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Flat knot 6.293

Min(phi) over symmetries of the knot is: [-4,0,0,1,1,2,1,2,1,4,4,0,0,1,0,0,1,1,0,-1,1]
Flat knots (up to 7 crossings) with same phi are :['6.293']
Arrow polynomial of the knot is: 4*K1**3 + 4*K1**2*K2 - 4*K1*K2 - 2*K1*K3 - K1 - K2 + K3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.229', '6.293']
Outer characteristic polynomial of the knot is: t^7+67t^5+65t^3+6t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.293']
2-strand cable arrow polynomial of the knot is: 1248*K1**4*K2 - 2464*K1**4 + 672*K1**3*K2*K3 - 448*K1**3*K3 - 128*K1**2*K2**4 + 896*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 5328*K1**2*K2**2 - 928*K1**2*K2*K4 + 5648*K1**2*K2 - 1312*K1**2*K3**2 - 64*K1**2*K3*K5 - 32*K1**2*K4**2 - 2880*K1**2 + 448*K1*K2**3*K3 + 224*K1*K2**2*K3*K4 - 1152*K1*K2**2*K3 - 512*K1*K2**2*K5 - 448*K1*K2*K3*K4 - 96*K1*K2*K3*K6 + 5920*K1*K2*K3 - 32*K1*K2*K4*K5 + 1856*K1*K3*K4 + 376*K1*K4*K5 + 32*K1*K5*K6 - 32*K2**6 - 32*K2**4*K4**2 + 192*K2**4*K4 - 1088*K2**4 + 96*K2**3*K3*K5 + 32*K2**3*K4*K6 - 96*K2**3*K6 - 624*K2**2*K3**2 - 32*K2**2*K3*K7 - 384*K2**2*K4**2 + 1880*K2**2*K4 - 96*K2**2*K5**2 - 8*K2**2*K6**2 - 2814*K2**2 + 864*K2*K3*K5 + 280*K2*K4*K6 + 40*K2*K5*K7 + 8*K3**2*K6 - 1612*K3**2 - 902*K4**2 - 268*K5**2 - 58*K6**2 + 3028
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.293']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16806', 'vk6.16815', 'vk6.16863', 'vk6.16870', 'vk6.18178', 'vk6.18182', 'vk6.18513', 'vk6.18517', 'vk6.23242', 'vk6.23251', 'vk6.24636', 'vk6.25058', 'vk6.25062', 'vk6.35236', 'vk6.35263', 'vk6.36770', 'vk6.37206', 'vk6.37214', 'vk6.42753', 'vk6.42766', 'vk6.44350', 'vk6.44354', 'vk6.54993', 'vk6.55028', 'vk6.55975', 'vk6.55987', 'vk6.59389', 'vk6.59405', 'vk6.60510', 'vk6.65637', 'vk6.68179', 'vk6.68188']
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 O1O2O3O4O5U1U5O6U4U3U6U2
R3 orbit {'O1O2O3O4O5U1U5O6U4U3U6U2'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U4U6U3U2O6U1U5
Gauss code of K* O1O2O3O4U5U4U2U1U6O5O6U3
Gauss code of -K* O1O2O3O4U2O5O6U5U4U3U1U6
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 -4 1 0 0 1 2],[ 4 0 4 3 2 1 2],[-1 -4 0 -1 -1 0 2],[ 0 -3 1 0 0 0 2],[ 0 -2 1 0 0 0 1],[-1 -1 0 0 0 0 0],[-2 -2 -2 -2 -1 0 0]]
Primitive based matrix [[ 0 2 1 1 0 0 -4],[-2 0 0 -2 -1 -2 -2],[-1 0 0 0 0 0 -1],[-1 2 0 0 -1 -1 -4],[ 0 1 0 1 0 0 -2],[ 0 2 0 1 0 0 -3],[ 4 2 1 4 2 3 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,-1,0,0,4,0,2,1,2,2,0,0,0,1,1,1,4,0,2,3]
Phi over symmetry [-4,0,0,1,1,2,1,2,1,4,4,0,0,1,0,0,1,1,0,-1,1]
Phi of -K [-4,0,0,1,1,2,1,2,1,4,4,0,0,1,0,0,1,1,0,-1,1]
Phi of K* [-2,-1,-1,0,0,4,-1,1,0,1,4,0,0,0,1,1,1,4,0,1,2]
Phi of -K* [-4,0,0,1,1,2,2,3,1,4,2,0,0,1,1,0,1,2,0,0,2]
Symmetry type of based matrix c
u-polynomial t^4-t^2-2t
Normalized Jones-Krushkal polynomial 8z^2+27z+23
Enhanced Jones-Krushkal polynomial 8w^3z^2+27w^2z+23w
Inner characteristic polynomial t^6+45t^4+18t^2+1
Outer characteristic polynomial t^7+67t^5+65t^3+6t
Flat arrow polynomial 4*K1**3 + 4*K1**2*K2 - 4*K1*K2 - 2*K1*K3 - K1 - K2 + K3
2-strand cable arrow polynomial 1248*K1**4*K2 - 2464*K1**4 + 672*K1**3*K2*K3 - 448*K1**3*K3 - 128*K1**2*K2**4 + 896*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 5328*K1**2*K2**2 - 928*K1**2*K2*K4 + 5648*K1**2*K2 - 1312*K1**2*K3**2 - 64*K1**2*K3*K5 - 32*K1**2*K4**2 - 2880*K1**2 + 448*K1*K2**3*K3 + 224*K1*K2**2*K3*K4 - 1152*K1*K2**2*K3 - 512*K1*K2**2*K5 - 448*K1*K2*K3*K4 - 96*K1*K2*K3*K6 + 5920*K1*K2*K3 - 32*K1*K2*K4*K5 + 1856*K1*K3*K4 + 376*K1*K4*K5 + 32*K1*K5*K6 - 32*K2**6 - 32*K2**4*K4**2 + 192*K2**4*K4 - 1088*K2**4 + 96*K2**3*K3*K5 + 32*K2**3*K4*K6 - 96*K2**3*K6 - 624*K2**2*K3**2 - 32*K2**2*K3*K7 - 384*K2**2*K4**2 + 1880*K2**2*K4 - 96*K2**2*K5**2 - 8*K2**2*K6**2 - 2814*K2**2 + 864*K2*K3*K5 + 280*K2*K4*K6 + 40*K2*K5*K7 + 8*K3**2*K6 - 1612*K3**2 - 902*K4**2 - 268*K5**2 - 58*K6**2 + 3028
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {2, 5}, {4}, {3}], [{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {1, 5}, {4}, {3}], [{5, 6}, {3, 4}, {1, 2}], [{5, 6}, {4}, {3}, {1, 2}]]
If K is slice False
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