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Flat knot 6.1733

Min(phi) over symmetries of the knot is: [-1,-1,-1,1,1,1,-1,-1,0,1,1,-1,1,1,1,1,0,1,0,-1,0]
Flat knots (up to 7 crossings) with same phi are :['6.1733']
Arrow polynomial of the knot is: 4*K1**3 - 4*K1**2 - 8*K1*K2 + K1 + 2*K2 + 3*K3 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.697', '6.1075', '6.1524', '6.1733']
Outer characteristic polynomial of the knot is: t^7+25t^5+49t^3+5t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1733']
2-strand cable arrow polynomial of the knot is: 1728*K1**4*K2 - 4704*K1**4 + 832*K1**3*K2*K3 - 1024*K1**3*K3 - 128*K1**2*K2**4 + 448*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 4992*K1**2*K2**2 - 896*K1**2*K2*K4 + 6776*K1**2*K2 - 736*K1**2*K3**2 - 32*K1**2*K4**2 - 1656*K1**2 + 384*K1*K2**3*K3 - 992*K1*K2**2*K3 - 256*K1*K2**2*K5 - 160*K1*K2*K3*K4 + 5280*K1*K2*K3 + 1320*K1*K3*K4 + 136*K1*K4*K5 - 32*K2**6 + 96*K2**4*K4 - 576*K2**4 - 32*K2**3*K6 - 384*K2**2*K3**2 - 128*K2**2*K4**2 + 1144*K2**2*K4 - 2530*K2**2 + 440*K2*K3*K5 + 104*K2*K4*K6 - 1376*K3**2 - 604*K4**2 - 136*K5**2 - 22*K6**2 + 2578
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1733']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.13813', 'vk6.13821', 'vk6.13822', 'vk6.13825', 'vk6.13834', 'vk6.13840', 'vk6.13841', 'vk6.13844', 'vk6.13847', 'vk6.13850', 'vk6.13851', 'vk6.13857', 'vk6.14884', 'vk6.14893', 'vk6.14894', 'vk6.14898', 'vk6.14900', 'vk6.14903', 'vk6.14907', 'vk6.14909', 'vk6.14917', 'vk6.14926', 'vk6.14929', 'vk6.14931', 'vk6.34239', 'vk6.34244', 'vk6.53830', 'vk6.53833', 'vk6.53839', 'vk6.53846', 'vk6.54375', 'vk6.54379']
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 O1O2O3U2U4O5O4U6U3O6U1U5
R3 orbit {'O1O2O3U2U4O5O4U6U3O6U1U5'}
R3 orbit length 1
Gauss code of -K O1O2O3U4U3O5U1U5O6O4U6U2
Gauss code of K* O1O2U1O3O4U3U5U2O5O6U4U6
Gauss code of -K* O1O2U3O4O3U5U1O5O6U4U6U2
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 -1 -1 1 1 1 -1],[ 1 0 -1 2 1 1 0],[ 1 1 0 1 1 1 0],[-1 -2 -1 0 -1 -1 -1],[-1 -1 -1 1 0 1 -2],[-1 -1 -1 1 -1 0 -1],[ 1 0 0 1 2 1 0]]
Primitive based matrix [[ 0 1 1 1 -1 -1 -1],[-1 0 1 1 -1 -1 -2],[-1 -1 0 1 -1 -1 -1],[-1 -1 -1 0 -1 -2 -1],[ 1 1 1 1 0 1 0],[ 1 1 1 2 -1 0 0],[ 1 2 1 1 0 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-1,-1,-1,1,1,1,-1,-1,1,1,2,-1,1,1,1,1,2,1,-1,0,0]
Phi over symmetry [-1,-1,-1,1,1,1,-1,-1,0,1,1,-1,1,1,1,1,0,1,0,-1,0]
Phi of -K [-1,-1,-1,1,1,1,-1,0,1,1,1,0,0,1,1,1,0,1,1,1,-1]
Phi of K* [-1,-1,-1,1,1,1,-1,-1,0,1,1,-1,1,1,1,1,0,1,0,-1,0]
Phi of -K* [-1,-1,-1,1,1,1,-1,0,1,1,2,0,1,1,1,1,2,1,-1,1,1]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 6z^2+23z+23
Enhanced Jones-Krushkal polynomial 6w^3z^2+23w^2z+23w
Inner characteristic polynomial t^6+19t^4+29t^2+1
Outer characteristic polynomial t^7+25t^5+49t^3+5t
Flat arrow polynomial 4*K1**3 - 4*K1**2 - 8*K1*K2 + K1 + 2*K2 + 3*K3 + 3
2-strand cable arrow polynomial 1728*K1**4*K2 - 4704*K1**4 + 832*K1**3*K2*K3 - 1024*K1**3*K3 - 128*K1**2*K2**4 + 448*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 4992*K1**2*K2**2 - 896*K1**2*K2*K4 + 6776*K1**2*K2 - 736*K1**2*K3**2 - 32*K1**2*K4**2 - 1656*K1**2 + 384*K1*K2**3*K3 - 992*K1*K2**2*K3 - 256*K1*K2**2*K5 - 160*K1*K2*K3*K4 + 5280*K1*K2*K3 + 1320*K1*K3*K4 + 136*K1*K4*K5 - 32*K2**6 + 96*K2**4*K4 - 576*K2**4 - 32*K2**3*K6 - 384*K2**2*K3**2 - 128*K2**2*K4**2 + 1144*K2**2*K4 - 2530*K2**2 + 440*K2*K3*K5 + 104*K2*K4*K6 - 1376*K3**2 - 604*K4**2 - 136*K5**2 - 22*K6**2 + 2578
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {1, 5}, {3, 4}], [{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {2, 5}, {1, 4}], [{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {2, 5}, {1, 3}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {2, 4}, {1, 3}]]
If K is slice False
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