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Flat knot 6.23

Min(phi) over symmetries of the knot is: [-5,-2,1,1,2,3,1,2,5,4,3,1,3,3,2,0,1,1,1,2,1]
Flat knots (up to 7 crossings) with same phi are :['6.23']
Arrow polynomial of the knot is: 8*K1**3 + 4*K1**2*K3 - 12*K1**2 - 12*K1*K2 - 2*K1*K4 - K1 + 6*K2 + 3*K3 + 7
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.23']
Outer characteristic polynomial of the knot is: t^7+130t^5+153t^3+7t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.23']
2-strand cable arrow polynomial of the knot is: -128*K1**6 - 256*K1**4*K2**2 + 576*K1**4*K2 - 2320*K1**4 + 512*K1**3*K2*K3 - 512*K1**3*K3 - 320*K1**2*K2**4 + 896*K1**2*K2**3 - 256*K1**2*K2**2*K3**2 + 64*K1**2*K2**2*K4 - 7936*K1**2*K2**2 + 192*K1**2*K2*K3**2 - 704*K1**2*K2*K4 + 10224*K1**2*K2 - 816*K1**2*K3**2 - 64*K1**2*K4**2 - 6624*K1**2 + 2624*K1*K2**3*K3 + 416*K1*K2**2*K3*K4 - 2112*K1*K2**2*K3 + 32*K1*K2**2*K4*K5 + 64*K1*K2**2*K5*K6 - 544*K1*K2**2*K5 + 224*K1*K2*K3**3 - 480*K1*K2*K3*K4 - 192*K1*K2*K3*K6 + 10128*K1*K2*K3 - 128*K1*K2*K4*K5 - 32*K1*K3**2*K5 + 1872*K1*K3*K4 + 168*K1*K4*K5 + 32*K1*K5*K6 - 64*K2**6 - 384*K2**4*K3**2 + 160*K2**4*K4 - 32*K2**4*K6**2 - 3216*K2**4 + 384*K2**3*K3*K5 + 128*K2**3*K4*K6 + 32*K2**3*K6*K8 - 96*K2**3*K6 - 128*K2**2*K3**4 + 64*K2**2*K3**2*K4 + 64*K2**2*K3**2*K6 - 2816*K2**2*K3**2 - 64*K2**2*K3*K7 - 464*K2**2*K4**2 - 32*K2**2*K4*K8 + 3176*K2**2*K4 - 224*K2**2*K5**2 - 128*K2**2*K6**2 - 8*K2**2*K8**2 - 4490*K2**2 + 32*K2*K3**3*K5 - 96*K2*K3**2*K4 + 1816*K2*K3*K5 + 424*K2*K4*K6 + 56*K2*K5*K7 + 32*K2*K6*K8 - 96*K3**4 - 32*K3**2*K4**2 + 80*K3**2*K6 - 2980*K3**2 + 32*K3*K4*K7 - 1092*K4**2 - 300*K5**2 - 78*K6**2 - 8*K7**2 - 4*K8**2 + 5750
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.23']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.19940', 'vk6.19982', 'vk6.21181', 'vk6.21249', 'vk6.26895', 'vk6.27001', 'vk6.28649', 'vk6.28725', 'vk6.38316', 'vk6.38407', 'vk6.40453', 'vk6.40588', 'vk6.45190', 'vk6.45297', 'vk6.47018', 'vk6.47077', 'vk6.56725', 'vk6.56788', 'vk6.57821', 'vk6.57920', 'vk6.61142', 'vk6.61280', 'vk6.62389', 'vk6.62471', 'vk6.66418', 'vk6.66484', 'vk6.67186', 'vk6.67275', 'vk6.69067', 'vk6.69142', 'vk6.69854', 'vk6.69899']
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 O1O2O3O4O5O6U1U3U5U6U4U2
R3 orbit {'O1O2O3O4O5O6U1U3U5U6U4U2'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5O6U5U3U1U2U4U6
Gauss code of K* O1O2O3O4O5O6U1U6U2U5U3U4
Gauss code of -K* O1O2O3O4O5O6U3U4U2U5U1U6
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 2 1 3],[ 5 0 5 1 4 2 3],[-1 -5 0 -3 1 0 2],[ 2 -1 3 0 3 1 2],[-2 -4 -1 -3 0 -1 1],[-1 -2 0 -1 1 0 1],[-3 -3 -2 -2 -1 -1 0]]
Primitive based matrix [[ 0 3 2 1 1 -2 -5],[-3 0 -1 -1 -2 -2 -3],[-2 1 0 -1 -1 -3 -4],[-1 1 1 0 0 -1 -2],[-1 2 1 0 0 -3 -5],[ 2 2 3 1 3 0 -1],[ 5 3 4 2 5 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-2,-1,-1,2,5,1,1,2,2,3,1,1,3,4,0,1,2,3,5,1]
Phi over symmetry [-5,-2,1,1,2,3,1,2,5,4,3,1,3,3,2,0,1,1,1,2,1]
Phi of -K [-5,-2,1,1,2,3,2,1,4,3,5,0,2,1,3,0,0,0,0,1,0]
Phi of K* [-3,-2,-1,-1,2,5,0,0,1,3,5,0,0,1,3,0,0,1,2,4,2]
Phi of -K* [-5,-2,1,1,2,3,1,2,5,4,3,1,3,3,2,0,1,1,1,2,1]
Symmetry type of based matrix c
u-polynomial t^5-t^3-2t
Normalized Jones-Krushkal polynomial 5z^2+25z+31
Enhanced Jones-Krushkal polynomial 5w^3z^2+25w^2z+31w
Inner characteristic polynomial t^6+86t^4+29t^2+1
Outer characteristic polynomial t^7+130t^5+153t^3+7t
Flat arrow polynomial 8*K1**3 + 4*K1**2*K3 - 12*K1**2 - 12*K1*K2 - 2*K1*K4 - K1 + 6*K2 + 3*K3 + 7
2-strand cable arrow polynomial -128*K1**6 - 256*K1**4*K2**2 + 576*K1**4*K2 - 2320*K1**4 + 512*K1**3*K2*K3 - 512*K1**3*K3 - 320*K1**2*K2**4 + 896*K1**2*K2**3 - 256*K1**2*K2**2*K3**2 + 64*K1**2*K2**2*K4 - 7936*K1**2*K2**2 + 192*K1**2*K2*K3**2 - 704*K1**2*K2*K4 + 10224*K1**2*K2 - 816*K1**2*K3**2 - 64*K1**2*K4**2 - 6624*K1**2 + 2624*K1*K2**3*K3 + 416*K1*K2**2*K3*K4 - 2112*K1*K2**2*K3 + 32*K1*K2**2*K4*K5 + 64*K1*K2**2*K5*K6 - 544*K1*K2**2*K5 + 224*K1*K2*K3**3 - 480*K1*K2*K3*K4 - 192*K1*K2*K3*K6 + 10128*K1*K2*K3 - 128*K1*K2*K4*K5 - 32*K1*K3**2*K5 + 1872*K1*K3*K4 + 168*K1*K4*K5 + 32*K1*K5*K6 - 64*K2**6 - 384*K2**4*K3**2 + 160*K2**4*K4 - 32*K2**4*K6**2 - 3216*K2**4 + 384*K2**3*K3*K5 + 128*K2**3*K4*K6 + 32*K2**3*K6*K8 - 96*K2**3*K6 - 128*K2**2*K3**4 + 64*K2**2*K3**2*K4 + 64*K2**2*K3**2*K6 - 2816*K2**2*K3**2 - 64*K2**2*K3*K7 - 464*K2**2*K4**2 - 32*K2**2*K4*K8 + 3176*K2**2*K4 - 224*K2**2*K5**2 - 128*K2**2*K6**2 - 8*K2**2*K8**2 - 4490*K2**2 + 32*K2*K3**3*K5 - 96*K2*K3**2*K4 + 1816*K2*K3*K5 + 424*K2*K4*K6 + 56*K2*K5*K7 + 32*K2*K6*K8 - 96*K3**4 - 32*K3**2*K4**2 + 80*K3**2*K6 - 2980*K3**2 + 32*K3*K4*K7 - 1092*K4**2 - 300*K5**2 - 78*K6**2 - 8*K7**2 - 4*K8**2 + 5750
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {4, 5}, {2, 3}], [{2, 6}, {1, 5}, {3, 4}], [{5, 6}, {3, 4}, {1, 2}], [{6}, {2, 5}, {3, 4}, {1}]]
If K is slice False
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