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Flat knot 6.503

Min(phi) over symmetries of the knot is: [-4,-1,-1,2,2,2,0,2,2,3,4,1,1,1,1,1,2,3,0,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.503']
Arrow polynomial of the knot is: 4*K1**3 - 4*K1*K2 - 2*K1*K3 - K1 + K2 + K3 + K4 + 1
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.71', '6.148', '6.170', '6.253', '6.259', '6.298', '6.439', '6.453', '6.499', '6.503']
Outer characteristic polynomial of the knot is: t^7+87t^5+175t^3
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.503']
2-strand cable arrow polynomial of the knot is: -144*K1**4 + 192*K1**3*K2*K3 - 992*K1**2*K2**2 + 640*K1**2*K2 - 208*K1**2*K3**2 - 592*K1**2 + 64*K1*K2**3*K3 + 1416*K1*K2*K3 + 240*K1*K3*K4 + 72*K1*K4*K5 + 24*K1*K5*K6 - 32*K2**6 + 64*K2**4*K4 - 880*K2**4 + 32*K2**3*K3*K5 - 352*K2**2*K3**2 - 48*K2**2*K4**2 + 848*K2**2*K4 - 48*K2**2*K5**2 - 8*K2**2*K6**2 - 674*K2**2 + 576*K2*K3*K5 + 56*K2*K4*K6 + 24*K2*K5*K7 + 8*K2*K6*K8 + 24*K3**2*K6 - 708*K3**2 - 366*K4**2 - 256*K5**2 - 46*K6**2 - 4*K7**2 - 2*K8**2 + 1078
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.503']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.19957', 'vk6.20093', 'vk6.21212', 'vk6.21373', 'vk6.26950', 'vk6.27158', 'vk6.28696', 'vk6.28845', 'vk6.38364', 'vk6.38558', 'vk6.40520', 'vk6.40753', 'vk6.45227', 'vk6.45455', 'vk6.47044', 'vk6.47195', 'vk6.56749', 'vk6.56910', 'vk6.57856', 'vk6.58046', 'vk6.61202', 'vk6.61443', 'vk6.62432', 'vk6.62598', 'vk6.66459', 'vk6.66614', 'vk6.67238', 'vk6.67403', 'vk6.69105', 'vk6.69262', 'vk6.69882', 'vk6.70001']
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 O1O2O3O4O5U1U6U2O6U5U4U3
R3 orbit {'O1O2O3O4O5U1U6U2O6U5U4U3'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U3U2U1O6U4U6U5
Gauss code of K* O1O2O3U2O4O5O6U1U3U6U5U4
Gauss code of -K* O1O2O3U4O5O4O6U3U2U1U5U6
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 -1 2 2 2 -1],[ 4 0 1 4 3 2 3],[ 1 -1 0 2 1 0 1],[-2 -4 -2 0 0 0 -2],[-2 -3 -1 0 0 0 -2],[-2 -2 0 0 0 0 -2],[ 1 -3 -1 2 2 2 0]]
Primitive based matrix [[ 0 2 2 2 -1 -1 -4],[-2 0 0 0 0 -2 -2],[-2 0 0 0 -1 -2 -3],[-2 0 0 0 -2 -2 -4],[ 1 0 1 2 0 1 -1],[ 1 2 2 2 -1 0 -3],[ 4 2 3 4 1 3 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,-2,1,1,4,0,0,0,2,2,0,1,2,3,2,2,4,-1,1,3]
Phi over symmetry [-4,-1,-1,2,2,2,0,2,2,3,4,1,1,1,1,1,2,3,0,0,0]
Phi of -K [-4,-1,-1,2,2,2,0,2,2,3,4,1,1,1,1,1,2,3,0,0,0]
Phi of K* [-2,-2,-2,1,1,4,0,0,1,1,2,0,1,2,3,1,3,4,-1,0,2]
Phi of -K* [-4,-1,-1,2,2,2,1,3,2,3,4,1,0,1,2,2,2,2,0,0,0]
Symmetry type of based matrix c
u-polynomial t^4-3t^2+2t
Normalized Jones-Krushkal polynomial 3z+7
Enhanced Jones-Krushkal polynomial -12w^3z+15w^2z+7w
Inner characteristic polynomial t^6+57t^4+92t^2
Outer characteristic polynomial t^7+87t^5+175t^3
Flat arrow polynomial 4*K1**3 - 4*K1*K2 - 2*K1*K3 - K1 + K2 + K3 + K4 + 1
2-strand cable arrow polynomial -144*K1**4 + 192*K1**3*K2*K3 - 992*K1**2*K2**2 + 640*K1**2*K2 - 208*K1**2*K3**2 - 592*K1**2 + 64*K1*K2**3*K3 + 1416*K1*K2*K3 + 240*K1*K3*K4 + 72*K1*K4*K5 + 24*K1*K5*K6 - 32*K2**6 + 64*K2**4*K4 - 880*K2**4 + 32*K2**3*K3*K5 - 352*K2**2*K3**2 - 48*K2**2*K4**2 + 848*K2**2*K4 - 48*K2**2*K5**2 - 8*K2**2*K6**2 - 674*K2**2 + 576*K2*K3*K5 + 56*K2*K4*K6 + 24*K2*K5*K7 + 8*K2*K6*K8 + 24*K3**2*K6 - 708*K3**2 - 366*K4**2 - 256*K5**2 - 46*K6**2 - 4*K7**2 - 2*K8**2 + 1078
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {3, 5}, {2, 4}], [{1, 6}, {3, 5}, {4}, {2}], [{2, 6}, {4, 5}, {1, 3}]]
If K is slice False
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