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Flat knot 6.336

Min(phi) over symmetries of the knot is: [-4,-2,-1,2,2,3,0,2,2,4,4,1,1,2,2,1,2,3,0,0,1]
Flat knots (up to 7 crossings) with same phi are :['6.336']
Arrow polynomial of the knot is: 4*K1**3 - 10*K1**2 - 10*K1*K2 + 2*K1 - 2*K2**2 + 5*K2 + 4*K3 + K4 + 7
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.336']
Outer characteristic polynomial of the knot is: t^7+103t^5+69t^3+5t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.336']
2-strand cable arrow polynomial of the knot is: 608*K1**4*K2 - 3504*K1**4 + 128*K1**3*K2*K3 + 32*K1**3*K3*K4 - 1088*K1**3*K3 + 224*K1**2*K2**3 - 4128*K1**2*K2**2 + 96*K1**2*K2*K3**2 - 672*K1**2*K2*K4 + 10912*K1**2*K2 - 1888*K1**2*K3**2 - 128*K1**2*K3*K5 - 224*K1**2*K4**2 - 8800*K1**2 + 192*K1*K2**3*K3 - 672*K1*K2**2*K3 - 128*K1*K2**2*K5 + 32*K1*K2*K3**3 - 224*K1*K2*K3*K4 + 9576*K1*K2*K3 + 3320*K1*K3*K4 + 408*K1*K4*K5 + 16*K1*K5*K6 - 32*K2**6 + 96*K2**4*K4 - 648*K2**4 - 32*K2**3*K6 - 704*K2**2*K3**2 - 136*K2**2*K4**2 + 1264*K2**2*K4 - 6420*K2**2 - 32*K2*K3**2*K4 + 800*K2*K3*K5 + 112*K2*K4*K6 - 240*K3**4 - 96*K3**2*K4**2 + 184*K3**2*K6 - 3760*K3**2 + 80*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 1354*K4**2 - 304*K5**2 - 60*K6**2 - 16*K7**2 - 2*K8**2 + 7210
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.336']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.73311', 'vk6.73313', 'vk6.73454', 'vk6.73456', 'vk6.74075', 'vk6.74079', 'vk6.74644', 'vk6.74648', 'vk6.75456', 'vk6.75458', 'vk6.76113', 'vk6.76117', 'vk6.78184', 'vk6.78186', 'vk6.78416', 'vk6.78418', 'vk6.79081', 'vk6.79085', 'vk6.80007', 'vk6.80009', 'vk6.80160', 'vk6.80162', 'vk6.80589', 'vk6.80593', 'vk6.83804', 'vk6.83820', 'vk6.85116', 'vk6.85127', 'vk6.86596', 'vk6.86619', 'vk6.87393', 'vk6.87395']
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 O1O2O3O4O5U3U1O6U2U4U6U5
R3 orbit {'O1O2O3O4O5U3U1O6U2U4U6U5'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U1U6U2U4O6U5U3
Gauss code of K* O1O2O3O4U5U1U6U2U4O6O5U3
Gauss code of -K* O1O2O3O4U2O5O6U1U3U6U4U5
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 -2 1 4 2],[ 3 0 1 0 3 4 2],[ 2 -1 0 0 2 4 2],[ 2 0 0 0 1 2 1],[-1 -3 -2 -1 0 2 1],[-4 -4 -4 -2 -2 0 0],[-2 -2 -2 -1 -1 0 0]]
Primitive based matrix [[ 0 4 2 1 -2 -2 -3],[-4 0 0 -2 -2 -4 -4],[-2 0 0 -1 -1 -2 -2],[-1 2 1 0 -1 -2 -3],[ 2 2 1 1 0 0 0],[ 2 4 2 2 0 0 -1],[ 3 4 2 3 0 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-4,-2,-1,2,2,3,0,2,2,4,4,1,1,2,2,1,2,3,0,0,1]
Phi over symmetry [-4,-2,-1,2,2,3,0,2,2,4,4,1,1,2,2,1,2,3,0,0,1]
Phi of -K [-3,-2,-2,1,2,4,0,1,1,3,3,0,1,2,2,2,3,4,0,1,2]
Phi of K* [-4,-2,-1,2,2,3,2,1,2,4,3,0,2,3,3,1,2,1,0,0,1]
Phi of -K* [-3,-2,-2,1,2,4,0,1,3,2,4,0,1,1,2,2,2,4,1,2,0]
Symmetry type of based matrix c
u-polynomial -t^4+t^3+t^2-t
Normalized Jones-Krushkal polynomial z^2+22z+41
Enhanced Jones-Krushkal polynomial w^3z^2+22w^2z+41w
Inner characteristic polynomial t^6+65t^4+16t^2
Outer characteristic polynomial t^7+103t^5+69t^3+5t
Flat arrow polynomial 4*K1**3 - 10*K1**2 - 10*K1*K2 + 2*K1 - 2*K2**2 + 5*K2 + 4*K3 + K4 + 7
2-strand cable arrow polynomial 608*K1**4*K2 - 3504*K1**4 + 128*K1**3*K2*K3 + 32*K1**3*K3*K4 - 1088*K1**3*K3 + 224*K1**2*K2**3 - 4128*K1**2*K2**2 + 96*K1**2*K2*K3**2 - 672*K1**2*K2*K4 + 10912*K1**2*K2 - 1888*K1**2*K3**2 - 128*K1**2*K3*K5 - 224*K1**2*K4**2 - 8800*K1**2 + 192*K1*K2**3*K3 - 672*K1*K2**2*K3 - 128*K1*K2**2*K5 + 32*K1*K2*K3**3 - 224*K1*K2*K3*K4 + 9576*K1*K2*K3 + 3320*K1*K3*K4 + 408*K1*K4*K5 + 16*K1*K5*K6 - 32*K2**6 + 96*K2**4*K4 - 648*K2**4 - 32*K2**3*K6 - 704*K2**2*K3**2 - 136*K2**2*K4**2 + 1264*K2**2*K4 - 6420*K2**2 - 32*K2*K3**2*K4 + 800*K2*K3*K5 + 112*K2*K4*K6 - 240*K3**4 - 96*K3**2*K4**2 + 184*K3**2*K6 - 3760*K3**2 + 80*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 1354*K4**2 - 304*K5**2 - 60*K6**2 - 16*K7**2 - 2*K8**2 + 7210
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {3, 5}, {2, 4}]]
If K is slice False
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