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Flat knot 6.71

Min(phi) over symmetries of the knot is: [-4,-3,-2,3,3,3,0,1,3,4,5,0,1,2,3,2,3,4,0,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.71']
Arrow polynomial of the knot is: 4*K1**3 - 4*K1*K2 - 2*K1*K3 - K1 + K2 + K3 + K4 + 1
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.71', '6.148', '6.170', '6.253', '6.259', '6.298', '6.439', '6.453', '6.499', '6.503']
Outer characteristic polynomial of the knot is: t^7+152t^5+350t^3
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.71']
2-strand cable arrow polynomial of the knot is: -288*K1**2*K2**2 + 152*K1**2*K2 - 416*K1**2*K3**2 - 868*K1**2 + 480*K1*K2**3*K3 + 1592*K1*K2*K3 + 704*K1*K3*K4 + 64*K1*K4*K5 + 8*K1*K5*K6 + 16*K1*K6*K7 - 288*K2**6 - 256*K2**4*K3**2 + 160*K2**4*K4 - 528*K2**4 + 384*K2**3*K3*K5 - 1040*K2**2*K3**2 - 80*K2**2*K4**2 + 480*K2**2*K4 - 192*K2**2*K5**2 - 8*K2**2*K6**2 - 490*K2**2 + 672*K2*K3*K5 + 112*K2*K4*K6 + 48*K2*K5*K7 + 8*K2*K6*K8 + 24*K3**2*K6 - 888*K3**2 - 450*K4**2 - 180*K5**2 - 62*K6**2 - 16*K7**2 - 2*K8**2 + 1146
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.71']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.81916', 'vk6.81919', 'vk6.81969', 'vk6.81972', 'vk6.81988', 'vk6.81990', 'vk6.82120', 'vk6.82126', 'vk6.82138', 'vk6.82141', 'vk6.82719', 'vk6.82721', 'vk6.82796', 'vk6.82797', 'vk6.83085', 'vk6.83086', 'vk6.83460', 'vk6.83465', 'vk6.84672', 'vk6.84677', 'vk6.84792', 'vk6.84802', 'vk6.84824', 'vk6.84828', 'vk6.86226', 'vk6.86229', 'vk6.88488', 'vk6.88492', 'vk6.88633', 'vk6.88641', 'vk6.89653', 'vk6.89655']
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 O1O2O3O4O5O6U2U3U1U6U5U4
R3 orbit {'O1O2O3O4O5O6U2U3U1U6U5U4'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5O6U3U2U1U6U4U5
Gauss code of K* O1O2O3O4O5O6U3U1U2U6U5U4
Gauss code of -K* O1O2O3O4O5O6U3U2U1U5U6U4
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 -3 -4 -2 3 3 3],[ 3 0 -1 1 5 4 3],[ 4 1 0 1 4 3 2],[ 2 -1 -1 0 3 2 1],[-3 -5 -4 -3 0 0 0],[-3 -4 -3 -2 0 0 0],[-3 -3 -2 -1 0 0 0]]
Primitive based matrix [[ 0 3 3 3 -2 -3 -4],[-3 0 0 0 -1 -3 -2],[-3 0 0 0 -2 -4 -3],[-3 0 0 0 -3 -5 -4],[ 2 1 2 3 0 -1 -1],[ 3 3 4 5 1 0 -1],[ 4 2 3 4 1 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-3,-3,2,3,4,0,0,1,3,2,0,2,4,3,3,5,4,1,1,1]
Phi over symmetry [-4,-3,-2,3,3,3,0,1,3,4,5,0,1,2,3,2,3,4,0,0,0]
Phi of -K [-4,-3,-2,3,3,3,0,1,3,4,5,0,1,2,3,2,3,4,0,0,0]
Phi of K* [-3,-3,-3,2,3,4,0,0,2,1,3,0,3,2,4,4,3,5,0,1,0]
Phi of -K* [-4,-3,-2,3,3,3,1,1,2,3,4,1,3,4,5,1,2,3,0,0,0]
Symmetry type of based matrix c
u-polynomial t^4-2t^3+t^2
Normalized Jones-Krushkal polynomial 3z+7
Enhanced Jones-Krushkal polynomial -8w^5z+16w^4z-10w^3z+5w^2z+7w
Inner characteristic polynomial t^6+96t^4+41t^2
Outer characteristic polynomial t^7+152t^5+350t^3
Flat arrow polynomial 4*K1**3 - 4*K1*K2 - 2*K1*K3 - K1 + K2 + K3 + K4 + 1
2-strand cable arrow polynomial -288*K1**2*K2**2 + 152*K1**2*K2 - 416*K1**2*K3**2 - 868*K1**2 + 480*K1*K2**3*K3 + 1592*K1*K2*K3 + 704*K1*K3*K4 + 64*K1*K4*K5 + 8*K1*K5*K6 + 16*K1*K6*K7 - 288*K2**6 - 256*K2**4*K3**2 + 160*K2**4*K4 - 528*K2**4 + 384*K2**3*K3*K5 - 1040*K2**2*K3**2 - 80*K2**2*K4**2 + 480*K2**2*K4 - 192*K2**2*K5**2 - 8*K2**2*K6**2 - 490*K2**2 + 672*K2*K3*K5 + 112*K2*K4*K6 + 48*K2*K5*K7 + 8*K2*K6*K8 + 24*K3**2*K6 - 888*K3**2 - 450*K4**2 - 180*K5**2 - 62*K6**2 - 16*K7**2 - 2*K8**2 + 1146
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {3, 5}, {1, 4}], [{3, 6}, {2, 5}, {1, 4}], [{5, 6}, {1, 4}, {2, 3}], [{6}, {2, 5}, {1, 4}, {3}]]
If K is slice False
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