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Flat knot 6.97

Min(phi) over symmetries of the knot is: [-4,-3,1,1,2,3,0,2,4,4,3,1,3,3,2,0,1,1,1,2,1]
Flat knots (up to 7 crossings) with same phi are :['6.97']
Arrow polynomial of the knot is: 4*K1**3 + 4*K1**2*K2 - 4*K1**2 - 4*K1*K2 - 2*K1*K3 - K1 + K2 + K3 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.97', '6.99', '6.287']
Outer characteristic polynomial of the knot is: t^7+102t^5+76t^3
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.97']
2-strand cable arrow polynomial of the knot is: -128*K1**6 + 256*K1**4*K2**3 - 640*K1**4*K2**2 + 640*K1**4*K2 - 528*K1**4 + 160*K1**3*K2*K3 - 384*K1**2*K2**4 + 448*K1**2*K2**3 - 880*K1**2*K2**2 + 744*K1**2*K2 - 16*K1**2*K3**2 - 292*K1**2 + 256*K1*K2**3*K3 + 568*K1*K2*K3 + 72*K1*K3*K4 + 24*K1*K4*K5 + 8*K1*K5*K6 - 32*K2**6 - 64*K2**4*K3**2 - 32*K2**4*K4**2 + 128*K2**4*K4 - 384*K2**4 + 96*K2**3*K3*K5 + 32*K2**3*K4*K6 - 176*K2**2*K3**2 - 112*K2**2*K4**2 + 184*K2**2*K4 - 48*K2**2*K5**2 - 8*K2**2*K6**2 - 186*K2**2 + 128*K2*K3*K5 + 48*K2*K4*K6 + 8*K2*K5*K7 - 200*K3**2 - 114*K4**2 - 52*K5**2 - 14*K6**2 + 488
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.97']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.17013', 'vk6.17254', 'vk6.19989', 'vk6.20243', 'vk6.21160', 'vk6.21549', 'vk6.23423', 'vk6.26875', 'vk6.27016', 'vk6.27464', 'vk6.28637', 'vk6.29061', 'vk6.35496', 'vk6.38304', 'vk6.38420', 'vk6.38874', 'vk6.40429', 'vk6.41073', 'vk6.42921', 'vk6.45166', 'vk6.45304', 'vk6.45637', 'vk6.47007', 'vk6.47377', 'vk6.55192', 'vk6.56721', 'vk6.56795', 'vk6.57813', 'vk6.58216', 'vk6.59571', 'vk6.61295', 'vk6.62385', 'vk6.64987', 'vk6.66414', 'vk6.66499', 'vk6.67178', 'vk6.67547', 'vk6.68273', 'vk6.69149', 'vk6.69851']
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 O1O2O3O4O5O6U2U6U1U5U3U4
R3 orbit {'O1O2O3O4O5U1O6U2U6U5U3U4', 'O1O2O3O4O5O6U2U6U1U5U3U4'}
R3 orbit length 2
Gauss code of -K O1O2O3O4O5O6U3U4U2U6U1U5
Gauss code of K* O1O2O3O4O5O6U3U1U5U6U4U2
Gauss code of -K* O1O2O3O4O5O6U5U3U1U2U6U4
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 -4 1 3 2 1],[ 3 0 -1 3 4 2 1],[ 4 1 0 3 4 2 1],[-1 -3 -3 0 1 0 0],[-3 -4 -4 -1 0 0 0],[-2 -2 -2 0 0 0 0],[-1 -1 -1 0 0 0 0]]
Primitive based matrix [[ 0 3 2 1 1 -3 -4],[-3 0 0 0 -1 -4 -4],[-2 0 0 0 0 -2 -2],[-1 0 0 0 0 -1 -1],[-1 1 0 0 0 -3 -3],[ 3 4 2 1 3 0 -1],[ 4 4 2 1 3 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-2,-1,-1,3,4,0,0,1,4,4,0,0,2,2,0,1,1,3,3,1]
Phi over symmetry [-4,-3,1,1,2,3,0,2,4,4,3,1,3,3,2,0,1,1,1,2,1]
Phi of -K [-4,-3,1,1,2,3,0,2,4,4,3,1,3,3,2,0,1,1,1,2,1]
Phi of K* [-3,-2,-1,-1,3,4,1,1,2,2,3,1,1,3,4,0,1,2,3,4,0]
Phi of -K* [-4,-3,1,1,2,3,1,1,3,2,4,1,3,2,4,0,0,0,0,1,0]
Symmetry type of based matrix c
u-polynomial t^4-t^2-2t
Normalized Jones-Krushkal polynomial 5z+11
Enhanced Jones-Krushkal polynomial -4w^3z+9w^2z+11w
Inner characteristic polynomial t^6+62t^4+11t^2
Outer characteristic polynomial t^7+102t^5+76t^3
Flat arrow polynomial 4*K1**3 + 4*K1**2*K2 - 4*K1**2 - 4*K1*K2 - 2*K1*K3 - K1 + K2 + K3 + 2
2-strand cable arrow polynomial -128*K1**6 + 256*K1**4*K2**3 - 640*K1**4*K2**2 + 640*K1**4*K2 - 528*K1**4 + 160*K1**3*K2*K3 - 384*K1**2*K2**4 + 448*K1**2*K2**3 - 880*K1**2*K2**2 + 744*K1**2*K2 - 16*K1**2*K3**2 - 292*K1**2 + 256*K1*K2**3*K3 + 568*K1*K2*K3 + 72*K1*K3*K4 + 24*K1*K4*K5 + 8*K1*K5*K6 - 32*K2**6 - 64*K2**4*K3**2 - 32*K2**4*K4**2 + 128*K2**4*K4 - 384*K2**4 + 96*K2**3*K3*K5 + 32*K2**3*K4*K6 - 176*K2**2*K3**2 - 112*K2**2*K4**2 + 184*K2**2*K4 - 48*K2**2*K5**2 - 8*K2**2*K6**2 - 186*K2**2 + 128*K2*K3*K5 + 48*K2*K4*K6 + 8*K2*K5*K7 - 200*K3**2 - 114*K4**2 - 52*K5**2 - 14*K6**2 + 488
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {3, 5}, {4}, {1}], [{3, 6}, {2, 5}, {1, 4}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {2, 4}, {1, 3}], [{6}, {5}, {1, 4}, {2, 3}], [{6}, {5}, {2, 4}, {1, 3}]]
If K is slice False
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