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Flat knot 6.743

Min(phi) over symmetries of the knot is: [-3,-2,-1,2,2,2,0,2,2,2,4,1,1,1,2,2,2,3,0,-1,0]
Flat knots (up to 7 crossings) with same phi are :['6.743']
Arrow polynomial of the knot is: 8*K1**3 - 12*K1**2 - 10*K1*K2 - K1 + 6*K2 + 3*K3 + 7
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.743', '6.836']
Outer characteristic polynomial of the knot is: t^7+79t^5+58t^3+3t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.743']
2-strand cable arrow polynomial of the knot is: -320*K1**4*K2**2 + 1504*K1**4*K2 - 3664*K1**4 - 128*K1**3*K2**2*K3 + 1056*K1**3*K2*K3 + 32*K1**3*K3*K4 - 640*K1**3*K3 + 640*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 7536*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 736*K1**2*K2*K4 + 11064*K1**2*K2 - 1040*K1**2*K3**2 - 32*K1**2*K4**2 - 7184*K1**2 + 576*K1*K2**3*K3 - 1664*K1*K2**2*K3 - 384*K1*K2**2*K5 - 288*K1*K2*K3*K4 + 10256*K1*K2*K3 + 1648*K1*K3*K4 + 128*K1*K4*K5 - 64*K2**6 + 192*K2**4*K4 - 1456*K2**4 - 64*K2**3*K6 - 880*K2**2*K3**2 - 168*K2**2*K4**2 + 2200*K2**2*K4 - 6210*K2**2 - 64*K2*K3**2*K4 + 888*K2*K3*K5 + 152*K2*K4*K6 - 32*K3**4 + 64*K3**2*K6 - 3252*K3**2 - 880*K4**2 - 204*K5**2 - 46*K6**2 + 6342
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.743']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.17111', 'vk6.17354', 'vk6.20570', 'vk6.21978', 'vk6.23502', 'vk6.23841', 'vk6.28033', 'vk6.29491', 'vk6.35655', 'vk6.36092', 'vk6.39446', 'vk6.41645', 'vk6.43015', 'vk6.43327', 'vk6.46030', 'vk6.47696', 'vk6.55250', 'vk6.55502', 'vk6.57452', 'vk6.58618', 'vk6.59652', 'vk6.60000', 'vk6.62123', 'vk6.63089', 'vk6.65056', 'vk6.65251', 'vk6.66986', 'vk6.67850', 'vk6.68317', 'vk6.68467', 'vk6.69602', 'vk6.70294']
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 O1O2O3O4U2O5U1U6U4O6U5U3
R3 orbit {'O1O2O3O4U2O5U1U6U4O6U5U3'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U2U5O6U1U6U4O5U3
Gauss code of K* O1O2O3U2O4O5U1U6U5U3O6U4
Gauss code of -K* O1O2O3U4O5U1U6U5U3O6O4U2
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 -3 -2 2 2 2 -1],[ 3 0 0 4 2 2 2],[ 2 0 0 2 1 1 1],[-2 -4 -2 0 0 1 -3],[-2 -2 -1 0 0 0 -2],[-2 -2 -1 -1 0 0 -2],[ 1 -2 -1 3 2 2 0]]
Primitive based matrix [[ 0 2 2 2 -1 -2 -3],[-2 0 1 0 -3 -2 -4],[-2 -1 0 0 -2 -1 -2],[-2 0 0 0 -2 -1 -2],[ 1 3 2 2 0 -1 -2],[ 2 2 1 1 1 0 0],[ 3 4 2 2 2 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,-2,1,2,3,-1,0,3,2,4,0,2,1,2,2,1,2,1,2,0]
Phi over symmetry [-3,-2,-1,2,2,2,0,2,2,2,4,1,1,1,2,2,2,3,0,-1,0]
Phi of -K [-3,-2,-1,2,2,2,1,0,1,3,3,0,2,3,3,0,1,1,-1,0,0]
Phi of K* [-2,-2,-2,1,2,3,-1,0,1,3,3,0,0,2,1,1,3,3,0,0,1]
Phi of -K* [-3,-2,-1,2,2,2,0,2,2,2,4,1,1,1,2,2,2,3,0,-1,0]
Symmetry type of based matrix c
u-polynomial t^3-2t^2+t
Normalized Jones-Krushkal polynomial 2z^2+23z+39
Enhanced Jones-Krushkal polynomial 2w^3z^2+23w^2z+39w
Inner characteristic polynomial t^6+53t^4+14t^2
Outer characteristic polynomial t^7+79t^5+58t^3+3t
Flat arrow polynomial 8*K1**3 - 12*K1**2 - 10*K1*K2 - K1 + 6*K2 + 3*K3 + 7
2-strand cable arrow polynomial -320*K1**4*K2**2 + 1504*K1**4*K2 - 3664*K1**4 - 128*K1**3*K2**2*K3 + 1056*K1**3*K2*K3 + 32*K1**3*K3*K4 - 640*K1**3*K3 + 640*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 7536*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 736*K1**2*K2*K4 + 11064*K1**2*K2 - 1040*K1**2*K3**2 - 32*K1**2*K4**2 - 7184*K1**2 + 576*K1*K2**3*K3 - 1664*K1*K2**2*K3 - 384*K1*K2**2*K5 - 288*K1*K2*K3*K4 + 10256*K1*K2*K3 + 1648*K1*K3*K4 + 128*K1*K4*K5 - 64*K2**6 + 192*K2**4*K4 - 1456*K2**4 - 64*K2**3*K6 - 880*K2**2*K3**2 - 168*K2**2*K4**2 + 2200*K2**2*K4 - 6210*K2**2 - 64*K2*K3**2*K4 + 888*K2*K3*K5 + 152*K2*K4*K6 - 32*K3**4 + 64*K3**2*K6 - 3252*K3**2 - 880*K4**2 - 204*K5**2 - 46*K6**2 + 6342
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {3, 5}, {2, 4}], [{3, 6}, {1, 5}, {2, 4}], [{5, 6}, {2, 4}, {1, 3}], [{6}, {1, 5}, {2, 4}, {3}]]
If K is slice False
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