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Flat knot 6.1735

Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,0,1,1,2,2,0,1,1,1,1,0,1,0,0,-1]
Flat knots (up to 7 crossings) with same phi are :['6.1735']
Arrow polynomial of the knot is: 4*K1**3 - 2*K1**2 - 8*K1*K2 + K1 + K2 + 3*K3 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.1030', '6.1062', '6.1226', '6.1508', '6.1525', '6.1596', '6.1724', '6.1729', '6.1735', '6.1738', '6.1789', '6.1809', '6.1921']
Outer characteristic polynomial of the knot is: t^7+24t^5+41t^3+8t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1724', '6.1735']
2-strand cable arrow polynomial of the knot is: 2688*K1**4*K2 - 5344*K1**4 + 1536*K1**3*K2*K3 - 1696*K1**3*K3 - 768*K1**2*K2**4 + 448*K1**2*K2**3 + 768*K1**2*K2**2*K4 - 6096*K1**2*K2**2 - 1056*K1**2*K2*K4 + 7176*K1**2*K2 - 1664*K1**2*K3**2 - 192*K1**2*K4**2 - 1676*K1**2 + 800*K1*K2**3*K3 - 672*K1*K2**2*K3 - 416*K1*K2**2*K5 - 448*K1*K2*K3*K4 + 6208*K1*K2*K3 + 1744*K1*K3*K4 + 256*K1*K4*K5 - 32*K2**6 + 96*K2**4*K4 - 312*K2**4 - 32*K2**3*K6 - 320*K2**2*K3**2 - 128*K2**2*K4**2 + 776*K2**2*K4 - 2538*K2**2 + 376*K2*K3*K5 + 104*K2*K4*K6 - 1532*K3**2 - 594*K4**2 - 120*K5**2 - 22*K6**2 + 2680
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1735']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.19918', 'vk6.19926', 'vk6.21147', 'vk6.21153', 'vk6.26849', 'vk6.26865', 'vk6.28621', 'vk6.28632', 'vk6.38285', 'vk6.38301', 'vk6.40415', 'vk6.40425', 'vk6.45156', 'vk6.45164', 'vk6.47001', 'vk6.47006', 'vk6.56701', 'vk6.56708', 'vk6.57789', 'vk6.57797', 'vk6.61108', 'vk6.61116', 'vk6.62359', 'vk6.62372', 'vk6.66391', 'vk6.66399', 'vk6.67153', 'vk6.67165', 'vk6.69050', 'vk6.69054', 'vk6.69838', 'vk6.69844']
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 O1O2O3U2U4O5O6U1U3O4U6U5
R3 orbit {'O1O2O3U2U4O5O6U1U3O4U6U5'}
R3 orbit length 1
Gauss code of -K O1O2O3U4U5O6U1U3O5O4U6U2
Gauss code of K* O1O2U3O4O5U1U6U2O6O3U5U4
Gauss code of -K* O1O2U3O4O5U2U1O3O6U4U6U5
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 -2 -1 1 0 1 1],[ 2 0 0 2 1 2 1],[ 1 0 0 1 0 1 1],[-1 -2 -1 0 -1 1 0],[ 0 -1 0 1 0 0 1],[-1 -2 -1 -1 0 0 0],[-1 -1 -1 0 -1 0 0]]
Primitive based matrix [[ 0 1 1 1 0 -1 -2],[-1 0 1 0 -1 -1 -2],[-1 -1 0 0 0 -1 -2],[-1 0 0 0 -1 -1 -1],[ 0 1 0 1 0 0 -1],[ 1 1 1 1 0 0 0],[ 2 2 2 1 1 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-1,-1,-1,0,1,2,-1,0,1,1,2,0,0,1,2,1,1,1,0,1,0]
Phi over symmetry [-2,-1,0,1,1,1,0,1,1,2,2,0,1,1,1,1,0,1,0,0,-1]
Phi of -K [-2,-1,0,1,1,1,1,1,1,1,2,1,1,1,1,0,1,0,-1,0,0]
Phi of K* [-1,-1,-1,0,1,2,-1,0,1,1,1,0,0,1,1,0,1,2,1,1,1]
Phi of -K* [-2,-1,0,1,1,1,0,1,1,2,2,0,1,1,1,1,0,1,0,0,-1]
Symmetry type of based matrix c
u-polynomial t^2-2t
Normalized Jones-Krushkal polynomial 5z^2+22z+25
Enhanced Jones-Krushkal polynomial 5w^3z^2+22w^2z+25w
Inner characteristic polynomial t^6+16t^4+24t^2+4
Outer characteristic polynomial t^7+24t^5+41t^3+8t
Flat arrow polynomial 4*K1**3 - 2*K1**2 - 8*K1*K2 + K1 + K2 + 3*K3 + 2
2-strand cable arrow polynomial 2688*K1**4*K2 - 5344*K1**4 + 1536*K1**3*K2*K3 - 1696*K1**3*K3 - 768*K1**2*K2**4 + 448*K1**2*K2**3 + 768*K1**2*K2**2*K4 - 6096*K1**2*K2**2 - 1056*K1**2*K2*K4 + 7176*K1**2*K2 - 1664*K1**2*K3**2 - 192*K1**2*K4**2 - 1676*K1**2 + 800*K1*K2**3*K3 - 672*K1*K2**2*K3 - 416*K1*K2**2*K5 - 448*K1*K2*K3*K4 + 6208*K1*K2*K3 + 1744*K1*K3*K4 + 256*K1*K4*K5 - 32*K2**6 + 96*K2**4*K4 - 312*K2**4 - 32*K2**3*K6 - 320*K2**2*K3**2 - 128*K2**2*K4**2 + 776*K2**2*K4 - 2538*K2**2 + 376*K2*K3*K5 + 104*K2*K4*K6 - 1532*K3**2 - 594*K4**2 - 120*K5**2 - 22*K6**2 + 2680
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {4, 5}, {2, 3}], [{1, 6}, {4, 5}, {3}, {2}], [{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {4, 5}, {1, 3}], [{3, 6}, {4, 5}, {1, 2}], [{4, 6}, {1, 5}, {2, 3}], [{5, 6}, {1, 4}, {2, 3}], [{6}, {4, 5}, {2, 3}, {1}]]
If K is slice False
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