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Flat knot 6.355

Min(phi) over symmetries of the knot is: [-4,-1,0,0,2,3,0,1,3,2,5,0,0,0,1,1,1,1,1,2,-1]
Flat knots (up to 7 crossings) with same phi are :['6.355']
Arrow polynomial of the knot is: 12*K1**3 + 4*K1**2*K2 - 8*K1**2 - 6*K1*K2 - 2*K1*K3 - 6*K1 + 3*K2 + 4
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.355']
Outer characteristic polynomial of the knot is: t^7+79t^5+110t^3
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.355']
2-strand cable arrow polynomial of the knot is: -192*K1**4*K2**2 + 224*K1**4*K2 - 640*K1**4 + 128*K1**3*K2**3*K3 + 224*K1**3*K2*K3 - 512*K1**2*K2**4 + 576*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 - 2576*K1**2*K2**2 + 2376*K1**2*K2 - 128*K1**2*K3**2 - 64*K1**2*K4**2 - 1584*K1**2 + 992*K1*K2**3*K3 + 32*K1*K2*K3**3 + 2320*K1*K2*K3 + 280*K1*K3*K4 + 64*K1*K4*K5 - 224*K2**6 - 192*K2**4*K3**2 - 32*K2**4*K4**2 + 224*K2**4*K4 - 1392*K2**4 + 128*K2**3*K3*K5 + 32*K2**3*K4*K6 - 896*K2**2*K3**2 - 136*K2**2*K4**2 + 816*K2**2*K4 - 16*K2**2*K5**2 - 8*K2**2*K6**2 - 552*K2**2 + 344*K2*K3*K5 + 56*K2*K4*K6 - 32*K3**2*K4**2 - 696*K3**2 + 16*K3*K4*K7 - 290*K4**2 - 48*K5**2 - 8*K6**2 + 1512
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.355']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.10138', 'vk6.10197', 'vk6.10342', 'vk6.10427', 'vk6.17663', 'vk6.17710', 'vk6.24226', 'vk6.24273', 'vk6.29917', 'vk6.29962', 'vk6.30027', 'vk6.30078', 'vk6.36494', 'vk6.36588', 'vk6.43587', 'vk6.43697', 'vk6.51620', 'vk6.51657', 'vk6.51704', 'vk6.51725', 'vk6.55701', 'vk6.55758', 'vk6.60267', 'vk6.60329', 'vk6.63331', 'vk6.63356', 'vk6.63379', 'vk6.63400', 'vk6.65409', 'vk6.65450', 'vk6.68547', 'vk6.68578']
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 O1O2O3O4O5U3U4O6U1U6U2U5
R3 orbit {'O1O2O3O4O5U3U4O6U1U6U2U5'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U1U4U6U5O6U2U3
Gauss code of K* O1O2O3O4U1U3U5U6U4O5O6U2
Gauss code of -K* O1O2O3O4U3O5O6U1U5U6U2U4
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 -2 0 4 1],[ 3 0 2 -1 1 5 1],[ 0 -2 0 -1 1 3 0],[ 2 1 1 0 1 2 0],[ 0 -1 -1 -1 0 1 0],[-4 -5 -3 -2 -1 0 0],[-1 -1 0 0 0 0 0]]
Primitive based matrix [[ 0 4 1 0 0 -2 -3],[-4 0 0 -1 -3 -2 -5],[-1 0 0 0 0 0 -1],[ 0 1 0 0 -1 -1 -1],[ 0 3 0 1 0 -1 -2],[ 2 2 0 1 1 0 1],[ 3 5 1 1 2 -1 0]]
If based matrix primitive True
Phi of primitive based matrix [-4,-1,0,0,2,3,0,1,3,2,5,0,0,0,1,1,1,1,1,2,-1]
Phi over symmetry [-4,-1,0,0,2,3,0,1,3,2,5,0,0,0,1,1,1,1,1,2,-1]
Phi of -K [-3,-2,0,0,1,4,2,1,2,3,2,1,1,3,4,-1,1,1,1,3,3]
Phi of K* [-4,-1,0,0,2,3,3,1,3,4,2,1,1,3,3,1,1,1,1,2,2]
Phi of -K* [-3,-2,0,0,1,4,-1,1,2,1,5,1,1,0,2,-1,0,1,0,3,0]
Symmetry type of based matrix c
u-polynomial -t^4+t^3+t^2-t
Normalized Jones-Krushkal polynomial 7z+15
Enhanced Jones-Krushkal polynomial -8w^3z+15w^2z+15w
Inner characteristic polynomial t^6+49t^4+45t^2
Outer characteristic polynomial t^7+79t^5+110t^3
Flat arrow polynomial 12*K1**3 + 4*K1**2*K2 - 8*K1**2 - 6*K1*K2 - 2*K1*K3 - 6*K1 + 3*K2 + 4
2-strand cable arrow polynomial -192*K1**4*K2**2 + 224*K1**4*K2 - 640*K1**4 + 128*K1**3*K2**3*K3 + 224*K1**3*K2*K3 - 512*K1**2*K2**4 + 576*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 - 2576*K1**2*K2**2 + 2376*K1**2*K2 - 128*K1**2*K3**2 - 64*K1**2*K4**2 - 1584*K1**2 + 992*K1*K2**3*K3 + 32*K1*K2*K3**3 + 2320*K1*K2*K3 + 280*K1*K3*K4 + 64*K1*K4*K5 - 224*K2**6 - 192*K2**4*K3**2 - 32*K2**4*K4**2 + 224*K2**4*K4 - 1392*K2**4 + 128*K2**3*K3*K5 + 32*K2**3*K4*K6 - 896*K2**2*K3**2 - 136*K2**2*K4**2 + 816*K2**2*K4 - 16*K2**2*K5**2 - 8*K2**2*K6**2 - 552*K2**2 + 344*K2*K3*K5 + 56*K2*K4*K6 - 32*K3**2*K4**2 - 696*K3**2 + 16*K3*K4*K7 - 290*K4**2 - 48*K5**2 - 8*K6**2 + 1512
Genus of based matrix 1
Fillings of based matrix [[{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {3, 5}, {1, 2}]]
If K is slice False
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