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Flat knot 6.298

Min(phi) over symmetries of the knot is: [-4,-2,-1,1,3,3,1,0,3,3,4,0,2,2,3,1,1,1,1,2,0]
Flat knots (up to 7 crossings) with same phi are :['6.298']
Arrow polynomial of the knot is: 4*K1**3 - 4*K1*K2 - 2*K1*K3 - K1 + K2 + K3 + K4 + 1
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.71', '6.148', '6.170', '6.253', '6.259', '6.298', '6.439', '6.453', '6.499', '6.503']
Outer characteristic polynomial of the knot is: t^7+100t^5+69t^3+6t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.298']
2-strand cable arrow polynomial of the knot is: -1024*K1**4*K2**2 + 1632*K1**4*K2 - 2672*K1**4 + 2048*K1**3*K2*K3 - 736*K1**3*K3 - 1280*K1**2*K2**4 + 1952*K1**2*K2**3 - 512*K1**2*K2**2*K3**2 + 256*K1**2*K2**2*K4 - 6400*K1**2*K2**2 - 1152*K1**2*K2*K4 + 6168*K1**2*K2 - 1168*K1**2*K3**2 - 96*K1**2*K3*K5 - 2968*K1**2 + 1728*K1*K2**3*K3 + 576*K1*K2**2*K3*K4 - 1568*K1*K2**2*K3 + 64*K1*K2**2*K4*K5 - 416*K1*K2**2*K5 + 256*K1*K2*K3**3 - 224*K1*K2*K3*K4 - 96*K1*K2*K3*K6 + 6240*K1*K2*K3 + 1552*K1*K3*K4 + 216*K1*K4*K5 + 8*K1*K5*K6 - 32*K2**6 + 64*K2**4*K4 - 976*K2**4 + 32*K2**3*K3*K5 - 896*K2**2*K3**2 - 240*K2**2*K4**2 + 1248*K2**2*K4 - 80*K2**2*K5**2 - 8*K2**2*K6**2 - 2498*K2**2 - 160*K2*K3**2*K4 - 32*K2*K3*K4*K5 + 632*K2*K3*K5 + 88*K2*K4*K6 + 32*K2*K5*K7 + 8*K2*K6*K8 - 64*K3**4 + 56*K3**2*K6 - 1708*K3**2 + 8*K3*K4*K7 - 614*K4**2 - 200*K5**2 - 22*K6**2 - 4*K7**2 - 2*K8**2 + 2982
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.298']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.19915', 'vk6.19960', 'vk6.21140', 'vk6.21217', 'vk6.26836', 'vk6.26957', 'vk6.28612', 'vk6.28699', 'vk6.38268', 'vk6.38369', 'vk6.40398', 'vk6.40528', 'vk6.45143', 'vk6.45234', 'vk6.46993', 'vk6.47047', 'vk6.56694', 'vk6.56758', 'vk6.57778', 'vk6.57873', 'vk6.61096', 'vk6.61225', 'vk6.62352', 'vk6.62443', 'vk6.66382', 'vk6.66468', 'vk6.67142', 'vk6.67253', 'vk6.69041', 'vk6.69115', 'vk6.69833', 'vk6.69887']
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 O1O2O3O4O5U2U1O6U3U6U4U5
R3 orbit {'O1O2O3O4O5U2U1O6U3U6U4U5'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U1U2U6U3O6U5U4
Gauss code of K* O1O2O3O4U5U6U1U3U4O6O5U2
Gauss code of -K* O1O2O3O4U3O5O6U1U2U4U6U5
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 -3 -1 2 4 1],[ 3 0 0 2 3 4 1],[ 3 0 0 1 2 3 1],[ 1 -2 -1 0 2 3 1],[-2 -3 -2 -2 0 1 0],[-4 -4 -3 -3 -1 0 0],[-1 -1 -1 -1 0 0 0]]
Primitive based matrix [[ 0 4 2 1 -1 -3 -3],[-4 0 -1 0 -3 -3 -4],[-2 1 0 0 -2 -2 -3],[-1 0 0 0 -1 -1 -1],[ 1 3 2 1 0 -1 -2],[ 3 3 2 1 1 0 0],[ 3 4 3 1 2 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-4,-2,-1,1,3,3,1,0,3,3,4,0,2,2,3,1,1,1,1,2,0]
Phi over symmetry [-4,-2,-1,1,3,3,1,0,3,3,4,0,2,2,3,1,1,1,1,2,0]
Phi of -K [-3,-3,-1,1,2,4,0,0,3,2,3,1,3,3,4,1,1,2,1,3,1]
Phi of K* [-4,-2,-1,1,3,3,1,3,2,3,4,1,1,2,3,1,3,3,0,1,0]
Phi of -K* [-3,-3,-1,1,2,4,0,1,1,2,3,2,1,3,4,1,2,3,0,0,1]
Symmetry type of based matrix c
u-polynomial -t^4+2t^3-t^2
Normalized Jones-Krushkal polynomial 7z^2+26z+25
Enhanced Jones-Krushkal polynomial 7w^3z^2+26w^2z+25w
Inner characteristic polynomial t^6+60t^4+16t^2+1
Outer characteristic polynomial t^7+100t^5+69t^3+6t
Flat arrow polynomial 4*K1**3 - 4*K1*K2 - 2*K1*K3 - K1 + K2 + K3 + K4 + 1
2-strand cable arrow polynomial -1024*K1**4*K2**2 + 1632*K1**4*K2 - 2672*K1**4 + 2048*K1**3*K2*K3 - 736*K1**3*K3 - 1280*K1**2*K2**4 + 1952*K1**2*K2**3 - 512*K1**2*K2**2*K3**2 + 256*K1**2*K2**2*K4 - 6400*K1**2*K2**2 - 1152*K1**2*K2*K4 + 6168*K1**2*K2 - 1168*K1**2*K3**2 - 96*K1**2*K3*K5 - 2968*K1**2 + 1728*K1*K2**3*K3 + 576*K1*K2**2*K3*K4 - 1568*K1*K2**2*K3 + 64*K1*K2**2*K4*K5 - 416*K1*K2**2*K5 + 256*K1*K2*K3**3 - 224*K1*K2*K3*K4 - 96*K1*K2*K3*K6 + 6240*K1*K2*K3 + 1552*K1*K3*K4 + 216*K1*K4*K5 + 8*K1*K5*K6 - 32*K2**6 + 64*K2**4*K4 - 976*K2**4 + 32*K2**3*K3*K5 - 896*K2**2*K3**2 - 240*K2**2*K4**2 + 1248*K2**2*K4 - 80*K2**2*K5**2 - 8*K2**2*K6**2 - 2498*K2**2 - 160*K2*K3**2*K4 - 32*K2*K3*K4*K5 + 632*K2*K3*K5 + 88*K2*K4*K6 + 32*K2*K5*K7 + 8*K2*K6*K8 - 64*K3**4 + 56*K3**2*K6 - 1708*K3**2 + 8*K3*K4*K7 - 614*K4**2 - 200*K5**2 - 22*K6**2 - 4*K7**2 - 2*K8**2 + 2982
Genus of based matrix 1
Fillings of based matrix [[{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {4, 5}, {1, 2}]]
If K is slice False
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