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Flat knot 6.1165

Min(phi) over symmetries of the knot is: [-3,0,0,1,1,1,1,2,1,1,3,0,0,1,1,1,1,1,0,0,-1]
Flat knots (up to 7 crossings) with same phi are :['6.1165']
Arrow polynomial of the knot is: -2*K1**2 - 6*K1*K2 - 2*K1*K3 + 3*K1 - 2*K2**2 + 2*K2 + 3*K3 + 2*K4 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.1165']
Outer characteristic polynomial of the knot is: t^7+38t^5+46t^3+5t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1165']
2-strand cable arrow polynomial of the knot is: -2640*K1**4 + 448*K1**3*K2*K3 + 128*K1**3*K3*K4 - 960*K1**3*K3 + 128*K1**2*K2**2*K4 - 1520*K1**2*K2**2 + 32*K1**2*K2*K3*K5 - 1600*K1**2*K2*K4 + 5384*K1**2*K2 - 736*K1**2*K3**2 - 64*K1**2*K3*K5 - 432*K1**2*K4**2 - 64*K1**2*K4*K6 - 32*K1**2*K5**2 - 3676*K1**2 - 256*K1*K2**2*K3 - 512*K1*K2**2*K5 - 192*K1*K2*K3*K4 + 5032*K1*K2*K3 - 64*K1*K2*K4*K5 - 32*K1*K3**2*K5 - 32*K1*K3*K4*K6 + 2592*K1*K3*K4 + 872*K1*K4*K5 + 128*K1*K5*K6 + 8*K1*K6*K7 - 72*K2**4 - 32*K2**3*K6 - 24*K2**2*K4**2 + 1152*K2**2*K4 - 16*K2**2*K5**2 - 8*K2**2*K6**2 - 3314*K2**2 - 32*K2*K3*K4*K5 + 688*K2*K3*K5 - 32*K2*K4**2*K6 + 200*K2*K4*K6 + 24*K2*K5*K7 + 16*K2*K6*K8 - 16*K3**4 - 32*K3**2*K4**2 + 64*K3**2*K6 - 2156*K3**2 + 64*K3*K4*K7 - 8*K4**4 + 16*K4**2*K8 - 1376*K4**2 - 508*K5**2 - 126*K6**2 - 20*K7**2 - 4*K8**2 + 3610
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1165']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4362', 'vk6.4393', 'vk6.5680', 'vk6.5711', 'vk6.7753', 'vk6.7784', 'vk6.9231', 'vk6.9262', 'vk6.10484', 'vk6.10565', 'vk6.10660', 'vk6.10707', 'vk6.10738', 'vk6.10851', 'vk6.14609', 'vk6.15321', 'vk6.15446', 'vk6.16232', 'vk6.17976', 'vk6.24420', 'vk6.30163', 'vk6.30244', 'vk6.30339', 'vk6.30470', 'vk6.33967', 'vk6.34368', 'vk6.34422', 'vk6.43847', 'vk6.50455', 'vk6.50486', 'vk6.54195', 'vk6.63423']
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 O1O2O3O4U1U4U5O6O5U3U6U2
R3 orbit {'O1O2O3O4U1U4U5O6O5U3U6U2'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U3U5U2O6O5U6U1U4
Gauss code of K* O1O2O3U4U3O5O4O6U1U6U5U2
Gauss code of -K* O1O2O3U4U2O4O5O6U5U3U1U6
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 -3 1 0 1 1 0],[ 3 0 3 2 1 3 1],[-1 -3 0 -1 0 0 0],[ 0 -2 1 0 0 0 0],[-1 -1 0 0 0 -1 0],[-1 -3 0 0 1 0 0],[ 0 -1 0 0 0 0 0]]
Primitive based matrix [[ 0 1 1 1 0 0 -3],[-1 0 1 0 0 0 -3],[-1 -1 0 0 0 0 -1],[-1 0 0 0 0 -1 -3],[ 0 0 0 0 0 0 -1],[ 0 0 0 1 0 0 -2],[ 3 3 1 3 1 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-1,-1,-1,0,0,3,-1,0,0,0,3,0,0,0,1,0,1,3,0,1,2]
Phi over symmetry [-3,0,0,1,1,1,1,2,1,1,3,0,0,1,1,1,1,1,0,0,-1]
Phi of -K [-3,0,0,1,1,1,1,2,1,1,3,0,0,1,1,1,1,1,0,0,-1]
Phi of K* [-1,-1,-1,0,0,3,-1,0,1,1,3,0,1,1,1,0,1,1,0,1,2]
Phi of -K* [-3,0,0,1,1,1,1,2,1,3,3,0,0,0,0,0,0,1,-1,0,0]
Symmetry type of based matrix c
u-polynomial t^3-3t
Normalized Jones-Krushkal polynomial 4z^2+21z+27
Enhanced Jones-Krushkal polynomial 4w^3z^2+21w^2z+27w
Inner characteristic polynomial t^6+26t^4+26t^2+1
Outer characteristic polynomial t^7+38t^5+46t^3+5t
Flat arrow polynomial -2*K1**2 - 6*K1*K2 - 2*K1*K3 + 3*K1 - 2*K2**2 + 2*K2 + 3*K3 + 2*K4 + 3
2-strand cable arrow polynomial -2640*K1**4 + 448*K1**3*K2*K3 + 128*K1**3*K3*K4 - 960*K1**3*K3 + 128*K1**2*K2**2*K4 - 1520*K1**2*K2**2 + 32*K1**2*K2*K3*K5 - 1600*K1**2*K2*K4 + 5384*K1**2*K2 - 736*K1**2*K3**2 - 64*K1**2*K3*K5 - 432*K1**2*K4**2 - 64*K1**2*K4*K6 - 32*K1**2*K5**2 - 3676*K1**2 - 256*K1*K2**2*K3 - 512*K1*K2**2*K5 - 192*K1*K2*K3*K4 + 5032*K1*K2*K3 - 64*K1*K2*K4*K5 - 32*K1*K3**2*K5 - 32*K1*K3*K4*K6 + 2592*K1*K3*K4 + 872*K1*K4*K5 + 128*K1*K5*K6 + 8*K1*K6*K7 - 72*K2**4 - 32*K2**3*K6 - 24*K2**2*K4**2 + 1152*K2**2*K4 - 16*K2**2*K5**2 - 8*K2**2*K6**2 - 3314*K2**2 - 32*K2*K3*K4*K5 + 688*K2*K3*K5 - 32*K2*K4**2*K6 + 200*K2*K4*K6 + 24*K2*K5*K7 + 16*K2*K6*K8 - 16*K3**4 - 32*K3**2*K4**2 + 64*K3**2*K6 - 2156*K3**2 + 64*K3*K4*K7 - 8*K4**4 + 16*K4**2*K8 - 1376*K4**2 - 508*K5**2 - 126*K6**2 - 20*K7**2 - 4*K8**2 + 3610
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {3, 5}, {2, 4}], [{4, 6}, {2, 5}, {1, 3}], [{4, 6}, {3, 5}, {1, 2}], [{5, 6}, {2, 4}, {1, 3}], [{5, 6}, {3, 4}, {1, 2}]]
If K is slice False
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