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Flat knot 6.1738

Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,0,1,1,2,2,0,0,1,2,1,1,0,-1,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.1738']
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+45t^3+8t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1729', '6.1738']
2-strand cable arrow polynomial of the knot is: 1856*K1**4*K2 - 3648*K1**4 + 672*K1**3*K2*K3 - 1728*K1**3*K3 - 128*K1**2*K2**4 + 448*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 3664*K1**2*K2**2 - 1152*K1**2*K2*K4 + 5744*K1**2*K2 - 736*K1**2*K3**2 - 32*K1**2*K4**2 - 2300*K1**2 + 352*K1*K2**3*K3 - 288*K1*K2**2*K3 - 192*K1*K2**2*K5 - 320*K1*K2*K3*K4 + 4816*K1*K2*K3 + 1296*K1*K3*K4 + 200*K1*K4*K5 - 32*K2**6 + 96*K2**4*K4 - 504*K2**4 - 256*K2**2*K3**2 - 128*K2**2*K4**2 + 968*K2**2*K4 - 2322*K2**2 + 464*K2*K3*K5 + 104*K2*K4*K6 - 1348*K3**2 - 642*K4**2 - 184*K5**2 - 30*K6**2 + 2488
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1738']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.13921', 'vk6.14018', 'vk6.14177', 'vk6.14416', 'vk6.14988', 'vk6.15111', 'vk6.15649', 'vk6.16103', 'vk6.16717', 'vk6.16746', 'vk6.16841', 'vk6.18795', 'vk6.19284', 'vk6.19576', 'vk6.23155', 'vk6.23220', 'vk6.25393', 'vk6.26469', 'vk6.33740', 'vk6.33817', 'vk6.34290', 'vk6.35144', 'vk6.37522', 'vk6.42727', 'vk6.44697', 'vk6.54116', 'vk6.54934', 'vk6.54964', 'vk6.56389', 'vk6.56619', 'vk6.59358', 'vk6.64586']
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 O1O2O3U2U4O5O6U5U3O4U1U6
R3 orbit {'O1O2O3U2U4O5O6U5U3O4U1U6'}
R3 orbit length 1
Gauss code of -K O1O2O3U4U3O5U1U6O4O6U5U2
Gauss code of K* O1O2U3O4O5U4U6U2O6O3U1U5
Gauss code of -K* O1O2U3O4O5U1U5O3O6U4U6U2
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 0 -1 2],[ 1 0 -1 2 0 0 2],[ 1 1 0 1 0 0 1],[-1 -2 -1 0 -1 0 1],[ 0 0 0 1 0 -1 1],[ 1 0 0 0 1 0 1],[-2 -2 -1 -1 -1 -1 0]]
Primitive based matrix [[ 0 2 1 0 -1 -1 -1],[-2 0 -1 -1 -1 -1 -2],[-1 1 0 -1 0 -1 -2],[ 0 1 1 0 -1 0 0],[ 1 1 0 1 0 0 0],[ 1 1 1 0 0 0 1],[ 1 2 2 0 0 -1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,0,1,1,1,1,1,1,1,2,1,0,1,2,1,0,0,0,0,-1]
Phi over symmetry [-2,-1,0,1,1,1,0,1,1,2,2,0,0,1,2,1,1,0,-1,0,0]
Phi of -K [-1,-1,-1,0,1,2,-1,0,1,1,2,0,1,0,1,0,2,2,0,1,0]
Phi of K* [-2,-1,0,1,1,1,0,1,1,2,2,0,0,1,2,1,1,0,-1,0,0]
Phi of -K* [-1,-1,-1,0,1,2,-1,0,0,2,2,0,0,1,1,1,0,1,1,1,1]
Symmetry type of based matrix c
u-polynomial -t^2+2t
Normalized Jones-Krushkal polynomial 6z^2+23z+23
Enhanced Jones-Krushkal polynomial 6w^3z^2+23w^2z+23w
Inner characteristic polynomial t^6+16t^4+24t^2+4
Outer characteristic polynomial t^7+24t^5+45t^3+8t
Flat arrow polynomial 4*K1**3 - 2*K1**2 - 8*K1*K2 + K1 + K2 + 3*K3 + 2
2-strand cable arrow polynomial 1856*K1**4*K2 - 3648*K1**4 + 672*K1**3*K2*K3 - 1728*K1**3*K3 - 128*K1**2*K2**4 + 448*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 3664*K1**2*K2**2 - 1152*K1**2*K2*K4 + 5744*K1**2*K2 - 736*K1**2*K3**2 - 32*K1**2*K4**2 - 2300*K1**2 + 352*K1*K2**3*K3 - 288*K1*K2**2*K3 - 192*K1*K2**2*K5 - 320*K1*K2*K3*K4 + 4816*K1*K2*K3 + 1296*K1*K3*K4 + 200*K1*K4*K5 - 32*K2**6 + 96*K2**4*K4 - 504*K2**4 - 256*K2**2*K3**2 - 128*K2**2*K4**2 + 968*K2**2*K4 - 2322*K2**2 + 464*K2*K3*K5 + 104*K2*K4*K6 - 1348*K3**2 - 642*K4**2 - 184*K5**2 - 30*K6**2 + 2488
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {4, 5}, {2, 3}], [{1, 6}, {4, 5}, {3}, {2}], [{2, 6}, {4, 5}, {1, 3}], [{3, 6}, {4, 5}, {1, 2}], [{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {2, 5}, {1, 3}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {2, 4}, {1, 3}], [{6}, {4, 5}, {2, 3}, {1}]]
If K is slice False
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