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Flat knot 6.1157

Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,0,2,0,1,3,0,1,0,1,0,0,1,0,-1,0]
Flat knots (up to 7 crossings) with same phi are :['6.1157', '7.24267']
Arrow polynomial of the knot is: -8*K1**4 + 8*K1**3 + 8*K1**2*K2 - 2*K1**2 - 6*K1*K2 - 3*K1 - 2*K2**2 + K2 + K3 + 4
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.405', '6.1157']
Outer characteristic polynomial of the knot is: t^7+34t^5+61t^3+10t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1157']
2-strand cable arrow polynomial of the knot is: -256*K1**6 - 576*K1**4*K2**2 + 3008*K1**4*K2 - 5520*K1**4 - 384*K1**3*K2**2*K3 + 1472*K1**3*K2*K3 + 128*K1**3*K3*K4 - 1184*K1**3*K3 + 384*K1**2*K2**5 - 1664*K1**2*K2**4 - 384*K1**2*K2**3*K4 + 4288*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 + 1152*K1**2*K2**2*K4 - 12096*K1**2*K2**2 + 256*K1**2*K2*K3**2 + 96*K1**2*K2*K4**2 - 1664*K1**2*K2*K4 + 9744*K1**2*K2 - 1552*K1**2*K3**2 - 64*K1**2*K3*K5 - 416*K1**2*K4**2 - 1948*K1**2 + 256*K1*K2**5*K3 - 256*K1*K2**3*K3*K4 + 2464*K1*K2**3*K3 + 128*K1*K2**2*K3*K4 - 2048*K1*K2**2*K3 - 576*K1*K2**2*K5 + 192*K1*K2*K3**3 + 64*K1*K2*K3*K4**2 - 640*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 8032*K1*K2*K3 - 32*K1*K2*K4*K5 - 32*K1*K3**2*K5 + 1416*K1*K3*K4 + 224*K1*K4*K5 - 128*K2**8 + 256*K2**6*K4 - 1088*K2**6 - 192*K2**4*K3**2 - 192*K2**4*K4**2 + 1472*K2**4*K4 - 3432*K2**4 + 32*K2**3*K3*K5 - 160*K2**3*K6 + 192*K2**2*K3**2*K4 - 1264*K2**2*K3**2 + 64*K2**2*K4**3 - 632*K2**2*K4**2 + 2256*K2**2*K4 - 974*K2**2 - 32*K2*K3*K4*K5 + 528*K2*K3*K5 + 168*K2*K4*K6 - 64*K3**4 - 48*K3**2*K4**2 + 16*K3**2*K6 - 1016*K3**2 + 8*K3*K4*K7 - 8*K4**4 - 334*K4**2 - 12*K5**2 - 2*K6**2 + 2732
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1157']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.316', 'vk6.354', 'vk6.415', 'vk6.710', 'vk6.757', 'vk6.825', 'vk6.864', 'vk6.1506', 'vk6.1569', 'vk6.1953', 'vk6.1991', 'vk6.2046', 'vk6.2481', 'vk6.2644', 'vk6.2728', 'vk6.3114', 'vk6.10253', 'vk6.10398', 'vk6.18315', 'vk6.18653', 'vk6.19401', 'vk6.19694', 'vk6.25207', 'vk6.25870', 'vk6.26185', 'vk6.36934', 'vk6.37401', 'vk6.37981', 'vk6.38038', 'vk6.44858', 'vk6.56097', 'vk6.65742']
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 O1O2O3O4U1U4U2O5O6U5U6U3
R3 orbit {'O1O2O3O4U1U4U2O5O6U5U6U3'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U2U5U6O5O6U3U1U4
Gauss code of K* O1O2O3U4U5O4O5O6U1U3U6U2
Gauss code of -K* O1O2O3U2U3O4O5O6U5U1U4U6
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 0 2 1 -1 1],[ 3 0 2 3 1 0 0],[ 0 -2 0 1 0 0 0],[-2 -3 -1 0 0 -1 1],[-1 -1 0 0 0 0 0],[ 1 0 0 1 0 0 1],[-1 0 0 -1 0 -1 0]]
Primitive based matrix [[ 0 2 1 1 0 -1 -3],[-2 0 1 0 -1 -1 -3],[-1 -1 0 0 0 -1 0],[-1 0 0 0 0 0 -1],[ 0 1 0 0 0 0 -2],[ 1 1 1 0 0 0 0],[ 3 3 0 1 2 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,-1,0,1,3,-1,0,1,1,3,0,0,1,0,0,0,1,0,2,0]
Phi over symmetry [-3,-1,0,1,1,2,0,2,0,1,3,0,1,0,1,0,0,1,0,-1,0]
Phi of -K [-3,-1,0,1,1,2,2,1,3,4,2,1,2,1,2,1,1,1,0,1,2]
Phi of K* [-2,-1,-1,0,1,3,1,2,1,2,2,0,1,2,3,1,1,4,1,1,2]
Phi of -K* [-3,-1,0,1,1,2,0,2,0,1,3,0,1,0,1,0,0,1,0,-1,0]
Symmetry type of based matrix c
u-polynomial t^3-t^2-t
Normalized Jones-Krushkal polynomial 6z^2+25z+27
Enhanced Jones-Krushkal polynomial -2w^4z^2+8w^3z^2+25w^2z+27w
Inner characteristic polynomial t^6+18t^4+26t^2+1
Outer characteristic polynomial t^7+34t^5+61t^3+10t
Flat arrow polynomial -8*K1**4 + 8*K1**3 + 8*K1**2*K2 - 2*K1**2 - 6*K1*K2 - 3*K1 - 2*K2**2 + K2 + K3 + 4
2-strand cable arrow polynomial -256*K1**6 - 576*K1**4*K2**2 + 3008*K1**4*K2 - 5520*K1**4 - 384*K1**3*K2**2*K3 + 1472*K1**3*K2*K3 + 128*K1**3*K3*K4 - 1184*K1**3*K3 + 384*K1**2*K2**5 - 1664*K1**2*K2**4 - 384*K1**2*K2**3*K4 + 4288*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 + 1152*K1**2*K2**2*K4 - 12096*K1**2*K2**2 + 256*K1**2*K2*K3**2 + 96*K1**2*K2*K4**2 - 1664*K1**2*K2*K4 + 9744*K1**2*K2 - 1552*K1**2*K3**2 - 64*K1**2*K3*K5 - 416*K1**2*K4**2 - 1948*K1**2 + 256*K1*K2**5*K3 - 256*K1*K2**3*K3*K4 + 2464*K1*K2**3*K3 + 128*K1*K2**2*K3*K4 - 2048*K1*K2**2*K3 - 576*K1*K2**2*K5 + 192*K1*K2*K3**3 + 64*K1*K2*K3*K4**2 - 640*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 8032*K1*K2*K3 - 32*K1*K2*K4*K5 - 32*K1*K3**2*K5 + 1416*K1*K3*K4 + 224*K1*K4*K5 - 128*K2**8 + 256*K2**6*K4 - 1088*K2**6 - 192*K2**4*K3**2 - 192*K2**4*K4**2 + 1472*K2**4*K4 - 3432*K2**4 + 32*K2**3*K3*K5 - 160*K2**3*K6 + 192*K2**2*K3**2*K4 - 1264*K2**2*K3**2 + 64*K2**2*K4**3 - 632*K2**2*K4**2 + 2256*K2**2*K4 - 974*K2**2 - 32*K2*K3*K4*K5 + 528*K2*K3*K5 + 168*K2*K4*K6 - 64*K3**4 - 48*K3**2*K4**2 + 16*K3**2*K6 - 1016*K3**2 + 8*K3*K4*K7 - 8*K4**4 - 334*K4**2 - 12*K5**2 - 2*K6**2 + 2732
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {3, 5}, {2, 4}], [{3, 6}, {1, 5}, {2, 4}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {2, 4}, {1, 3}], [{5, 6}, {2, 4}, {3}, {1}], [{5, 6}, {3, 4}, {1, 2}]]
If K is slice False
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