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Flat knot 6.54

Min(phi) over symmetries of the knot is: [-5,0,0,1,1,3,2,3,1,5,4,0,0,1,1,0,1,2,0,0,2]
Flat knots (up to 7 crossings) with same phi are :['6.54']
Arrow polynomial of the knot is: 4*K1**3 - 2*K1*K2 - K1 - 2*K2*K3 + K5 + 1
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.54']
Outer characteristic polynomial of the knot is: t^7+102t^5+85t^3+7t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.54']
2-strand cable arrow polynomial of the knot is: 768*K1**4*K2 - 2240*K1**4 + 832*K1**3*K2*K3 - 768*K1**3*K3 - 128*K1**2*K2**4 + 384*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 2944*K1**2*K2**2 - 448*K1**2*K2*K4 + 5680*K1**2*K2 - 2432*K1**2*K3**2 - 192*K1**2*K3*K5 - 32*K1**2*K4**2 - 4352*K1**2 + 256*K1*K2**3*K3 - 1088*K1*K2**2*K3 - 64*K1*K2**2*K5 + 256*K1*K2*K3**3 - 832*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 6144*K1*K2*K3 - 128*K1*K3**2*K5 - 64*K1*K3*K4*K6 + 3280*K1*K3*K4 + 688*K1*K4*K5 + 32*K1*K5*K6 + 80*K1*K6*K7 - 2*K10**2 + 8*K10*K4*K6 - 32*K2**6 + 32*K2**4*K4 - 560*K2**4 + 128*K2**2*K3**2*K4 - 640*K2**2*K3**2 - 8*K2**2*K4**2 + 1424*K2**2*K4 - 3650*K2**2 - 64*K2*K3**2*K4 - 128*K2*K3*K4*K5 + 960*K2*K3*K5 - 32*K2*K4**2*K6 + 200*K2*K4*K6 + 32*K2*K5*K7 + 16*K2*K6*K8 - 256*K3**4 - 256*K3**2*K4**2 + 256*K3**2*K6 - 2504*K3**2 + 240*K3*K4*K7 - 8*K4**2*K6**2 + 16*K4**2*K8 - 1504*K4**2 - 440*K5**2 - 172*K6**2 - 80*K7**2 - 8*K8**2 + 4278
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.54']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.20147', 'vk6.20171', 'vk6.21437', 'vk6.21454', 'vk6.27263', 'vk6.27295', 'vk6.28923', 'vk6.28953', 'vk6.38682', 'vk6.38728', 'vk6.40906', 'vk6.47266', 'vk6.47293', 'vk6.56972', 'vk6.57005', 'vk6.58124', 'vk6.62671', 'vk6.67470', 'vk6.70031', 'vk6.70051']
The R3 orbit of minmal crossing diagrams contains:
The diagrammatic symmetry type of this knot is +.
The reverse -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 O1O2O3O4O5O6U1U6U4U3U5U2
R3 orbit {'O1O2O3O4O5O6U1U6U4U3U5U2'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5O6U5U2U4U3U1U6
Gauss code of K* Same
Gauss code of -K* O1O2O3O4O5O6U5U2U4U3U1U6
Diagrammatic symmetry type +
Flat genus of the diagram 3
If K is checkerboard colorable False
If K is almost classical False
Based matrix from Gauss code [[ 0 -5 1 0 0 3 1],[ 5 0 5 3 2 4 1],[-1 -5 0 -1 -1 2 0],[ 0 -3 1 0 0 2 0],[ 0 -2 1 0 0 1 0],[-3 -4 -2 -2 -1 0 0],[-1 -1 0 0 0 0 0]]
Primitive based matrix [[ 0 3 1 1 0 0 -5],[-3 0 0 -2 -1 -2 -4],[-1 0 0 0 0 0 -1],[-1 2 0 0 -1 -1 -5],[ 0 1 0 1 0 0 -2],[ 0 2 0 1 0 0 -3],[ 5 4 1 5 2 3 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-1,-1,0,0,5,0,2,1,2,4,0,0,0,1,1,1,5,0,2,3]
Phi over symmetry [-5,0,0,1,1,3,2,3,1,5,4,0,0,1,1,0,1,2,0,0,2]
Phi of -K [-5,0,0,1,1,3,2,3,1,5,4,0,0,1,1,0,1,2,0,0,2]
Phi of K* [-3,-1,-1,0,0,5,0,2,1,2,4,0,0,0,1,1,1,5,0,2,3]
Phi of -K* [-5,0,0,1,1,3,2,3,1,5,4,0,0,1,1,0,1,2,0,0,2]
Symmetry type of based matrix +
u-polynomial t^5-t^3-2t
Normalized Jones-Krushkal polynomial 7z^2+27z+27
Enhanced Jones-Krushkal polynomial 7w^3z^2+27w^2z+27w
Inner characteristic polynomial t^6+66t^4+23t^2+1
Outer characteristic polynomial t^7+102t^5+85t^3+7t
Flat arrow polynomial 4*K1**3 - 2*K1*K2 - K1 - 2*K2*K3 + K5 + 1
2-strand cable arrow polynomial 768*K1**4*K2 - 2240*K1**4 + 832*K1**3*K2*K3 - 768*K1**3*K3 - 128*K1**2*K2**4 + 384*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 2944*K1**2*K2**2 - 448*K1**2*K2*K4 + 5680*K1**2*K2 - 2432*K1**2*K3**2 - 192*K1**2*K3*K5 - 32*K1**2*K4**2 - 4352*K1**2 + 256*K1*K2**3*K3 - 1088*K1*K2**2*K3 - 64*K1*K2**2*K5 + 256*K1*K2*K3**3 - 832*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 6144*K1*K2*K3 - 128*K1*K3**2*K5 - 64*K1*K3*K4*K6 + 3280*K1*K3*K4 + 688*K1*K4*K5 + 32*K1*K5*K6 + 80*K1*K6*K7 - 2*K10**2 + 8*K10*K4*K6 - 32*K2**6 + 32*K2**4*K4 - 560*K2**4 + 128*K2**2*K3**2*K4 - 640*K2**2*K3**2 - 8*K2**2*K4**2 + 1424*K2**2*K4 - 3650*K2**2 - 64*K2*K3**2*K4 - 128*K2*K3*K4*K5 + 960*K2*K3*K5 - 32*K2*K4**2*K6 + 200*K2*K4*K6 + 32*K2*K5*K7 + 16*K2*K6*K8 - 256*K3**4 - 256*K3**2*K4**2 + 256*K3**2*K6 - 2504*K3**2 + 240*K3*K4*K7 - 8*K4**2*K6**2 + 16*K4**2*K8 - 1504*K4**2 - 440*K5**2 - 172*K6**2 - 80*K7**2 - 8*K8**2 + 4278
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}], [{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {4, 5}, {1, 3}], [{2, 6}, {5}, {3, 4}, {1}], [{5, 6}, {3, 4}, {1, 2}], [{5, 6}, {4}, {3}, {1, 2}]]
If K is slice False
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