polynomial -128*K1**6 - 384*K1**4*K2**2 + 3904*K1**4*K2 - 7584*K1**4 + 352*K1**3*K2*K3 - 2144*K1**3*K3 - 192*K1**2*K2**4 + 1248*K1**2*K2**3 + 192*K1**2*K2**2*K4 - 6208*K1**2*K2**2 - 480*K1**2*K2*K4 + 11968*K1**2*K2 - 480*K1**2*K3**2 - 32*K1**2*K3*K5 - 48*K1**2*K4**2 - 4600*K1**2 + 128*K1*K2**3*K3 - 576*K1*K2**2*K3 - 64*K1*K2**2*K5 - 32*K1*K2*K3*K4 + 5816*K1*K2*K3 + 664*K1*K3*K4 + 56*K1*K4*K5 - 32*K2**6 + 32*K2**4*K4 - 608*K2**4 - 32*K2**2*K3**2 - 8*K2**2*K4**2 + 448*K2**2*K4 - 3976*K2**2 + 8*K2*K3*K5 - 1344*K3**2 - 200*K4**2 - 8*K5**2 + 4342
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {3, 5}, {2, 4}], [{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {5}, {1, 4}, {3}]]
If K is slice False
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