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Flat knot 6.451

Min(phi) over symmetries of the knot is: [-3,-2,0,1,1,3,0,2,1,3,4,1,0,2,2,0,1,2,-1,1,2]
Flat knots (up to 7 crossings) with same phi are :['6.451']
Arrow polynomial of the knot is: 4*K1**3 + 4*K1**2*K2 - 8*K1**2 - 4*K1*K2 - 2*K1*K3 - K1 - 2*K2**2 + 3*K2 + K3 + K4 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.365', '6.451']
Outer characteristic polynomial of the knot is: t^7+74t^5+140t^3+11t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.451']
2-strand cable arrow polynomial of the knot is: -64*K1**6 + 448*K1**4*K2 - 2272*K1**4 + 832*K1**3*K2*K3 - 704*K1**3*K3 - 128*K1**2*K2**4 + 192*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 4656*K1**2*K2**2 + 128*K1**2*K2*K3**2 + 64*K1**2*K2*K4**2 - 608*K1**2*K2*K4 + 6560*K1**2*K2 - 1696*K1**2*K3**2 - 160*K1**2*K4**2 - 4884*K1**2 + 736*K1*K2**3*K3 - 544*K1*K2**2*K3 - 384*K1*K2**2*K5 + 192*K1*K2*K3**3 + 64*K1*K2*K3*K4**2 - 384*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 8136*K1*K2*K3 - 96*K1*K2*K4*K5 - 32*K1*K2*K4*K7 + 32*K1*K3**3*K4 + 2384*K1*K3*K4 + 344*K1*K4*K5 + 32*K1*K5*K6 - 32*K2**6 - 64*K2**4*K3**2 - 32*K2**4*K4**2 + 96*K2**4*K4 - 608*K2**4 + 32*K2**3*K3*K5 + 32*K2**3*K4*K6 + 64*K2**2*K3**2*K4 - 1072*K2**2*K3**2 - 32*K2**2*K3*K7 + 32*K2**2*K4**3 - 240*K2**2*K4**2 - 32*K2**2*K4*K8 + 1032*K2**2*K4 - 8*K2**2*K6**2 - 4074*K2**2 - 224*K2*K3**2*K4 + 936*K2*K3*K5 + 272*K2*K4*K6 + 8*K2*K6*K8 - 192*K3**4 - 112*K3**2*K4**2 + 192*K3**2*K6 - 2904*K3**2 + 48*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 998*K4**2 - 268*K5**2 - 86*K6**2 - 2*K8**2 + 4638
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.451']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11530', 'vk6.11863', 'vk6.12880', 'vk6.13189', 'vk6.20340', 'vk6.21682', 'vk6.27643', 'vk6.29188', 'vk6.31309', 'vk6.31706', 'vk6.32467', 'vk6.32884', 'vk6.39076', 'vk6.41333', 'vk6.45832', 'vk6.47501', 'vk6.52317', 'vk6.52579', 'vk6.53161', 'vk6.53463', 'vk6.57211', 'vk6.58432', 'vk6.61824', 'vk6.62956', 'vk6.63822', 'vk6.63956', 'vk6.64268', 'vk6.64466', 'vk6.66826', 'vk6.67695', 'vk6.69465', 'vk6.70188']
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 O1O2O3O4O5U6U3O6U1U5U2U4
R3 orbit {'O1O2O3O4O5U6U3O6U1U5U2U4'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U2U4U1U5O6U3U6
Gauss code of K* O1O2O3O4U1U3U5U4U2O6O5U6
Gauss code of -K* O1O2O3O4U5O6O5U3U1U6U2U4
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 0 -1 3 2 -1],[ 3 0 2 1 4 2 2],[ 0 -2 0 0 2 1 -1],[ 1 -1 0 0 1 0 1],[-3 -4 -2 -1 0 0 -3],[-2 -2 -1 0 0 0 -2],[ 1 -2 1 -1 3 2 0]]
Primitive based matrix [[ 0 3 2 0 -1 -1 -3],[-3 0 0 -2 -1 -3 -4],[-2 0 0 -1 0 -2 -2],[ 0 2 1 0 0 -1 -2],[ 1 1 0 0 0 1 -1],[ 1 3 2 1 -1 0 -2],[ 3 4 2 2 1 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-2,0,1,1,3,0,2,1,3,4,1,0,2,2,0,1,2,-1,1,2]
Phi over symmetry [-3,-2,0,1,1,3,0,2,1,3,4,1,0,2,2,0,1,2,-1,1,2]
Phi of -K [-3,-1,-1,0,2,3,0,1,1,3,2,1,0,1,1,1,3,3,1,1,1]
Phi of K* [-3,-2,0,1,1,3,1,1,1,3,2,1,1,3,3,0,1,1,-1,0,1]
Phi of -K* [-3,-1,-1,0,2,3,1,2,2,2,4,1,0,0,1,1,2,3,1,2,0]
Symmetry type of based matrix c
u-polynomial -t^2+2t
Normalized Jones-Krushkal polynomial 4z^2+25z+35
Enhanced Jones-Krushkal polynomial 4w^3z^2+25w^2z+35w
Inner characteristic polynomial t^6+50t^4+79t^2+1
Outer characteristic polynomial t^7+74t^5+140t^3+11t
Flat arrow polynomial 4*K1**3 + 4*K1**2*K2 - 8*K1**2 - 4*K1*K2 - 2*K1*K3 - K1 - 2*K2**2 + 3*K2 + K3 + K4 + 5
2-strand cable arrow polynomial -64*K1**6 + 448*K1**4*K2 - 2272*K1**4 + 832*K1**3*K2*K3 - 704*K1**3*K3 - 128*K1**2*K2**4 + 192*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 4656*K1**2*K2**2 + 128*K1**2*K2*K3**2 + 64*K1**2*K2*K4**2 - 608*K1**2*K2*K4 + 6560*K1**2*K2 - 1696*K1**2*K3**2 - 160*K1**2*K4**2 - 4884*K1**2 + 736*K1*K2**3*K3 - 544*K1*K2**2*K3 - 384*K1*K2**2*K5 + 192*K1*K2*K3**3 + 64*K1*K2*K3*K4**2 - 384*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 8136*K1*K2*K3 - 96*K1*K2*K4*K5 - 32*K1*K2*K4*K7 + 32*K1*K3**3*K4 + 2384*K1*K3*K4 + 344*K1*K4*K5 + 32*K1*K5*K6 - 32*K2**6 - 64*K2**4*K3**2 - 32*K2**4*K4**2 + 96*K2**4*K4 - 608*K2**4 + 32*K2**3*K3*K5 + 32*K2**3*K4*K6 + 64*K2**2*K3**2*K4 - 1072*K2**2*K3**2 - 32*K2**2*K3*K7 + 32*K2**2*K4**3 - 240*K2**2*K4**2 - 32*K2**2*K4*K8 + 1032*K2**2*K4 - 8*K2**2*K6**2 - 4074*K2**2 - 224*K2*K3**2*K4 + 936*K2*K3*K5 + 272*K2*K4*K6 + 8*K2*K6*K8 - 192*K3**4 - 112*K3**2*K4**2 + 192*K3**2*K6 - 2904*K3**2 + 48*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 998*K4**2 - 268*K5**2 - 86*K6**2 - 2*K8**2 + 4638
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {4, 5}, {1, 3}]]
If K is slice False
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