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Flat knot 6.871

Min(phi) over symmetries of the knot is: [-3,-2,0,1,2,2,0,2,1,2,3,2,1,1,2,1,1,1,0,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.871']
Arrow polynomial of the knot is: -6*K1*K2 - 2*K1*K3 + 3*K1 - 2*K2**2 + K2 + 3*K3 + 2*K4 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.466', '6.871', '6.1186']
Outer characteristic polynomial of the knot is: t^7+67t^5+63t^3+5t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.871']
2-strand cable arrow polynomial of the knot is: -256*K1**4*K2**2 + 480*K1**4*K2 - 1424*K1**4 + 736*K1**3*K2*K3 + 96*K1**3*K3*K4 - 1248*K1**3*K3 + 160*K1**2*K2**3 - 1888*K1**2*K2**2 - 1600*K1**2*K2*K4 + 4960*K1**2*K2 - 864*K1**2*K3**2 - 128*K1**2*K3*K5 - 304*K1**2*K4**2 - 96*K1**2*K4*K6 - 4776*K1**2 + 128*K1*K2**3*K3 + 128*K1*K2**2*K3*K4 - 384*K1*K2**2*K3 + 64*K1*K2**2*K4*K5 - 64*K1*K2**2*K5 - 224*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 5712*K1*K2*K3 - 32*K1*K2*K4*K5 - 64*K1*K2*K4*K7 - 128*K1*K3**2*K5 - 32*K1*K3*K4*K6 + 2704*K1*K3*K4 + 760*K1*K4*K5 + 160*K1*K5*K6 + 24*K1*K6*K7 - 96*K2**4 - 192*K2**2*K3**2 - 216*K2**2*K4**2 + 1168*K2**2*K4 - 80*K2**2*K5**2 - 8*K2**2*K6**2 - 3466*K2**2 - 32*K2*K3**2*K4 - 96*K2*K3*K4*K5 + 568*K2*K3*K5 - 32*K2*K4**2*K6 + 264*K2*K4*K6 + 104*K2*K5*K7 + 16*K2*K6*K8 - 16*K3**4 - 32*K3**2*K4**2 + 136*K3**2*K6 - 2496*K3**2 + 80*K3*K4*K7 - 8*K4**4 + 16*K4**2*K8 - 1422*K4**2 - 444*K5**2 - 142*K6**2 - 36*K7**2 - 4*K8**2 + 3928
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.871']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4664', 'vk6.4951', 'vk6.6122', 'vk6.6609', 'vk6.8135', 'vk6.8537', 'vk6.9517', 'vk6.9872', 'vk6.20368', 'vk6.21711', 'vk6.27676', 'vk6.29222', 'vk6.39116', 'vk6.41372', 'vk6.45864', 'vk6.47527', 'vk6.48704', 'vk6.48907', 'vk6.49472', 'vk6.49691', 'vk6.50732', 'vk6.50931', 'vk6.51207', 'vk6.51408', 'vk6.57229', 'vk6.58456', 'vk6.61843', 'vk6.62980', 'vk6.66844', 'vk6.67714', 'vk6.69480', 'vk6.70204']
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 O1O2O3O4U1U5O6O5U2U6U4U3
R3 orbit {'O1O2O3O4U1U5O6O5U2U6U4U3'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U2U1U5U3O6O5U6U4
Gauss code of K* O1O2O3O4U5U1U4U3O5O6U2U6
Gauss code of -K* O1O2O3O4U5U3O5O6U2U1U4U6
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 -2 2 2 1 0],[ 3 0 1 3 2 3 1],[ 2 -1 0 3 2 2 0],[-2 -3 -3 0 0 -1 -1],[-2 -2 -2 0 0 -1 -1],[-1 -3 -2 1 1 0 0],[ 0 -1 0 1 1 0 0]]
Primitive based matrix [[ 0 2 2 1 0 -2 -3],[-2 0 0 -1 -1 -2 -2],[-2 0 0 -1 -1 -3 -3],[-1 1 1 0 0 -2 -3],[ 0 1 1 0 0 0 -1],[ 2 2 3 2 0 0 -1],[ 3 2 3 3 1 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,-1,0,2,3,0,1,1,2,2,1,1,3,3,0,2,3,0,1,1]
Phi over symmetry [-3,-2,0,1,2,2,0,2,1,2,3,2,1,1,2,1,1,1,0,0,0]
Phi of -K [-3,-2,0,1,2,2,0,2,1,2,3,2,1,1,2,1,1,1,0,0,0]
Phi of K* [-2,-2,-1,0,2,3,0,0,1,1,2,0,1,2,3,1,1,1,2,2,0]
Phi of -K* [-3,-2,0,1,2,2,1,1,3,2,3,0,2,2,3,0,1,1,1,1,0]
Symmetry type of based matrix c
u-polynomial t^3-t^2-t
Normalized Jones-Krushkal polynomial 5z^2+24z+29
Enhanced Jones-Krushkal polynomial 5w^3z^2+24w^2z+29w
Inner characteristic polynomial t^6+45t^4+34t^2
Outer characteristic polynomial t^7+67t^5+63t^3+5t
Flat arrow polynomial -6*K1*K2 - 2*K1*K3 + 3*K1 - 2*K2**2 + K2 + 3*K3 + 2*K4 + 2
2-strand cable arrow polynomial -256*K1**4*K2**2 + 480*K1**4*K2 - 1424*K1**4 + 736*K1**3*K2*K3 + 96*K1**3*K3*K4 - 1248*K1**3*K3 + 160*K1**2*K2**3 - 1888*K1**2*K2**2 - 1600*K1**2*K2*K4 + 4960*K1**2*K2 - 864*K1**2*K3**2 - 128*K1**2*K3*K5 - 304*K1**2*K4**2 - 96*K1**2*K4*K6 - 4776*K1**2 + 128*K1*K2**3*K3 + 128*K1*K2**2*K3*K4 - 384*K1*K2**2*K3 + 64*K1*K2**2*K4*K5 - 64*K1*K2**2*K5 - 224*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 5712*K1*K2*K3 - 32*K1*K2*K4*K5 - 64*K1*K2*K4*K7 - 128*K1*K3**2*K5 - 32*K1*K3*K4*K6 + 2704*K1*K3*K4 + 760*K1*K4*K5 + 160*K1*K5*K6 + 24*K1*K6*K7 - 96*K2**4 - 192*K2**2*K3**2 - 216*K2**2*K4**2 + 1168*K2**2*K4 - 80*K2**2*K5**2 - 8*K2**2*K6**2 - 3466*K2**2 - 32*K2*K3**2*K4 - 96*K2*K3*K4*K5 + 568*K2*K3*K5 - 32*K2*K4**2*K6 + 264*K2*K4*K6 + 104*K2*K5*K7 + 16*K2*K6*K8 - 16*K3**4 - 32*K3**2*K4**2 + 136*K3**2*K6 - 2496*K3**2 + 80*K3*K4*K7 - 8*K4**4 + 16*K4**2*K8 - 1422*K4**2 - 444*K5**2 - 142*K6**2 - 36*K7**2 - 4*K8**2 + 3928
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {3, 5}, {2, 4}], [{2, 6}, {1, 5}, {3, 4}], [{3, 6}, {1, 5}, {2, 4}], [{5, 6}, {2, 4}, {1, 3}]]
If K is slice False
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