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Flat knot 6.21

Min(phi) over symmetries of the knot is: [-5,-2,1,1,1,4,1,2,3,5,4,1,2,3,3,0,0,1,0,2,3]
Flat knots (up to 7 crossings) with same phi are :['6.21']
Arrow polynomial of the knot is: 4*K1**2*K3 - 4*K1*K2 - 2*K1*K3 + 2*K1 - 2*K2*K3 + K2 + K4 + 1
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.21']
Outer characteristic polynomial of the knot is: t^7+140t^5+85t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.21']
2-strand cable arrow polynomial of the knot is: -864*K1**3*K3 - 384*K1**2*K2**2 + 640*K1**2*K2*K3**2 + 64*K1**2*K2*K6**2 + 2832*K1**2*K2 - 1888*K1**2*K3**2 - 192*K1**2*K3*K5 - 64*K1**2*K4*K6 - 112*K1**2*K6**2 - 3784*K1**2 + 512*K1*K2**3*K3 - 1216*K1*K2**2*K3 + 256*K1*K2*K3**3 - 672*K1*K2*K3*K4 + 32*K1*K2*K3*K6**2 - 160*K1*K2*K3*K6 + 5464*K1*K2*K3 - 64*K1*K2*K5*K6 - 128*K1*K3**2*K5 - 32*K1*K3*K4*K6 + 2048*K1*K3*K4 + 160*K1*K4*K5 + 184*K1*K5*K6 + 64*K1*K6*K7 - 256*K2**4*K3**2 - 32*K2**4*K6**2 - 208*K2**4 + 128*K2**3*K3*K5 + 64*K2**3*K4*K6 - 32*K2**3*K6 + 256*K2**2*K3**2*K4 - 1376*K2**2*K3**2 - 64*K2**2*K3*K7 - 48*K2**2*K4**2 + 32*K2**2*K4*K6**2 + 624*K2**2*K4 - 104*K2**2*K6**2 - 2688*K2**2 - 64*K2*K3**2*K4 - 64*K2*K3*K4*K5 + 1072*K2*K3*K5 - 32*K2*K4**2*K6 + 248*K2*K4*K6 + 32*K2*K6*K8 - 320*K3**4 - 64*K3**2*K4**2 - 32*K3**2*K6**2 + 328*K3**2*K6 - 2088*K3**2 + 40*K3*K4*K7 + 8*K3*K6*K9 - 8*K4**2*K6**2 - 566*K4**2 - 196*K5**2 - 144*K6**2 - 4*K7**2 - 2*K8**2 + 2862
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.21']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.19920', 'vk6.19928', 'vk6.21149', 'vk6.21154', 'vk6.26851', 'vk6.26867', 'vk6.28623', 'vk6.28633', 'vk6.38283', 'vk6.38299', 'vk6.40413', 'vk6.40424', 'vk6.45154', 'vk6.45162', 'vk6.46999', 'vk6.47005', 'vk6.56703', 'vk6.56710', 'vk6.57791', 'vk6.57798', 'vk6.61110', 'vk6.61118', 'vk6.62361', 'vk6.62373', 'vk6.66389', 'vk6.66397', 'vk6.67151', 'vk6.67164', 'vk6.69048', 'vk6.69052', 'vk6.69836', 'vk6.69843']
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 O1O2O3O4O5O6U1U3U5U4U6U2
R3 orbit {'O1O2O3O4O5O6U1U3U5U4U6U2'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5O6U5U1U3U2U4U6
Gauss code of K* O1O2O3O4O5O6U1U6U2U4U3U5
Gauss code of -K* O1O2O3O4O5O6U2U4U3U5U1U6
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 -5 1 -2 1 1 4],[ 5 0 5 1 3 2 4],[-1 -5 0 -3 0 0 3],[ 2 -1 3 0 2 1 3],[-1 -3 0 -2 0 0 2],[-1 -2 0 -1 0 0 1],[-4 -4 -3 -3 -2 -1 0]]
Primitive based matrix [[ 0 4 1 1 1 -2 -5],[-4 0 -1 -2 -3 -3 -4],[-1 1 0 0 0 -1 -2],[-1 2 0 0 0 -2 -3],[-1 3 0 0 0 -3 -5],[ 2 3 1 2 3 0 -1],[ 5 4 2 3 5 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-4,-1,-1,-1,2,5,1,2,3,3,4,0,0,1,2,0,2,3,3,5,1]
Phi over symmetry [-5,-2,1,1,1,4,1,2,3,5,4,1,2,3,3,0,0,1,0,2,3]
Phi of -K [-5,-2,1,1,1,4,2,1,3,4,5,0,1,2,3,0,0,0,0,1,2]
Phi of K* [-4,-1,-1,-1,2,5,0,1,2,3,5,0,0,0,1,0,1,3,2,4,2]
Phi of -K* [-5,-2,1,1,1,4,1,2,3,5,4,1,2,3,3,0,0,1,0,2,3]
Symmetry type of based matrix c
u-polynomial t^5-t^4+t^2-3t
Normalized Jones-Krushkal polynomial 5z^2+22z+25
Enhanced Jones-Krushkal polynomial 5w^3z^2+22w^2z+25w
Inner characteristic polynomial t^6+92t^4+8t^2
Outer characteristic polynomial t^7+140t^5+85t^3+4t
Flat arrow polynomial 4*K1**2*K3 - 4*K1*K2 - 2*K1*K3 + 2*K1 - 2*K2*K3 + K2 + K4 + 1
2-strand cable arrow polynomial -864*K1**3*K3 - 384*K1**2*K2**2 + 640*K1**2*K2*K3**2 + 64*K1**2*K2*K6**2 + 2832*K1**2*K2 - 1888*K1**2*K3**2 - 192*K1**2*K3*K5 - 64*K1**2*K4*K6 - 112*K1**2*K6**2 - 3784*K1**2 + 512*K1*K2**3*K3 - 1216*K1*K2**2*K3 + 256*K1*K2*K3**3 - 672*K1*K2*K3*K4 + 32*K1*K2*K3*K6**2 - 160*K1*K2*K3*K6 + 5464*K1*K2*K3 - 64*K1*K2*K5*K6 - 128*K1*K3**2*K5 - 32*K1*K3*K4*K6 + 2048*K1*K3*K4 + 160*K1*K4*K5 + 184*K1*K5*K6 + 64*K1*K6*K7 - 256*K2**4*K3**2 - 32*K2**4*K6**2 - 208*K2**4 + 128*K2**3*K3*K5 + 64*K2**3*K4*K6 - 32*K2**3*K6 + 256*K2**2*K3**2*K4 - 1376*K2**2*K3**2 - 64*K2**2*K3*K7 - 48*K2**2*K4**2 + 32*K2**2*K4*K6**2 + 624*K2**2*K4 - 104*K2**2*K6**2 - 2688*K2**2 - 64*K2*K3**2*K4 - 64*K2*K3*K4*K5 + 1072*K2*K3*K5 - 32*K2*K4**2*K6 + 248*K2*K4*K6 + 32*K2*K6*K8 - 320*K3**4 - 64*K3**2*K4**2 - 32*K3**2*K6**2 + 328*K3**2*K6 - 2088*K3**2 + 40*K3*K4*K7 + 8*K3*K6*K9 - 8*K4**2*K6**2 - 566*K4**2 - 196*K5**2 - 144*K6**2 - 4*K7**2 - 2*K8**2 + 2862
Genus of based matrix 1
Fillings of based matrix [[{3, 6}, {4, 5}, {1, 2}], [{5, 6}, {1, 4}, {2, 3}]]
If K is slice False
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