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Flat knot 6.311

Min(phi) over symmetries of the knot is: [-3,-3,0,1,2,3,-1,1,2,2,4,2,2,1,3,0,0,1,1,1,-1]
Flat knots (up to 7 crossings) with same phi are :['6.311']
Arrow polynomial of the knot is: 4*K1**3 - 6*K1**2 - 6*K1*K2 + 3*K2 + 2*K3 + 4
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.311', '6.528', '6.536', '6.817', '6.982', '6.984', '6.1284']
Outer characteristic polynomial of the knot is: t^7+98t^5+79t^3+7t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.311']
2-strand cable arrow polynomial of the knot is: -64*K1**4*K2**2 + 64*K1**4*K2 - 80*K1**4 + 128*K1**3*K2**3*K3 - 128*K1**3*K2**2*K3 + 128*K1**3*K2*K3 - 192*K1**3*K3 - 1920*K1**2*K2**4 - 128*K1**2*K2**3*K4 + 2912*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 - 5856*K1**2*K2**2 + 192*K1**2*K2*K3**2 - 256*K1**2*K2*K4 + 4672*K1**2*K2 - 208*K1**2*K3**2 - 16*K1**2*K4**2 - 4040*K1**2 - 128*K1*K2**4*K3 + 3008*K1*K2**3*K3 + 224*K1*K2**2*K3*K4 - 1344*K1*K2**2*K3 - 256*K1*K2**2*K5 - 288*K1*K2*K3*K4 + 5712*K1*K2*K3 + 600*K1*K3*K4 + 96*K1*K4*K5 + 24*K1*K5*K6 - 32*K2**6 + 160*K2**4*K4 - 2680*K2**4 - 1472*K2**2*K3**2 - 120*K2**2*K4**2 + 1632*K2**2*K4 - 1636*K2**2 + 680*K2*K3*K5 + 24*K2*K4*K6 - 32*K3**4 + 32*K3**2*K6 - 1772*K3**2 - 446*K4**2 - 164*K5**2 - 28*K6**2 + 3124
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.311']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.71574', 'vk6.71688', 'vk6.72105', 'vk6.72311', 'vk6.74032', 'vk6.74593', 'vk6.76073', 'vk6.76785', 'vk6.77190', 'vk6.77289', 'vk6.77490', 'vk6.77650', 'vk6.79015', 'vk6.79591', 'vk6.80554', 'vk6.81004', 'vk6.81106', 'vk6.81139', 'vk6.81160', 'vk6.81213', 'vk6.81319', 'vk6.81463', 'vk6.82254', 'vk6.83498', 'vk6.83831', 'vk6.83972', 'vk6.85395', 'vk6.86325', 'vk6.87109', 'vk6.88027', 'vk6.88326', 'vk6.88960']
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 O1O2O3O4O5U2U3O6U4U1U5U6
R3 orbit {'O1O2O3O4O5U2U3O6U4U1U5U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U6U1U5U2O6U3U4
Gauss code of K* O1O2O3O4U2U5U6U1U3O5O6U4
Gauss code of -K* O1O2O3O4U1O5O6U2U4U5U6U3
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 -2 -3 -1 0 3 3],[ 2 0 -2 0 2 4 3],[ 3 2 0 1 2 3 2],[ 1 0 -1 0 1 2 2],[ 0 -2 -2 -1 0 1 2],[-3 -4 -3 -2 -1 0 1],[-3 -3 -2 -2 -2 -1 0]]
Primitive based matrix [[ 0 3 3 0 -1 -2 -3],[-3 0 1 -1 -2 -4 -3],[-3 -1 0 -2 -2 -3 -2],[ 0 1 2 0 -1 -2 -2],[ 1 2 2 1 0 0 -1],[ 2 4 3 2 0 0 -2],[ 3 3 2 2 1 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-3,0,1,2,3,-1,1,2,4,3,2,2,3,2,1,2,2,0,1,2]
Phi over symmetry [-3,-3,0,1,2,3,-1,1,2,2,4,2,2,1,3,0,0,1,1,1,-1]
Phi of -K [-3,-2,-1,0,3,3,-1,1,1,3,4,1,0,1,2,0,2,2,2,1,-1]
Phi of K* [-3,-3,0,1,2,3,-1,1,2,2,4,2,2,1,3,0,0,1,1,1,-1]
Phi of -K* [-3,-2,-1,0,3,3,2,1,2,2,3,0,2,3,4,1,2,2,2,1,-1]
Symmetry type of based matrix c
u-polynomial -t^3+t^2+t
Normalized Jones-Krushkal polynomial 4z^2+17z+19
Enhanced Jones-Krushkal polynomial -2w^4z^2+6w^3z^2-6w^3z+23w^2z+19w
Inner characteristic polynomial t^6+66t^4+26t^2
Outer characteristic polynomial t^7+98t^5+79t^3+7t
Flat arrow polynomial 4*K1**3 - 6*K1**2 - 6*K1*K2 + 3*K2 + 2*K3 + 4
2-strand cable arrow polynomial -64*K1**4*K2**2 + 64*K1**4*K2 - 80*K1**4 + 128*K1**3*K2**3*K3 - 128*K1**3*K2**2*K3 + 128*K1**3*K2*K3 - 192*K1**3*K3 - 1920*K1**2*K2**4 - 128*K1**2*K2**3*K4 + 2912*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 - 5856*K1**2*K2**2 + 192*K1**2*K2*K3**2 - 256*K1**2*K2*K4 + 4672*K1**2*K2 - 208*K1**2*K3**2 - 16*K1**2*K4**2 - 4040*K1**2 - 128*K1*K2**4*K3 + 3008*K1*K2**3*K3 + 224*K1*K2**2*K3*K4 - 1344*K1*K2**2*K3 - 256*K1*K2**2*K5 - 288*K1*K2*K3*K4 + 5712*K1*K2*K3 + 600*K1*K3*K4 + 96*K1*K4*K5 + 24*K1*K5*K6 - 32*K2**6 + 160*K2**4*K4 - 2680*K2**4 - 1472*K2**2*K3**2 - 120*K2**2*K4**2 + 1632*K2**2*K4 - 1636*K2**2 + 680*K2*K3*K5 + 24*K2*K4*K6 - 32*K3**4 + 32*K3**2*K6 - 1772*K3**2 - 446*K4**2 - 164*K5**2 - 28*K6**2 + 3124
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {4, 5}, {1, 3}]]
If K is slice False
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