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Flat knot 6.281

Min(phi) over symmetries of the knot is: [-4,-2,1,1,2,2,0,2,4,2,4,2,3,1,2,0,0,0,1,1,-1]
Flat knots (up to 7 crossings) with same phi are :['6.281']
Arrow polynomial of the knot is: 4*K1**3 - 10*K1**2 - 8*K1*K2 + K1 - 2*K2**2 + 5*K2 + 3*K3 + K4 + 7
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.223', '6.281', '6.497']
Outer characteristic polynomial of the knot is: t^7+81t^5+106t^3
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.281']
2-strand cable arrow polynomial of the knot is: -192*K1**4*K2**2 + 288*K1**4*K2 - 1296*K1**4 + 128*K1**3*K2**3*K3 + 416*K1**3*K2*K3 - 448*K1**2*K2**4 + 544*K1**2*K2**3 - 256*K1**2*K2**2*K3**2 - 3424*K1**2*K2**2 + 4360*K1**2*K2 - 672*K1**2*K3**2 - 128*K1**2*K4**2 - 3568*K1**2 + 992*K1*K2**3*K3 + 192*K1*K2*K3**3 + 3872*K1*K2*K3 + 32*K1*K3**3*K4 + 1264*K1*K3*K4 + 288*K1*K4*K5 + 24*K1*K5*K6 + 16*K1*K6*K7 - 32*K2**6 + 64*K2**4*K4 - 1096*K2**4 - 1024*K2**2*K3**2 - 64*K2**2*K4**2 + 512*K2**2*K4 - 1926*K2**2 + 568*K2*K3*K5 + 48*K2*K4*K6 - 144*K3**4 - 96*K3**2*K4**2 + 120*K3**2*K6 - 1660*K3**2 + 88*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 722*K4**2 - 252*K5**2 - 74*K6**2 - 32*K7**2 - 2*K8**2 + 3290
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.281']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16505', 'vk6.16598', 'vk6.18099', 'vk6.18435', 'vk6.22932', 'vk6.23029', 'vk6.24546', 'vk6.24963', 'vk6.34907', 'vk6.35014', 'vk6.36681', 'vk6.37103', 'vk6.42470', 'vk6.42583', 'vk6.43957', 'vk6.44272', 'vk6.54732', 'vk6.54829', 'vk6.55923', 'vk6.56215', 'vk6.59192', 'vk6.59257', 'vk6.60449', 'vk6.60810', 'vk6.64746', 'vk6.64803', 'vk6.65565', 'vk6.65875', 'vk6.68036', 'vk6.68101', 'vk6.68643', 'vk6.68856']
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 O1O2O3O4O5U1U5O6U2U4U6U3
R3 orbit {'O1O2O3O4O5U1U5O6U2U4U6U3'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U3U6U2U4O6U1U5
Gauss code of K* O1O2O3O4U5U1U4U2U6O5O6U3
Gauss code of -K* O1O2O3O4U2O5O6U5U3U1U4U6
Diagrammatic symmetry type c
Flat genus of the diagram 2
If K is checkerboard colorable False
If K is almost classical False
Based matrix from Gauss code [[ 0 -4 -2 2 1 1 2],[ 4 0 2 4 3 1 2],[ 2 -2 0 3 1 0 2],[-2 -4 -3 0 -1 0 1],[-1 -3 -1 1 0 0 1],[-1 -1 0 0 0 0 0],[-2 -2 -2 -1 -1 0 0]]
Primitive based matrix [[ 0 2 2 1 1 -2 -4],[-2 0 1 0 -1 -3 -4],[-2 -1 0 0 -1 -2 -2],[-1 0 0 0 0 0 -1],[-1 1 1 0 0 -1 -3],[ 2 3 2 0 1 0 -2],[ 4 4 2 1 3 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,-1,-1,2,4,-1,0,1,3,4,0,1,2,2,0,0,1,1,3,2]
Phi over symmetry [-4,-2,1,1,2,2,0,2,4,2,4,2,3,1,2,0,0,0,1,1,-1]
Phi of -K [-4,-2,1,1,2,2,0,2,4,2,4,2,3,1,2,0,0,0,1,1,-1]
Phi of K* [-2,-2,-1,-1,2,4,-1,0,1,2,4,0,1,1,2,0,2,2,3,4,0]
Phi of -K* [-4,-2,1,1,2,2,2,1,3,2,4,0,1,2,3,0,0,0,1,1,-1]
Symmetry type of based matrix c
u-polynomial t^4-t^2-2t
Normalized Jones-Krushkal polynomial 13z+27
Enhanced Jones-Krushkal polynomial -4w^3z+17w^2z+27w
Inner characteristic polynomial t^6+51t^4+31t^2
Outer characteristic polynomial t^7+81t^5+106t^3
Flat arrow polynomial 4*K1**3 - 10*K1**2 - 8*K1*K2 + K1 - 2*K2**2 + 5*K2 + 3*K3 + K4 + 7
2-strand cable arrow polynomial -192*K1**4*K2**2 + 288*K1**4*K2 - 1296*K1**4 + 128*K1**3*K2**3*K3 + 416*K1**3*K2*K3 - 448*K1**2*K2**4 + 544*K1**2*K2**3 - 256*K1**2*K2**2*K3**2 - 3424*K1**2*K2**2 + 4360*K1**2*K2 - 672*K1**2*K3**2 - 128*K1**2*K4**2 - 3568*K1**2 + 992*K1*K2**3*K3 + 192*K1*K2*K3**3 + 3872*K1*K2*K3 + 32*K1*K3**3*K4 + 1264*K1*K3*K4 + 288*K1*K4*K5 + 24*K1*K5*K6 + 16*K1*K6*K7 - 32*K2**6 + 64*K2**4*K4 - 1096*K2**4 - 1024*K2**2*K3**2 - 64*K2**2*K4**2 + 512*K2**2*K4 - 1926*K2**2 + 568*K2*K3*K5 + 48*K2*K4*K6 - 144*K3**4 - 96*K3**2*K4**2 + 120*K3**2*K6 - 1660*K3**2 + 88*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 722*K4**2 - 252*K5**2 - 74*K6**2 - 32*K7**2 - 2*K8**2 + 3290
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{4, 6}, {3, 5}, {1, 2}], [{6}, {2, 5}, {3, 4}, {1}]]
If K is slice False
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