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Flat knot 6.1156

Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,0,2,1,3,1,0,0,0,1,0,1,1,0,0,2]
Flat knots (up to 7 crossings) with same phi are :['6.1156']
Arrow polynomial of the knot is: 4*K1**3 + 4*K1**2*K2 - 10*K1**2 - 6*K1*K2 - 4*K1*K3 + 5*K2 + 2*K3 + K4 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.1156']
Outer characteristic polynomial of the knot is: t^7+38t^5+74t^3+6t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1156']
2-strand cable arrow polynomial of the knot is: -192*K1**6 - 256*K1**4*K2**2 + 928*K1**4*K2 - 2944*K1**4 + 896*K1**3*K2*K3 + 32*K1**3*K3*K4 - 736*K1**3*K3 - 256*K1**2*K2**4 + 352*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 + 96*K1**2*K2**2*K4 - 4704*K1**2*K2**2 + 160*K1**2*K2*K3**2 - 672*K1**2*K2*K4 + 8008*K1**2*K2 - 1856*K1**2*K3**2 - 128*K1**2*K3*K5 - 240*K1**2*K4**2 - 5588*K1**2 + 992*K1*K2**3*K3 + 224*K1*K2**2*K3*K4 - 1056*K1*K2**2*K3 - 448*K1*K2**2*K5 + 288*K1*K2*K3**3 - 384*K1*K2*K3*K4 - 128*K1*K2*K3*K6 + 7952*K1*K2*K3 - 128*K1*K2*K4*K5 - 32*K1*K2*K4*K7 - 32*K1*K3**2*K5 + 2792*K1*K3*K4 + 584*K1*K4*K5 + 56*K1*K5*K6 + 8*K1*K6*K7 - 32*K2**6 - 64*K2**4*K3**2 - 32*K2**4*K4**2 + 96*K2**4*K4 - 792*K2**4 + 96*K2**3*K3*K5 + 64*K2**3*K4*K6 - 32*K2**3*K6 - 1104*K2**2*K3**2 - 32*K2**2*K3*K7 - 216*K2**2*K4**2 - 32*K2**2*K4*K8 + 1440*K2**2*K4 - 16*K2**2*K5**2 - 16*K2**2*K6**2 - 4580*K2**2 - 128*K2*K3**2*K4 - 32*K2*K3*K4*K5 + 984*K2*K3*K5 + 288*K2*K4*K6 + 64*K2*K5*K7 + 16*K2*K6*K8 - 160*K3**4 + 144*K3**2*K6 - 2852*K3**2 + 48*K3*K4*K7 - 1250*K4**2 - 368*K5**2 - 100*K6**2 - 40*K7**2 - 2*K8**2 + 5178
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1156']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4205', 'vk6.4285', 'vk6.5465', 'vk6.5578', 'vk6.7572', 'vk6.7663', 'vk6.9077', 'vk6.9157', 'vk6.11174', 'vk6.12260', 'vk6.12369', 'vk6.19377', 'vk6.19670', 'vk6.19782', 'vk6.26161', 'vk6.26221', 'vk6.26577', 'vk6.26664', 'vk6.30764', 'vk6.31969', 'vk6.38165', 'vk6.38205', 'vk6.44826', 'vk6.44950', 'vk6.48515', 'vk6.49213', 'vk6.49320', 'vk6.50307', 'vk6.52746', 'vk6.63580', 'vk6.66329', 'vk6.66345']
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 O1O2O3O4U1U4U2O5O6U5U3U6
R3 orbit {'O1O2O3O4U1U4U2O5O6U5U3U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U5U2U6O5O6U3U1U4
Gauss code of K* O1O2O3U4U5O4O6O5U1U3U6U2
Gauss code of -K* O1O2O3U1U3O4O5O6U5U2U4U6
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 1 1 -1 2],[ 3 0 2 3 1 0 1],[ 0 -2 0 1 0 0 1],[-1 -3 -1 0 0 0 2],[-1 -1 0 0 0 0 0],[ 1 0 0 0 0 0 1],[-2 -1 -1 -2 0 -1 0]]
Primitive based matrix [[ 0 2 1 1 0 -1 -3],[-2 0 0 -2 -1 -1 -1],[-1 0 0 0 0 0 -1],[-1 2 0 0 -1 0 -3],[ 0 1 0 1 0 0 -2],[ 1 1 0 0 0 0 0],[ 3 1 1 3 2 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,-1,0,1,3,0,2,1,1,1,0,0,0,1,1,0,3,0,2,0]
Phi over symmetry [-3,-1,0,1,1,2,0,2,1,3,1,0,0,0,1,0,1,1,0,0,2]
Phi of -K [-3,-1,0,1,1,2,2,1,1,3,4,1,2,2,2,0,1,1,0,-1,1]
Phi of K* [-2,-1,-1,0,1,3,-1,1,1,2,4,0,0,2,1,1,2,3,1,1,2]
Phi of -K* [-3,-1,0,1,1,2,0,2,1,3,1,0,0,0,1,0,1,1,0,0,2]
Symmetry type of based matrix c
u-polynomial t^3-t^2-t
Normalized Jones-Krushkal polynomial 2z^2+21z+35
Enhanced Jones-Krushkal polynomial 2w^3z^2+21w^2z+35w
Inner characteristic polynomial t^6+22t^4+25t^2+1
Outer characteristic polynomial t^7+38t^5+74t^3+6t
Flat arrow polynomial 4*K1**3 + 4*K1**2*K2 - 10*K1**2 - 6*K1*K2 - 4*K1*K3 + 5*K2 + 2*K3 + K4 + 5
2-strand cable arrow polynomial -192*K1**6 - 256*K1**4*K2**2 + 928*K1**4*K2 - 2944*K1**4 + 896*K1**3*K2*K3 + 32*K1**3*K3*K4 - 736*K1**3*K3 - 256*K1**2*K2**4 + 352*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 + 96*K1**2*K2**2*K4 - 4704*K1**2*K2**2 + 160*K1**2*K2*K3**2 - 672*K1**2*K2*K4 + 8008*K1**2*K2 - 1856*K1**2*K3**2 - 128*K1**2*K3*K5 - 240*K1**2*K4**2 - 5588*K1**2 + 992*K1*K2**3*K3 + 224*K1*K2**2*K3*K4 - 1056*K1*K2**2*K3 - 448*K1*K2**2*K5 + 288*K1*K2*K3**3 - 384*K1*K2*K3*K4 - 128*K1*K2*K3*K6 + 7952*K1*K2*K3 - 128*K1*K2*K4*K5 - 32*K1*K2*K4*K7 - 32*K1*K3**2*K5 + 2792*K1*K3*K4 + 584*K1*K4*K5 + 56*K1*K5*K6 + 8*K1*K6*K7 - 32*K2**6 - 64*K2**4*K3**2 - 32*K2**4*K4**2 + 96*K2**4*K4 - 792*K2**4 + 96*K2**3*K3*K5 + 64*K2**3*K4*K6 - 32*K2**3*K6 - 1104*K2**2*K3**2 - 32*K2**2*K3*K7 - 216*K2**2*K4**2 - 32*K2**2*K4*K8 + 1440*K2**2*K4 - 16*K2**2*K5**2 - 16*K2**2*K6**2 - 4580*K2**2 - 128*K2*K3**2*K4 - 32*K2*K3*K4*K5 + 984*K2*K3*K5 + 288*K2*K4*K6 + 64*K2*K5*K7 + 16*K2*K6*K8 - 160*K3**4 + 144*K3**2*K6 - 2852*K3**2 + 48*K3*K4*K7 - 1250*K4**2 - 368*K5**2 - 100*K6**2 - 40*K7**2 - 2*K8**2 + 5178
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {4, 5}, {2, 3}], [{1, 6}, {4, 5}, {3}, {2}], [{2, 6}, {4, 5}, {1, 3}], [{3, 6}, {4, 5}, {1, 2}], [{3, 6}, {4, 5}, {2}, {1}], [{4, 6}, {1, 5}, {2, 3}], [{6}, {4, 5}, {1, 3}, {2}]]
If K is slice False
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