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Flat knot 6.370

Min(phi) over symmetries of the knot is: [-3,-1,2,2,0,2,3,0,1,-1]
Flat knots (up to 7 crossings) with same phi are :['6.370']
Arrow polynomial of the knot is: -4*K1**2 - 2*K1*K2 + K1 + 2*K2 + K3 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.136', '6.207', '6.342', '6.370', '6.376', '6.442', '6.456', '6.539', '6.631', '6.636', '6.674', '6.679', '6.705', '6.740', '6.760', '6.794', '6.795', '6.1369']
Outer characteristic polynomial of the knot is: t^5+33t^3+10t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.370']
2-strand cable arrow polynomial of the knot is: -64*K1**6 + 64*K1**4*K2 - 624*K1**4 - 416*K1**2*K2**2 + 928*K1**2*K2 - 336*K1**2*K3**2 - 48*K1**2*K4**2 - 500*K1**2 + 952*K1*K2*K3 + 448*K1*K3*K4 + 40*K1*K4*K5 - 48*K2**4 - 64*K2**2*K3**2 - 8*K2**2*K4**2 + 112*K2**2*K4 - 574*K2**2 + 88*K2*K3*K5 + 8*K2*K4*K6 - 432*K3**2 - 180*K4**2 - 36*K5**2 - 2*K6**2 + 690
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.370']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4133', 'vk6.4166', 'vk6.5371', 'vk6.5404', 'vk6.5467', 'vk6.5580', 'vk6.7493', 'vk6.7661', 'vk6.8994', 'vk6.9027', 'vk6.11186', 'vk6.12272', 'vk6.12381', 'vk6.12431', 'vk6.12462', 'vk6.13363', 'vk6.13586', 'vk6.13619', 'vk6.14275', 'vk6.14722', 'vk6.14746', 'vk6.15181', 'vk6.15878', 'vk6.15902', 'vk6.26197', 'vk6.26640', 'vk6.30836', 'vk6.30867', 'vk6.31305', 'vk6.31702', 'vk6.32020', 'vk6.32051', 'vk6.32463', 'vk6.32880', 'vk6.33079', 'vk6.33112', 'vk6.38175', 'vk6.38181', 'vk6.39077', 'vk6.44832', 'vk6.44926', 'vk6.45833', 'vk6.49322', 'vk6.52321', 'vk6.53165', 'vk6.53535', 'vk6.58431', 'vk6.62955']
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 O1O2O3O4O5U3U5O6U2U6U1U4
R3 orbit {'O1O2O3O4O5U3U5O6U2U6U1U4', 'O1O2O3O4U2U5U3O6U1U6O5U4', 'O1O2O3O4O5U3U5U1O6U2U6U4'}
R3 orbit length 3
Gauss code of -K O1O2O3O4O5U2U5U6U4O6U1U3
Gauss code of K* O1O2O3O4U3U1U5U4U6O5O6U2
Gauss code of -K* O1O2O3O4U3O5O6U5U1U6U4U2
Diagrammatic symmetry type c
Flat genus of the diagram 2
If K is checkerboard colorable False
If K is almost classical False
Based matrix from Gauss code [[ 0 -1 -2 -2 3 1 1],[ 1 0 -1 -1 3 1 1],[ 2 1 0 -1 3 1 1],[ 2 1 1 0 2 1 0],[-3 -3 -3 -2 0 0 0],[-1 -1 -1 -1 0 0 0],[-1 -1 -1 0 0 0 0]]
Primitive based matrix [[ 0 3 1 -2 -2],[-3 0 0 -2 -3],[-1 0 0 0 -1],[ 2 2 0 0 1],[ 2 3 1 -1 0]]
If based matrix primitive False
Phi of primitive based matrix [-3,-1,2,2,0,2,3,0,1,-1]
Phi over symmetry [-3,-1,2,2,0,2,3,0,1,-1]
Phi of -K [-2,-2,1,3,-1,3,3,2,2,2]
Phi of K* [-3,-1,2,2,2,2,3,2,3,-1]
Phi of -K* [-2,-2,1,3,-1,1,3,0,2,0]
Symmetry type of based matrix c
u-polynomial -t^3+2t^2-t
Normalized Jones-Krushkal polynomial 9z+19
Enhanced Jones-Krushkal polynomial 9w^2z+19w
Inner characteristic polynomial t^4+15t^2+4
Outer characteristic polynomial t^5+33t^3+10t
Flat arrow polynomial -4*K1**2 - 2*K1*K2 + K1 + 2*K2 + K3 + 3
2-strand cable arrow polynomial -64*K1**6 + 64*K1**4*K2 - 624*K1**4 - 416*K1**2*K2**2 + 928*K1**2*K2 - 336*K1**2*K3**2 - 48*K1**2*K4**2 - 500*K1**2 + 952*K1*K2*K3 + 448*K1*K3*K4 + 40*K1*K4*K5 - 48*K2**4 - 64*K2**2*K3**2 - 8*K2**2*K4**2 + 112*K2**2*K4 - 574*K2**2 + 88*K2*K3*K5 + 8*K2*K4*K6 - 432*K3**2 - 180*K4**2 - 36*K5**2 - 2*K6**2 + 690
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {5}, {3, 4}, {1}], [{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {5}, {2, 4}, {1}], [{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {1, 5}, {3}, {2}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {1, 4}, {3}, {2}]]
If K is slice False
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