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Flat knot 6.1677

Min(phi) over symmetries of the knot is: [-2,-2,0,1,1,2,-1,1,0,2,2,1,1,2,1,0,1,1,-1,0,1]
Flat knots (up to 7 crossings) with same phi are :['6.1677']
Arrow polynomial of the knot is: 12*K1**3 - 10*K1**2 - 8*K1*K2 - 5*K1 + 5*K2 + K3 + 6
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.1677']
Outer characteristic polynomial of the knot is: t^7+49t^5+66t^3+11t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1677']
2-strand cable arrow polynomial of the knot is: -1152*K1**4*K2**2 + 2624*K1**4*K2 - 3760*K1**4 - 384*K1**3*K2**2*K3 + 1248*K1**3*K2*K3 - 800*K1**3*K3 - 896*K1**2*K2**4 + 4192*K1**2*K2**3 - 256*K1**2*K2**2*K3**2 + 192*K1**2*K2**2*K4 - 13024*K1**2*K2**2 + 64*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 960*K1**2*K2*K4 + 11560*K1**2*K2 - 400*K1**2*K3**2 - 32*K1**2*K3*K5 - 5424*K1**2 + 2528*K1*K2**3*K3 + 192*K1*K2**2*K3*K4 - 2080*K1*K2**2*K3 - 416*K1*K2**2*K5 - 96*K1*K2*K3*K4 + 8856*K1*K2*K3 - 32*K1*K2*K4*K5 + 624*K1*K3*K4 + 24*K1*K4*K5 - 96*K2**6 + 192*K2**4*K4 - 3224*K2**4 - 32*K2**3*K6 - 1088*K2**2*K3**2 - 96*K2**2*K4**2 + 1936*K2**2*K4 - 2766*K2**2 + 320*K2*K3*K5 + 24*K2*K4*K6 - 1632*K3**2 - 286*K4**2 - 24*K5**2 - 2*K6**2 + 4348
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1677']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.13372', 'vk6.13443', 'vk6.13634', 'vk6.13754', 'vk6.14161', 'vk6.14398', 'vk6.15627', 'vk6.16085', 'vk6.16462', 'vk6.16477', 'vk6.17640', 'vk6.22861', 'vk6.22892', 'vk6.24189', 'vk6.33123', 'vk6.33160', 'vk6.33224', 'vk6.33283', 'vk6.34838', 'vk6.34869', 'vk6.36444', 'vk6.42432', 'vk6.42447', 'vk6.43542', 'vk6.53548', 'vk6.53583', 'vk6.53616', 'vk6.53682', 'vk6.54718', 'vk6.55686', 'vk6.60236', 'vk6.64580']
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 O1O2O3U2O4U5U6U3O5O6U1U4
R3 orbit {'O1O2O3U2O4U5U6U3O5O6U1U4'}
R3 orbit length 1
Gauss code of -K O1O2O3U4U3O5O6U1U5U6O4U2
Gauss code of K* O1O2O3U1U2O4O5U4U6U3O6U5
Gauss code of -K* O1O2O3U4O5U1U5U6O4O6U2U3
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 2 2 -2 0],[ 1 0 -1 3 2 -1 1],[ 1 1 0 1 1 0 0],[-2 -3 -1 0 -1 -2 -1],[-2 -2 -1 1 0 -3 -1],[ 2 1 0 2 3 0 1],[ 0 -1 0 1 1 -1 0]]
Primitive based matrix [[ 0 2 2 0 -1 -1 -2],[-2 0 1 -1 -1 -2 -3],[-2 -1 0 -1 -1 -3 -2],[ 0 1 1 0 0 -1 -1],[ 1 1 1 0 0 1 0],[ 1 2 3 1 -1 0 -1],[ 2 3 2 1 0 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,0,1,1,2,-1,1,1,2,3,1,1,3,2,0,1,1,-1,0,1]
Phi over symmetry [-2,-2,0,1,1,2,-1,1,0,2,2,1,1,2,1,0,1,1,-1,0,1]
Phi of -K [-2,-1,-1,0,2,2,0,1,1,1,2,1,0,1,0,1,2,2,1,1,-1]
Phi of K* [-2,-2,0,1,1,2,-1,1,0,2,2,1,1,2,1,0,1,1,-1,0,1]
Phi of -K* [-2,-1,-1,0,2,2,0,1,1,2,3,1,0,1,1,1,3,2,1,1,-1]
Symmetry type of based matrix c
u-polynomial -t^2+2t
Normalized Jones-Krushkal polynomial 5z^2+26z+33
Enhanced Jones-Krushkal polynomial 5w^3z^2+26w^2z+33w
Inner characteristic polynomial t^6+35t^4+41t^2+4
Outer characteristic polynomial t^7+49t^5+66t^3+11t
Flat arrow polynomial 12*K1**3 - 10*K1**2 - 8*K1*K2 - 5*K1 + 5*K2 + K3 + 6
2-strand cable arrow polynomial -1152*K1**4*K2**2 + 2624*K1**4*K2 - 3760*K1**4 - 384*K1**3*K2**2*K3 + 1248*K1**3*K2*K3 - 800*K1**3*K3 - 896*K1**2*K2**4 + 4192*K1**2*K2**3 - 256*K1**2*K2**2*K3**2 + 192*K1**2*K2**2*K4 - 13024*K1**2*K2**2 + 64*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 960*K1**2*K2*K4 + 11560*K1**2*K2 - 400*K1**2*K3**2 - 32*K1**2*K3*K5 - 5424*K1**2 + 2528*K1*K2**3*K3 + 192*K1*K2**2*K3*K4 - 2080*K1*K2**2*K3 - 416*K1*K2**2*K5 - 96*K1*K2*K3*K4 + 8856*K1*K2*K3 - 32*K1*K2*K4*K5 + 624*K1*K3*K4 + 24*K1*K4*K5 - 96*K2**6 + 192*K2**4*K4 - 3224*K2**4 - 32*K2**3*K6 - 1088*K2**2*K3**2 - 96*K2**2*K4**2 + 1936*K2**2*K4 - 2766*K2**2 + 320*K2*K3*K5 + 24*K2*K4*K6 - 1632*K3**2 - 286*K4**2 - 24*K5**2 - 2*K6**2 + 4348
Genus of based matrix 1
Fillings of based matrix [[{5, 6}, {3, 4}, {1, 2}], [{6}, {5}, {3, 4}, {1, 2}]]
If K is slice False
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