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Flat knot 6.450

Min(phi) over symmetries of the knot is: [-3,-1,-1,1,1,3,0,1,1,2,3,1,0,1,1,1,2,3,0,0,1]
Flat knots (up to 7 crossings) with same phi are :['6.450']
Arrow polynomial of the knot is: 4*K1**3 + 8*K1**2*K2 - 12*K1**2 - 4*K1*K2 - 4*K1*K3 - K1 + 4*K2 + K3 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.450']
Outer characteristic polynomial of the knot is: t^7+71t^5+90t^3+9t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.450']
2-strand cable arrow polynomial of the knot is: 256*K1**4*K2**3 - 1024*K1**4*K2**2 + 1280*K1**4*K2 - 1760*K1**4 + 384*K1**3*K2*K3 + 32*K1**3*K3*K4 - 128*K1**3*K3 - 1152*K1**2*K2**4 - 384*K1**2*K2**3*K4 + 3104*K1**2*K2**3 - 8320*K1**2*K2**2 - 800*K1**2*K2*K4 + 7288*K1**2*K2 - 160*K1**2*K3**2 - 80*K1**2*K4**2 - 4312*K1**2 - 128*K1*K2**3*K3*K4 + 2400*K1*K2**3*K3 + 512*K1*K2**2*K3*K4 - 1312*K1*K2**2*K3 + 32*K1*K2**2*K4*K5 - 352*K1*K2**2*K5 - 320*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 6192*K1*K2*K3 - 32*K1*K2*K4*K5 + 840*K1*K3*K4 + 144*K1*K4*K5 - 32*K2**6 - 64*K2**4*K3**2 - 192*K2**4*K4**2 + 800*K2**4*K4 - 2976*K2**4 + 192*K2**3*K3*K5 + 128*K2**3*K4*K6 - 64*K2**3*K6 - 1232*K2**2*K3**2 - 720*K2**2*K4**2 + 2088*K2**2*K4 - 112*K2**2*K5**2 - 16*K2**2*K6**2 - 2126*K2**2 + 568*K2*K3*K5 + 216*K2*K4*K6 + 24*K2*K5*K7 + 8*K3**2*K6 - 1456*K3**2 - 648*K4**2 - 96*K5**2 - 26*K6**2 + 3566
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.450']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11599', 'vk6.11602', 'vk6.11950', 'vk6.11955', 'vk6.12945', 'vk6.12948', 'vk6.13256', 'vk6.20420', 'vk6.20434', 'vk6.21787', 'vk6.27776', 'vk6.27802', 'vk6.29298', 'vk6.31394', 'vk6.31405', 'vk6.32572', 'vk6.32583', 'vk6.32956', 'vk6.39204', 'vk6.39226', 'vk6.41428', 'vk6.47555', 'vk6.53190', 'vk6.53205', 'vk6.53503', 'vk6.57289', 'vk6.57295', 'vk6.61963', 'vk6.61973', 'vk6.64283', 'vk6.64294', 'vk6.64493']
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 O1O2O3O4O5U6U3O6U1U4U5U2
R3 orbit {'O1O2O3O4O5U6U3O6U1U4U5U2'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U4U1U2U5O6U3U6
Gauss code of K* O1O2O3O4U1U4U5U2U3O6O5U6
Gauss code of -K* O1O2O3O4U5O6O5U2U3U6U1U4
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 1 -1 1 3 -1],[ 3 0 3 1 2 3 2],[-1 -3 0 -1 0 2 -2],[ 1 -1 1 0 0 1 1],[-1 -2 0 0 0 1 -1],[-3 -3 -2 -1 -1 0 -3],[ 1 -2 2 -1 1 3 0]]
Primitive based matrix [[ 0 3 1 1 -1 -1 -3],[-3 0 -1 -2 -1 -3 -3],[-1 1 0 0 0 -1 -2],[-1 2 0 0 -1 -2 -3],[ 1 1 0 1 0 1 -1],[ 1 3 1 2 -1 0 -2],[ 3 3 2 3 1 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-1,-1,1,1,3,1,2,1,3,3,0,0,1,2,1,2,3,-1,1,2]
Phi over symmetry [-3,-1,-1,1,1,3,0,1,1,2,3,1,0,1,1,1,2,3,0,0,1]
Phi of -K [-3,-1,-1,1,1,3,0,1,1,2,3,1,0,1,1,1,2,3,0,0,1]
Phi of K* [-3,-1,-1,1,1,3,0,1,1,3,3,0,0,1,1,1,2,2,-1,0,1]
Phi of -K* [-3,-1,-1,1,1,3,1,2,2,3,3,1,0,1,1,1,2,3,0,1,2]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 4z^2+21z+27
Enhanced Jones-Krushkal polynomial 4w^3z^2-4w^3z+25w^2z+27w
Inner characteristic polynomial t^6+49t^4+54t^2+1
Outer characteristic polynomial t^7+71t^5+90t^3+9t
Flat arrow polynomial 4*K1**3 + 8*K1**2*K2 - 12*K1**2 - 4*K1*K2 - 4*K1*K3 - K1 + 4*K2 + K3 + 5
2-strand cable arrow polynomial 256*K1**4*K2**3 - 1024*K1**4*K2**2 + 1280*K1**4*K2 - 1760*K1**4 + 384*K1**3*K2*K3 + 32*K1**3*K3*K4 - 128*K1**3*K3 - 1152*K1**2*K2**4 - 384*K1**2*K2**3*K4 + 3104*K1**2*K2**3 - 8320*K1**2*K2**2 - 800*K1**2*K2*K4 + 7288*K1**2*K2 - 160*K1**2*K3**2 - 80*K1**2*K4**2 - 4312*K1**2 - 128*K1*K2**3*K3*K4 + 2400*K1*K2**3*K3 + 512*K1*K2**2*K3*K4 - 1312*K1*K2**2*K3 + 32*K1*K2**2*K4*K5 - 352*K1*K2**2*K5 - 320*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 6192*K1*K2*K3 - 32*K1*K2*K4*K5 + 840*K1*K3*K4 + 144*K1*K4*K5 - 32*K2**6 - 64*K2**4*K3**2 - 192*K2**4*K4**2 + 800*K2**4*K4 - 2976*K2**4 + 192*K2**3*K3*K5 + 128*K2**3*K4*K6 - 64*K2**3*K6 - 1232*K2**2*K3**2 - 720*K2**2*K4**2 + 2088*K2**2*K4 - 112*K2**2*K5**2 - 16*K2**2*K6**2 - 2126*K2**2 + 568*K2*K3*K5 + 216*K2*K4*K6 + 24*K2*K5*K7 + 8*K3**2*K6 - 1456*K3**2 - 648*K4**2 - 96*K5**2 - 26*K6**2 + 3566
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{2, 6}, {1, 5}, {3, 4}], [{4, 6}, {1, 5}, {2, 3}], [{5, 6}, {3, 4}, {1, 2}]]
If K is slice False
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