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Flat knot 6.901

Min(phi) over symmetries of the knot is: [-3,-2,0,1,1,3,0,2,1,3,4,1,1,2,2,0,0,2,-1,1,2]
Flat knots (up to 7 crossings) with same phi are :['6.901']
Arrow polynomial of the knot is: -4*K1**2 - 4*K1*K2 - 2*K1*K3 + 2*K1 - 2*K2**2 + 3*K2 + 2*K3 + 2*K4 + 4
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.901']
Outer characteristic polynomial of the knot is: t^7+74t^5+107t^3+8t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.901']
2-strand cable arrow polynomial of the knot is: 576*K1**4*K2 - 1600*K1**4 + 704*K1**3*K2*K3 + 64*K1**3*K3*K4 - 1088*K1**3*K3 + 32*K1**3*K4*K5 + 128*K1**2*K2**2*K4 - 2752*K1**2*K2**2 + 96*K1**2*K2*K3**2 - 448*K1**2*K2*K4 + 6904*K1**2*K2 - 1888*K1**2*K3**2 - 64*K1**2*K3*K5 - 272*K1**2*K4**2 - 96*K1**2*K5**2 - 6976*K1**2 + 96*K1*K2**3*K3 - 1184*K1*K2**2*K3 - 256*K1*K2**2*K5 + 256*K1*K2*K3**3 + 96*K1*K2*K3*K4**2 - 480*K1*K2*K3*K4 - 128*K1*K2*K3*K6 + 8232*K1*K2*K3 + 3144*K1*K3*K4 + 792*K1*K4*K5 + 160*K1*K5*K6 + 8*K1*K6*K7 - 208*K2**4 - 32*K2**3*K6 - 784*K2**2*K3**2 - 80*K2**2*K4**2 + 1408*K2**2*K4 - 8*K2**2*K6**2 - 5640*K2**2 - 128*K2*K3**2*K4 - 64*K2*K3*K4*K5 + 1320*K2*K3*K5 + 168*K2*K4*K6 + 16*K2*K5*K7 + 16*K2*K6*K8 - 224*K3**4 - 112*K3**2*K4**2 + 224*K3**2*K6 - 3684*K3**2 + 96*K3*K4*K7 - 8*K4**4 + 16*K4**2*K8 - 1550*K4**2 - 688*K5**2 - 160*K6**2 - 28*K7**2 - 12*K8**2 + 6088
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.901']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11015', 'vk6.11094', 'vk6.12185', 'vk6.12292', 'vk6.18193', 'vk6.18530', 'vk6.24651', 'vk6.25079', 'vk6.30588', 'vk6.30683', 'vk6.31858', 'vk6.31904', 'vk6.36787', 'vk6.37239', 'vk6.44030', 'vk6.44372', 'vk6.51826', 'vk6.51893', 'vk6.52698', 'vk6.52792', 'vk6.55988', 'vk6.56262', 'vk6.60522', 'vk6.60866', 'vk6.63509', 'vk6.63553', 'vk6.63991', 'vk6.64035', 'vk6.65653', 'vk6.65934', 'vk6.68700', 'vk6.68911']
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 O1O2O3O4U2U5O6O5U1U3U6U4
R3 orbit {'O1O2O3O4U2U5O6O5U1U3U6U4'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U1U5U2U4O6O5U6U3
Gauss code of K* O1O2O3O4U1U5U2U4O5O6U3U6
Gauss code of -K* O1O2O3O4U5U2O5O6U1U3U6U4
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 -2 0 3 1 1],[ 3 0 0 2 4 3 1],[ 2 0 0 1 2 2 1],[ 0 -2 -1 0 2 0 0],[-3 -4 -2 -2 0 -2 -1],[-1 -3 -2 0 2 0 1],[-1 -1 -1 0 1 -1 0]]
Primitive based matrix [[ 0 3 1 1 0 -2 -3],[-3 0 -1 -2 -2 -2 -4],[-1 1 0 -1 0 -1 -1],[-1 2 1 0 0 -2 -3],[ 0 2 0 0 0 -1 -2],[ 2 2 1 2 1 0 0],[ 3 4 1 3 2 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-1,-1,0,2,3,1,2,2,2,4,1,0,1,1,0,2,3,1,2,0]
Phi over symmetry [-3,-2,0,1,1,3,0,2,1,3,4,1,1,2,2,0,0,2,-1,1,2]
Phi of -K [-3,-2,0,1,1,3,1,1,1,3,2,1,1,2,3,1,1,1,-1,0,1]
Phi of K* [-3,-1,-1,0,2,3,0,1,1,3,2,1,1,1,1,1,2,3,1,1,1]
Phi of -K* [-3,-2,0,1,1,3,0,2,1,3,4,1,1,2,2,0,0,2,-1,1,2]
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+50t^4+58t^2+1
Outer characteristic polynomial t^7+74t^5+107t^3+8t
Flat arrow polynomial -4*K1**2 - 4*K1*K2 - 2*K1*K3 + 2*K1 - 2*K2**2 + 3*K2 + 2*K3 + 2*K4 + 4
2-strand cable arrow polynomial 576*K1**4*K2 - 1600*K1**4 + 704*K1**3*K2*K3 + 64*K1**3*K3*K4 - 1088*K1**3*K3 + 32*K1**3*K4*K5 + 128*K1**2*K2**2*K4 - 2752*K1**2*K2**2 + 96*K1**2*K2*K3**2 - 448*K1**2*K2*K4 + 6904*K1**2*K2 - 1888*K1**2*K3**2 - 64*K1**2*K3*K5 - 272*K1**2*K4**2 - 96*K1**2*K5**2 - 6976*K1**2 + 96*K1*K2**3*K3 - 1184*K1*K2**2*K3 - 256*K1*K2**2*K5 + 256*K1*K2*K3**3 + 96*K1*K2*K3*K4**2 - 480*K1*K2*K3*K4 - 128*K1*K2*K3*K6 + 8232*K1*K2*K3 + 3144*K1*K3*K4 + 792*K1*K4*K5 + 160*K1*K5*K6 + 8*K1*K6*K7 - 208*K2**4 - 32*K2**3*K6 - 784*K2**2*K3**2 - 80*K2**2*K4**2 + 1408*K2**2*K4 - 8*K2**2*K6**2 - 5640*K2**2 - 128*K2*K3**2*K4 - 64*K2*K3*K4*K5 + 1320*K2*K3*K5 + 168*K2*K4*K6 + 16*K2*K5*K7 + 16*K2*K6*K8 - 224*K3**4 - 112*K3**2*K4**2 + 224*K3**2*K6 - 3684*K3**2 + 96*K3*K4*K7 - 8*K4**4 + 16*K4**2*K8 - 1550*K4**2 - 688*K5**2 - 160*K6**2 - 28*K7**2 - 12*K8**2 + 6088
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{2, 6}, {1, 5}, {3, 4}]]
If K is slice False
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