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Flat knot 6.1153

Min(phi) over symmetries of the knot is: [-3,0,0,1,1,1,0,2,1,2,2,1,1,0,0,1,0,0,-1,-1,0]
Flat knots (up to 7 crossings) with same phi are :['6.1153']
Arrow polynomial of the knot is: 4*K1**3 + 4*K1**2*K2 - 8*K1**2 - 2*K1*K2 - 4*K1*K3 - 2*K1 + 4*K2 + K4 + 4
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.1150', '6.1153']
Outer characteristic polynomial of the knot is: t^7+46t^5+61t^3
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1153']
2-strand cable arrow polynomial of the knot is: -192*K1**4*K2**2 + 224*K1**4*K2 - 1152*K1**4 + 128*K1**3*K2**3*K3 + 416*K1**3*K2*K3 - 384*K1**2*K2**4 + 288*K1**2*K2**3 - 256*K1**2*K2**2*K3**2 - 2384*K1**2*K2**2 + 2496*K1**2*K2 - 608*K1**2*K3**2 - 32*K1**2*K4**2 - 1856*K1**2 + 1024*K1*K2**3*K3 + 192*K1*K2*K3**3 + 3496*K1*K2*K3 + 616*K1*K3*K4 + 64*K1*K4*K5 + 24*K1*K5*K6 - 32*K2**6 - 64*K2**4*K3**2 - 32*K2**4*K4**2 + 96*K2**4*K4 - 832*K2**4 + 96*K2**3*K3*K5 + 64*K2**3*K4*K6 - 1152*K2**2*K3**2 - 136*K2**2*K4**2 + 424*K2**2*K4 - 48*K2**2*K5**2 - 48*K2**2*K6**2 - 1344*K2**2 + 496*K2*K3*K5 + 120*K2*K4*K6 + 8*K2*K5*K7 + 16*K2*K6*K8 - 96*K3**4 + 72*K3**2*K6 - 1312*K3**2 - 368*K4**2 - 104*K5**2 - 72*K6**2 - 2*K8**2 + 2144
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1153']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11234', 'vk6.11313', 'vk6.12495', 'vk6.12606', 'vk6.18220', 'vk6.18557', 'vk6.24687', 'vk6.25106', 'vk6.30912', 'vk6.31035', 'vk6.32096', 'vk6.32215', 'vk6.36808', 'vk6.37267', 'vk6.44051', 'vk6.44393', 'vk6.51996', 'vk6.52091', 'vk6.52873', 'vk6.52920', 'vk6.56013', 'vk6.56289', 'vk6.60557', 'vk6.60898', 'vk6.63651', 'vk6.63696', 'vk6.64079', 'vk6.64124', 'vk6.65673', 'vk6.65960', 'vk6.68721', 'vk6.68931']
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 O1O2O3O4U1U3U5O6O5U4U2U6
R3 orbit {'O1O2O3O4U1U3U5O6O5U4U2U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U5U3U1O6O5U6U2U4
Gauss code of K* O1O2O3U4U3O5O6O4U1U6U2U5
Gauss code of -K* O1O2O3U4U1O4O5O6U3U5U2U6
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 -3 0 0 1 1 1],[ 3 0 3 1 2 3 2],[ 0 -3 0 -1 1 0 1],[ 0 -1 1 0 1 0 1],[-1 -2 -1 -1 0 -1 0],[-1 -3 0 0 1 0 1],[-1 -2 -1 -1 0 -1 0]]
Primitive based matrix [[ 0 1 1 1 0 0 -3],[-1 0 1 1 0 0 -3],[-1 -1 0 0 -1 -1 -2],[-1 -1 0 0 -1 -1 -2],[ 0 0 1 1 0 1 -1],[ 0 0 1 1 -1 0 -3],[ 3 3 2 2 1 3 0]]
If based matrix primitive True
Phi of primitive based matrix [-1,-1,-1,0,0,3,-1,-1,0,0,3,0,1,1,2,1,1,2,-1,1,3]
Phi over symmetry [-3,0,0,1,1,1,0,2,1,2,2,1,1,0,0,1,0,0,-1,-1,0]
Phi of -K [-3,0,0,1,1,1,0,2,1,2,2,1,1,0,0,1,0,0,-1,-1,0]
Phi of K* [-1,-1,-1,0,0,3,-1,0,0,0,2,1,1,1,1,0,0,2,-1,0,2]
Phi of -K* [-3,0,0,1,1,1,1,3,2,2,3,1,1,1,0,1,1,0,0,-1,-1]
Symmetry type of based matrix c
u-polynomial t^3-3t
Normalized Jones-Krushkal polynomial 12z+25
Enhanced Jones-Krushkal polynomial -4w^3z+16w^2z+25w
Inner characteristic polynomial t^6+34t^4+19t^2
Outer characteristic polynomial t^7+46t^5+61t^3
Flat arrow polynomial 4*K1**3 + 4*K1**2*K2 - 8*K1**2 - 2*K1*K2 - 4*K1*K3 - 2*K1 + 4*K2 + K4 + 4
2-strand cable arrow polynomial -192*K1**4*K2**2 + 224*K1**4*K2 - 1152*K1**4 + 128*K1**3*K2**3*K3 + 416*K1**3*K2*K3 - 384*K1**2*K2**4 + 288*K1**2*K2**3 - 256*K1**2*K2**2*K3**2 - 2384*K1**2*K2**2 + 2496*K1**2*K2 - 608*K1**2*K3**2 - 32*K1**2*K4**2 - 1856*K1**2 + 1024*K1*K2**3*K3 + 192*K1*K2*K3**3 + 3496*K1*K2*K3 + 616*K1*K3*K4 + 64*K1*K4*K5 + 24*K1*K5*K6 - 32*K2**6 - 64*K2**4*K3**2 - 32*K2**4*K4**2 + 96*K2**4*K4 - 832*K2**4 + 96*K2**3*K3*K5 + 64*K2**3*K4*K6 - 1152*K2**2*K3**2 - 136*K2**2*K4**2 + 424*K2**2*K4 - 48*K2**2*K5**2 - 48*K2**2*K6**2 - 1344*K2**2 + 496*K2*K3*K5 + 120*K2*K4*K6 + 8*K2*K5*K7 + 16*K2*K6*K8 - 96*K3**4 + 72*K3**2*K6 - 1312*K3**2 - 368*K4**2 - 104*K5**2 - 72*K6**2 - 2*K8**2 + 2144
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {4, 5}, {2, 3}], [{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {4, 5}, {1, 2}], [{4, 6}, {1, 5}, {2, 3}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {3, 4}, {1, 2}], [{6}, {1, 5}, {4}, {2, 3}]]
If K is slice False
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