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Flat knot 6.879

Min(phi) over symmetries of the knot is: [-3,0,0,1,1,1,1,2,1,1,2,0,0,1,0,0,1,1,0,-1,-1]
Flat knots (up to 7 crossings) with same phi are :['6.879']
Arrow polynomial of the knot is: 4*K1**2*K2 - 4*K1**2 - 2*K1*K2 - 4*K1*K3 + K1 + 2*K2 + K3 + K4 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.115', '6.407', '6.413', '6.448', '6.844', '6.879', '6.888', '6.926', '6.934', '6.1140', '6.1143', '6.1161', '6.1177']
Outer characteristic polynomial of the knot is: t^7+44t^5+59t^3+16t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.879']
2-strand cable arrow polynomial of the knot is: 384*K1**4*K2 - 1632*K1**4 - 128*K1**3*K2**2*K3 + 576*K1**3*K2*K3 + 32*K1**3*K3*K4 - 640*K1**3*K3 + 1024*K1**2*K2**3 + 96*K1**2*K2**2*K4 - 4656*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 832*K1**2*K2*K4 + 6784*K1**2*K2 - 384*K1**2*K3**2 - 64*K1**2*K4**2 - 4892*K1**2 + 800*K1*K2**3*K3 - 1376*K1*K2**2*K3 - 288*K1*K2**2*K5 - 416*K1*K2*K3*K4 + 6408*K1*K2*K3 + 1224*K1*K3*K4 + 256*K1*K4*K5 + 40*K1*K5*K6 - 32*K2**4*K4**2 + 320*K2**4*K4 - 1280*K2**4 + 32*K2**3*K3*K5 + 64*K2**3*K4*K6 - 64*K2**3*K6 - 768*K2**2*K3**2 - 744*K2**2*K4**2 + 2344*K2**2*K4 - 48*K2**2*K5**2 - 48*K2**2*K6**2 - 4270*K2**2 + 816*K2*K3*K5 + 520*K2*K4*K6 + 24*K2*K5*K7 + 16*K2*K6*K8 + 40*K3**2*K6 - 2184*K3**2 - 1072*K4**2 - 288*K5**2 - 138*K6**2 - 4*K7**2 - 2*K8**2 + 4384
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.879']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11651', 'vk6.11655', 'vk6.12002', 'vk6.12006', 'vk6.12997', 'vk6.13001', 'vk6.13268', 'vk6.20440', 'vk6.20442', 'vk6.21801', 'vk6.27816', 'vk6.27818', 'vk6.29326', 'vk6.29328', 'vk6.31457', 'vk6.32640', 'vk6.32644', 'vk6.32982', 'vk6.32986', 'vk6.39242', 'vk6.39244', 'vk6.47567', 'vk6.52375', 'vk6.53266', 'vk6.53270', 'vk6.57301', 'vk6.57303', 'vk6.61987', 'vk6.64331', 'vk6.64335', 'vk6.64509', 'vk6.66890']
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 O1O2O3O4U1U5O6O5U4U3U6U2
R3 orbit {'O1O2O3O4U1U5O6O5U4U3U6U2'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U3U5U2U1O6O5U6U4
Gauss code of K* O1O2O3O4U5U4U2U1O5O6U3U6
Gauss code of -K* O1O2O3O4U5U2O5O6U4U3U1U6
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 0 0 1 1],[ 3 0 3 2 1 3 2],[-1 -3 0 -1 -1 0 1],[ 0 -2 1 0 0 0 1],[ 0 -1 1 0 0 0 0],[-1 -3 0 0 0 0 1],[-1 -2 -1 -1 0 -1 0]]
Primitive based matrix [[ 0 1 1 1 0 0 -3],[-1 0 1 0 0 0 -3],[-1 -1 0 -1 0 -1 -2],[-1 0 1 0 -1 -1 -3],[ 0 0 0 1 0 0 -1],[ 0 0 1 1 0 0 -2],[ 3 3 2 3 1 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-1,-1,-1,0,0,3,-1,0,0,0,3,1,0,1,2,1,1,3,0,1,2]
Phi over symmetry [-3,0,0,1,1,1,1,2,1,1,2,0,0,1,0,0,1,1,0,-1,-1]
Phi of -K [-3,0,0,1,1,1,1,2,1,1,2,0,0,1,0,0,1,1,0,-1,-1]
Phi of K* [-1,-1,-1,0,0,3,-1,-1,0,1,2,0,0,0,1,1,1,1,0,1,2]
Phi of -K* [-3,0,0,1,1,1,1,2,2,3,3,0,0,0,1,1,0,1,-1,-1,0]
Symmetry type of based matrix c
u-polynomial t^3-3t
Normalized Jones-Krushkal polynomial 5z^2+22z+25
Enhanced Jones-Krushkal polynomial 5w^3z^2-4w^3z+26w^2z+25w
Inner characteristic polynomial t^6+32t^4+35t^2+4
Outer characteristic polynomial t^7+44t^5+59t^3+16t
Flat arrow polynomial 4*K1**2*K2 - 4*K1**2 - 2*K1*K2 - 4*K1*K3 + K1 + 2*K2 + K3 + K4 + 2
2-strand cable arrow polynomial 384*K1**4*K2 - 1632*K1**4 - 128*K1**3*K2**2*K3 + 576*K1**3*K2*K3 + 32*K1**3*K3*K4 - 640*K1**3*K3 + 1024*K1**2*K2**3 + 96*K1**2*K2**2*K4 - 4656*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 832*K1**2*K2*K4 + 6784*K1**2*K2 - 384*K1**2*K3**2 - 64*K1**2*K4**2 - 4892*K1**2 + 800*K1*K2**3*K3 - 1376*K1*K2**2*K3 - 288*K1*K2**2*K5 - 416*K1*K2*K3*K4 + 6408*K1*K2*K3 + 1224*K1*K3*K4 + 256*K1*K4*K5 + 40*K1*K5*K6 - 32*K2**4*K4**2 + 320*K2**4*K4 - 1280*K2**4 + 32*K2**3*K3*K5 + 64*K2**3*K4*K6 - 64*K2**3*K6 - 768*K2**2*K3**2 - 744*K2**2*K4**2 + 2344*K2**2*K4 - 48*K2**2*K5**2 - 48*K2**2*K6**2 - 4270*K2**2 + 816*K2*K3*K5 + 520*K2*K4*K6 + 24*K2*K5*K7 + 16*K2*K6*K8 + 40*K3**2*K6 - 2184*K3**2 - 1072*K4**2 - 288*K5**2 - 138*K6**2 - 4*K7**2 - 2*K8**2 + 4384
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {2, 5}, {4}, {3}], [{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {1, 5}, {4}, {3}], [{5, 6}, {3, 4}, {1, 2}], [{5, 6}, {4}, {3}, {1, 2}]]
If K is slice False
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