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Flat knot 6.32

Min(phi) over symmetries of the knot is: [-5,-1,-1,0,3,4,1,2,4,3,5,0,1,1,2,1,2,3,2,3,0]
Flat knots (up to 7 crossings) with same phi are :['6.32']
Arrow polynomial of the knot is: 4*K1*K2**2 - 6*K1*K2 + 2*K1 - 2*K2**2 - 2*K2*K3 + 2*K3 + K4 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.32']
Outer characteristic polynomial of the knot is: t^7+140t^5+48t^3
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.32']
2-strand cable arrow polynomial of the knot is: -112*K1**2*K3**2 - 464*K1**2*K4**2 - 872*K1**2 + 608*K1*K2*K3 + 64*K1*K3*K4**3 + 1184*K1*K3*K4 + 32*K1*K4**3*K5 + 504*K1*K4*K5 + 8*K1*K5*K6 + 8*K1*K7*K8 - 32*K2**2*K4**4 + 64*K2**2*K4**3 - 472*K2**2*K4**2 + 704*K2**2*K4 - 884*K2**2 + 128*K2*K3*K5 + 32*K2*K4**3*K6 + 280*K2*K4*K6 + 16*K2*K5*K7 + 8*K2*K6*K8 - 16*K3**4 - 96*K3**2*K4**2 + 24*K3**2*K6 - 796*K3**2 + 40*K3*K4*K7 - 88*K4**4 - 32*K4**2*K5**2 - 8*K4**2*K6**2 + 8*K4**2*K8 - 796*K4**2 + 8*K4*K5*K9 - 220*K5**2 - 52*K6**2 - 16*K7**2 - 10*K8**2 + 1244
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.32']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.73320', 'vk6.73461', 'vk6.74024', 'vk6.74578', 'vk6.75212', 'vk6.75468', 'vk6.76054', 'vk6.76775', 'vk6.78201', 'vk6.78431', 'vk6.79005', 'vk6.79574', 'vk6.80020', 'vk6.80171', 'vk6.80539', 'vk6.80996', 'vk6.81889', 'vk6.82352', 'vk6.82374', 'vk6.82603', 'vk6.83624', 'vk6.83663', 'vk6.84305', 'vk6.84357', 'vk6.84472', 'vk6.84580', 'vk6.84635', 'vk6.85221', 'vk6.85592', 'vk6.86759', 'vk6.88701', 'vk6.88985']
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 O1O2O3O4O5O6U1U4U3U6U2U5
R3 orbit {'O1O2O3O4O5O6U1U4U3U6U2U5'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5O6U2U5U1U4U3U6
Gauss code of K* O1O2O3O4O5O6U1U5U3U2U6U4
Gauss code of -K* O1O2O3O4O5O6U3U1U5U4U2U6
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 -5 0 -1 -1 4 3],[ 5 0 4 2 1 5 3],[ 0 -4 0 -1 -1 3 2],[ 1 -2 1 0 0 3 2],[ 1 -1 1 0 0 2 1],[-4 -5 -3 -3 -2 0 0],[-3 -3 -2 -2 -1 0 0]]
Primitive based matrix [[ 0 4 3 0 -1 -1 -5],[-4 0 0 -3 -2 -3 -5],[-3 0 0 -2 -1 -2 -3],[ 0 3 2 0 -1 -1 -4],[ 1 2 1 1 0 0 -1],[ 1 3 2 1 0 0 -2],[ 5 5 3 4 1 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-4,-3,0,1,1,5,0,3,2,3,5,2,1,2,3,1,1,4,0,1,2]
Phi over symmetry [-5,-1,-1,0,3,4,1,2,4,3,5,0,1,1,2,1,2,3,2,3,0]
Phi of -K [-5,-1,-1,0,3,4,2,3,1,5,4,0,0,2,2,0,3,3,1,1,1]
Phi of K* [-4,-3,0,1,1,5,1,1,2,3,4,1,2,3,5,0,0,1,0,2,3]
Phi of -K* [-5,-1,-1,0,3,4,1,2,4,3,5,0,1,1,2,1,2,3,2,3,0]
Symmetry type of based matrix c
u-polynomial t^5-t^4-t^3+2t
Normalized Jones-Krushkal polynomial 5z+11
Enhanced Jones-Krushkal polynomial -8w^3z+13w^2z+11w
Inner characteristic polynomial t^6+88t^4+12t^2
Outer characteristic polynomial t^7+140t^5+48t^3
Flat arrow polynomial 4*K1*K2**2 - 6*K1*K2 + 2*K1 - 2*K2**2 - 2*K2*K3 + 2*K3 + K4 + 2
2-strand cable arrow polynomial -112*K1**2*K3**2 - 464*K1**2*K4**2 - 872*K1**2 + 608*K1*K2*K3 + 64*K1*K3*K4**3 + 1184*K1*K3*K4 + 32*K1*K4**3*K5 + 504*K1*K4*K5 + 8*K1*K5*K6 + 8*K1*K7*K8 - 32*K2**2*K4**4 + 64*K2**2*K4**3 - 472*K2**2*K4**2 + 704*K2**2*K4 - 884*K2**2 + 128*K2*K3*K5 + 32*K2*K4**3*K6 + 280*K2*K4*K6 + 16*K2*K5*K7 + 8*K2*K6*K8 - 16*K3**4 - 96*K3**2*K4**2 + 24*K3**2*K6 - 796*K3**2 + 40*K3*K4*K7 - 88*K4**4 - 32*K4**2*K5**2 - 8*K4**2*K6**2 + 8*K4**2*K8 - 796*K4**2 + 8*K4*K5*K9 - 220*K5**2 - 52*K6**2 - 16*K7**2 - 10*K8**2 + 1244
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {3, 5}, {4}, {1}], [{3, 6}, {1, 5}, {2, 4}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {4}, {2, 3}, {1}]]
If K is slice False
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