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Flat knot 6.93

Min(phi) over symmetries of the knot is: [-4,-1,0,0,2,3,0,1,3,4,3,0,1,1,1,1,1,2,1,2,2]
Flat knots (up to 7 crossings) with same phi are :['6.93']
Arrow polynomial of the knot is: 16*K1**3 + 4*K1**2*K2 - 8*K1**2 - 10*K1*K2 - 2*K1*K3 - 7*K1 + 3*K2 + K3 + 4
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.93']
Outer characteristic polynomial of the knot is: t^7+89t^5+105t^3
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.93']
2-strand cable arrow polynomial of the knot is: 256*K1**4*K2**3 - 640*K1**4*K2**2 + 704*K1**4*K2 - 832*K1**4 + 96*K1**3*K2*K3 + 32*K1**3*K3*K4 - 768*K1**2*K2**4 + 1280*K1**2*K2**3 - 3360*K1**2*K2**2 + 2816*K1**2*K2 - 96*K1**2*K3**2 - 96*K1**2*K4**2 - 1928*K1**2 + 896*K1*K2**3*K3 + 2528*K1*K2*K3 + 480*K1*K3*K4 + 104*K1*K4*K5 + 16*K1*K5*K6 - 512*K2**6 - 320*K2**4*K3**2 - 32*K2**4*K4**2 + 768*K2**4*K4 - 2256*K2**4 + 384*K2**3*K3*K5 + 32*K2**3*K4*K6 - 1040*K2**2*K3**2 - 472*K2**2*K4**2 + 1360*K2**2*K4 - 144*K2**2*K5**2 - 8*K2**2*K6**2 - 446*K2**2 + 592*K2*K3*K5 + 176*K2*K4*K6 + 16*K2*K5*K7 - 32*K3**2*K4**2 - 820*K3**2 + 32*K3*K4*K7 - 578*K4**2 - 148*K5**2 - 42*K6**2 - 8*K7**2 + 2008
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.93']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11455', 'vk6.11755', 'vk6.12770', 'vk6.13111', 'vk6.20677', 'vk6.22115', 'vk6.28182', 'vk6.29605', 'vk6.31210', 'vk6.31553', 'vk6.32380', 'vk6.32789', 'vk6.39630', 'vk6.41869', 'vk6.46234', 'vk6.47839', 'vk6.52211', 'vk6.52479', 'vk6.53042', 'vk6.53363', 'vk6.57608', 'vk6.58767', 'vk6.62268', 'vk6.63209', 'vk6.63777', 'vk6.63889', 'vk6.64203', 'vk6.64390', 'vk6.67066', 'vk6.67931', 'vk6.69682', 'vk6.70363']
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 O1O2O3O4O5O6U2U5U6U3U1U4
R3 orbit {'O1O2O3O4O5O6U2U5U6U3U1U4'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5O6U3U6U4U1U2U5
Gauss code of K* O1O2O3O4O5O6U5U1U4U6U2U3
Gauss code of -K* O1O2O3O4O5O6U4U5U1U3U6U2
Diagrammatic symmetry type c
Flat genus of the diagram 2
If K is checkerboard colorable False
If K is almost classical False
Based matrix from Gauss code [[ 0 -1 -4 0 3 0 2],[ 1 0 -3 1 3 0 2],[ 4 3 0 3 4 1 2],[ 0 -1 -3 0 1 -1 1],[-3 -3 -4 -1 0 -1 1],[ 0 0 -1 1 1 0 1],[-2 -2 -2 -1 -1 -1 0]]
Primitive based matrix [[ 0 3 2 0 0 -1 -4],[-3 0 1 -1 -1 -3 -4],[-2 -1 0 -1 -1 -2 -2],[ 0 1 1 0 1 0 -1],[ 0 1 1 -1 0 -1 -3],[ 1 3 2 0 1 0 -3],[ 4 4 2 1 3 3 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-2,0,0,1,4,-1,1,1,3,4,1,1,2,2,-1,0,1,1,3,3]
Phi over symmetry [-4,-1,0,0,2,3,0,1,3,4,3,0,1,1,1,1,1,2,1,2,2]
Phi of -K [-4,-1,0,0,2,3,0,1,3,4,3,0,1,1,1,1,1,2,1,2,2]
Phi of K* [-3,-2,0,0,1,4,2,2,2,1,3,1,1,1,4,-1,0,1,1,3,0]
Phi of -K* [-4,-1,0,0,2,3,3,1,3,2,4,0,1,2,3,1,1,1,1,1,-1]
Symmetry type of based matrix c
u-polynomial t^4-t^3-t^2+t
Normalized Jones-Krushkal polynomial 7z+15
Enhanced Jones-Krushkal polynomial -10w^3z+17w^2z+15w
Inner characteristic polynomial t^6+59t^4+24t^2
Outer characteristic polynomial t^7+89t^5+105t^3
Flat arrow polynomial 16*K1**3 + 4*K1**2*K2 - 8*K1**2 - 10*K1*K2 - 2*K1*K3 - 7*K1 + 3*K2 + K3 + 4
2-strand cable arrow polynomial 256*K1**4*K2**3 - 640*K1**4*K2**2 + 704*K1**4*K2 - 832*K1**4 + 96*K1**3*K2*K3 + 32*K1**3*K3*K4 - 768*K1**2*K2**4 + 1280*K1**2*K2**3 - 3360*K1**2*K2**2 + 2816*K1**2*K2 - 96*K1**2*K3**2 - 96*K1**2*K4**2 - 1928*K1**2 + 896*K1*K2**3*K3 + 2528*K1*K2*K3 + 480*K1*K3*K4 + 104*K1*K4*K5 + 16*K1*K5*K6 - 512*K2**6 - 320*K2**4*K3**2 - 32*K2**4*K4**2 + 768*K2**4*K4 - 2256*K2**4 + 384*K2**3*K3*K5 + 32*K2**3*K4*K6 - 1040*K2**2*K3**2 - 472*K2**2*K4**2 + 1360*K2**2*K4 - 144*K2**2*K5**2 - 8*K2**2*K6**2 - 446*K2**2 + 592*K2*K3*K5 + 176*K2*K4*K6 + 16*K2*K5*K7 - 32*K3**2*K4**2 - 820*K3**2 + 32*K3*K4*K7 - 578*K4**2 - 148*K5**2 - 42*K6**2 - 8*K7**2 + 2008
Genus of based matrix 1
Fillings of based matrix [[{3, 6}, {1, 5}, {2, 4}], [{5, 6}, {1, 4}, {2, 3}]]
If K is slice False
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