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Flat knot 6.1229

Min(phi) over symmetries of the knot is: [-2,-2,1,1,1,1,0,0,0,1,2,1,2,0,2,0,0,-1,0,-1,0]
Flat knots (up to 7 crossings) with same phi are :['6.1229']
Arrow polynomial of the knot is: 4*K1**3 + 4*K1**2*K2 - 4*K1*K2 - 4*K1*K3 - K1 + K3 + K4
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.565', '6.1229', '6.1243', '6.1920']
Outer characteristic polynomial of the knot is: t^7+28t^5+76t^3+8t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1205', '6.1229']
2-strand cable arrow polynomial of the knot is: 1792*K1**4*K2 - 2976*K1**4 - 384*K1**3*K2**2*K3 + 1536*K1**3*K2*K3 - 1088*K1**3*K3 - 128*K1**2*K2**4 + 1280*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 + 256*K1**2*K2**2*K4 - 6672*K1**2*K2**2 + 768*K1**2*K2*K3**2 - 1024*K1**2*K2*K4 + 6864*K1**2*K2 - 2208*K1**2*K3**2 - 128*K1**2*K3*K5 - 32*K1**2*K4**2 - 3176*K1**2 + 1088*K1*K2**3*K3 + 256*K1*K2**2*K3*K4 - 2464*K1*K2**2*K3 - 448*K1*K2**2*K5 + 384*K1*K2*K3**3 - 832*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 8144*K1*K2*K3 - 192*K1*K2*K4*K5 + 2368*K1*K3*K4 + 192*K1*K4*K5 + 48*K1*K5*K6 - 32*K2**6 - 32*K2**4*K4**2 + 192*K2**4*K4 - 1648*K2**4 + 64*K2**3*K3*K5 + 64*K2**3*K4*K6 - 64*K2**3*K6 - 2032*K2**2*K3**2 - 32*K2**2*K3*K7 - 304*K2**2*K4**2 - 32*K2**2*K4*K8 + 2080*K2**2*K4 - 64*K2**2*K5**2 - 16*K2**2*K6**2 - 2966*K2**2 - 224*K2*K3**2*K4 + 1472*K2*K3*K5 + 272*K2*K4*K6 + 48*K2*K5*K7 + 16*K2*K6*K8 - 256*K3**4 + 192*K3**2*K6 - 1972*K3**2 - 724*K4**2 - 220*K5**2 - 74*K6**2 - 8*K7**2 - 2*K8**2 + 3268
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1229']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.3260', 'vk6.3290', 'vk6.3292', 'vk6.3394', 'vk6.3419', 'vk6.3423', 'vk6.3467', 'vk6.3519', 'vk6.4607', 'vk6.5898', 'vk6.6025', 'vk6.7947', 'vk6.8070', 'vk6.9385', 'vk6.17845', 'vk6.17860', 'vk6.19058', 'vk6.19884', 'vk6.24362', 'vk6.25672', 'vk6.25687', 'vk6.26328', 'vk6.26773', 'vk6.37778', 'vk6.43787', 'vk6.43801', 'vk6.45065', 'vk6.48112', 'vk6.48117', 'vk6.48142', 'vk6.48201', 'vk6.50659']
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 O1O2O3O4U3U2U4O5O6U5U1U6
R3 orbit {'O1O2O3O4U3U2U4O5O6U5U1U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U5U4U6O5O6U1U3U2
Gauss code of K* O1O2O3U4U5O4O6O5U6U2U1U3
Gauss code of -K* O1O2O3U1U3O4O5O6U4U6U5U2
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 -1 -1 -1 2 -1 2],[ 1 0 -1 -1 2 0 2],[ 1 1 0 0 2 0 0],[ 1 1 0 0 1 0 0],[-2 -2 -2 -1 0 0 0],[ 1 0 0 0 0 0 1],[-2 -2 0 0 0 -1 0]]
Primitive based matrix [[ 0 2 2 -1 -1 -1 -1],[-2 0 0 0 0 -1 -2],[-2 0 0 -1 -2 0 -2],[ 1 0 1 0 0 0 1],[ 1 0 2 0 0 0 1],[ 1 1 0 0 0 0 0],[ 1 2 2 -1 -1 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,1,1,1,1,0,0,0,1,2,1,2,0,2,0,0,-1,0,-1,0]
Phi over symmetry [-2,-2,1,1,1,1,0,0,0,1,2,1,2,0,2,0,0,-1,0,-1,0]
Phi of -K [-1,-1,-1,-1,2,2,-1,0,0,1,3,0,1,1,1,0,3,2,2,3,0]
Phi of K* [-2,-2,1,1,1,1,0,1,1,2,3,1,3,3,2,-1,-1,0,0,0,0]
Phi of -K* [-1,-1,-1,-1,2,2,-1,-1,0,2,2,0,0,0,1,0,0,2,1,0,0]
Symmetry type of based matrix c
u-polynomial -2t^2+4t
Normalized Jones-Krushkal polynomial 9z^2+30z+25
Enhanced Jones-Krushkal polynomial 9w^3z^2+30w^2z+25w
Inner characteristic polynomial t^6+16t^4+32t^2+1
Outer characteristic polynomial t^7+28t^5+76t^3+8t
Flat arrow polynomial 4*K1**3 + 4*K1**2*K2 - 4*K1*K2 - 4*K1*K3 - K1 + K3 + K4
2-strand cable arrow polynomial 1792*K1**4*K2 - 2976*K1**4 - 384*K1**3*K2**2*K3 + 1536*K1**3*K2*K3 - 1088*K1**3*K3 - 128*K1**2*K2**4 + 1280*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 + 256*K1**2*K2**2*K4 - 6672*K1**2*K2**2 + 768*K1**2*K2*K3**2 - 1024*K1**2*K2*K4 + 6864*K1**2*K2 - 2208*K1**2*K3**2 - 128*K1**2*K3*K5 - 32*K1**2*K4**2 - 3176*K1**2 + 1088*K1*K2**3*K3 + 256*K1*K2**2*K3*K4 - 2464*K1*K2**2*K3 - 448*K1*K2**2*K5 + 384*K1*K2*K3**3 - 832*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 8144*K1*K2*K3 - 192*K1*K2*K4*K5 + 2368*K1*K3*K4 + 192*K1*K4*K5 + 48*K1*K5*K6 - 32*K2**6 - 32*K2**4*K4**2 + 192*K2**4*K4 - 1648*K2**4 + 64*K2**3*K3*K5 + 64*K2**3*K4*K6 - 64*K2**3*K6 - 2032*K2**2*K3**2 - 32*K2**2*K3*K7 - 304*K2**2*K4**2 - 32*K2**2*K4*K8 + 2080*K2**2*K4 - 64*K2**2*K5**2 - 16*K2**2*K6**2 - 2966*K2**2 - 224*K2*K3**2*K4 + 1472*K2*K3*K5 + 272*K2*K4*K6 + 48*K2*K5*K7 + 16*K2*K6*K8 - 256*K3**4 + 192*K3**2*K6 - 1972*K3**2 - 724*K4**2 - 220*K5**2 - 74*K6**2 - 8*K7**2 - 2*K8**2 + 3268
Genus of based matrix 1
Fillings of based matrix [[{4, 6}, {2, 5}, {1, 3}], [{5, 6}, {2, 4}, {1, 3}]]
If K is slice False
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