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Flat knot 6.31

Min(phi) over symmetries of the knot is: [-5,-1,-1,1,2,4,1,2,5,3,4,0,2,1,2,2,2,3,1,3,1]
Flat knots (up to 7 crossings) with same phi are :['6.31']
Arrow polynomial of the knot is: 4*K1**2*K3 - 6*K1*K2 - 2*K1*K3 - 2*K1*K4 + 2*K1 + K2 + 2*K3 + K4 + 1
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.31']
Outer characteristic polynomial of the knot is: t^7+140t^5+105t^3+7t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.31']
2-strand cable arrow polynomial of the knot is: 128*K1**4*K2 - 368*K1**4 - 512*K1**3*K2**2*K3 + 896*K1**3*K2*K3 - 864*K1**3*K3 + 512*K1**2*K2**3*K3**2 + 1984*K1**2*K2**3 - 768*K1**2*K2**2*K3**2 - 4352*K1**2*K2**2 + 384*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 928*K1**2*K2*K4 + 3936*K1**2*K2 - 624*K1**2*K3**2 - 3848*K1**2 - 1024*K1*K2**4*K3 - 256*K1*K2**3*K3*K4 + 2784*K1*K2**3*K3 + 928*K1*K2**2*K3*K4 - 1024*K1*K2**2*K3 - 96*K1*K2**2*K5 - 320*K1*K2*K3*K4 + 5328*K1*K2*K3 - 32*K1*K3**2*K5 + 1360*K1*K3*K4 + 208*K1*K4*K5 + 56*K1*K5*K6 - 768*K2**4*K3**2 + 512*K2**4*K4 - 32*K2**4*K6**2 - 1936*K2**4 + 576*K2**3*K3*K5 + 64*K2**3*K4*K6 + 32*K2**3*K5*K7 + 32*K2**3*K6*K8 - 1728*K2**2*K3**2 - 536*K2**2*K4**2 + 1184*K2**2*K4 - 176*K2**2*K5**2 - 72*K2**2*K6**2 - 32*K2**2*K7**2 - 8*K2**2*K8**2 - 1712*K2**2 - 64*K2*K3**2*K4 + 808*K2*K3*K5 + 272*K2*K4*K6 + 88*K2*K5*K7 + 32*K2*K6*K8 + 8*K2*K7*K9 + 64*K3**2*K6 - 1988*K3**2 + 8*K3*K4*K7 - 858*K4**2 - 320*K5**2 - 128*K6**2 - 28*K7**2 - 6*K8**2 + 3190
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.31']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.19912', 'vk6.19966', 'vk6.21133', 'vk6.21226', 'vk6.26827', 'vk6.26969', 'vk6.28607', 'vk6.28704', 'vk6.38261', 'vk6.38379', 'vk6.40383', 'vk6.40543', 'vk6.45126', 'vk6.45253', 'vk6.46984', 'vk6.47055', 'vk6.56688', 'vk6.56764', 'vk6.57765', 'vk6.57881', 'vk6.61081', 'vk6.61237', 'vk6.62344', 'vk6.62448', 'vk6.66381', 'vk6.66470', 'vk6.67139', 'vk6.67255', 'vk6.69036', 'vk6.69119', 'vk6.69828', 'vk6.69889']
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 O1O2O3O4O5O6U1U4U3U5U6U2
R3 orbit {'O1O2O3O4O5O6U1U4U3U5U6U2'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5O6U5U1U2U4U3U6
Gauss code of K* O1O2O3O4O5O6U1U6U3U2U4U5
Gauss code of -K* O1O2O3O4O5O6U2U3U5U4U1U6
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 -1 -1 2 4],[ 5 0 5 2 1 3 4],[-1 -5 0 -2 -2 1 3],[ 1 -2 2 0 0 2 3],[ 1 -1 2 0 0 1 2],[-2 -3 -1 -2 -1 0 1],[-4 -4 -3 -3 -2 -1 0]]
Primitive based matrix [[ 0 4 2 1 -1 -1 -5],[-4 0 -1 -3 -2 -3 -4],[-2 1 0 -1 -1 -2 -3],[-1 3 1 0 -2 -2 -5],[ 1 2 1 2 0 0 -1],[ 1 3 2 2 0 0 -2],[ 5 4 3 5 1 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-4,-2,-1,1,1,5,1,3,2,3,4,1,1,2,3,2,2,5,0,1,2]
Phi over symmetry [-5,-1,-1,1,2,4,1,2,5,3,4,0,2,1,2,2,2,3,1,3,1]
Phi of -K [-5,-1,-1,1,2,4,2,3,1,4,5,0,0,1,2,0,2,3,0,0,1]
Phi of K* [-4,-2,-1,1,1,5,1,0,2,3,5,0,1,2,4,0,0,1,0,2,3]
Phi of -K* [-5,-1,-1,1,2,4,1,2,5,3,4,0,2,1,2,2,2,3,1,3,1]
Symmetry type of based matrix c
u-polynomial t^5-t^4-t^2+t
Normalized Jones-Krushkal polynomial 4z^2+17z+19
Enhanced Jones-Krushkal polynomial -2w^4z^2+6w^3z^2-6w^3z+23w^2z+19w
Inner characteristic polynomial t^6+92t^4+28t^2
Outer characteristic polynomial t^7+140t^5+105t^3+7t
Flat arrow polynomial 4*K1**2*K3 - 6*K1*K2 - 2*K1*K3 - 2*K1*K4 + 2*K1 + K2 + 2*K3 + K4 + 1
2-strand cable arrow polynomial 128*K1**4*K2 - 368*K1**4 - 512*K1**3*K2**2*K3 + 896*K1**3*K2*K3 - 864*K1**3*K3 + 512*K1**2*K2**3*K3**2 + 1984*K1**2*K2**3 - 768*K1**2*K2**2*K3**2 - 4352*K1**2*K2**2 + 384*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 928*K1**2*K2*K4 + 3936*K1**2*K2 - 624*K1**2*K3**2 - 3848*K1**2 - 1024*K1*K2**4*K3 - 256*K1*K2**3*K3*K4 + 2784*K1*K2**3*K3 + 928*K1*K2**2*K3*K4 - 1024*K1*K2**2*K3 - 96*K1*K2**2*K5 - 320*K1*K2*K3*K4 + 5328*K1*K2*K3 - 32*K1*K3**2*K5 + 1360*K1*K3*K4 + 208*K1*K4*K5 + 56*K1*K5*K6 - 768*K2**4*K3**2 + 512*K2**4*K4 - 32*K2**4*K6**2 - 1936*K2**4 + 576*K2**3*K3*K5 + 64*K2**3*K4*K6 + 32*K2**3*K5*K7 + 32*K2**3*K6*K8 - 1728*K2**2*K3**2 - 536*K2**2*K4**2 + 1184*K2**2*K4 - 176*K2**2*K5**2 - 72*K2**2*K6**2 - 32*K2**2*K7**2 - 8*K2**2*K8**2 - 1712*K2**2 - 64*K2*K3**2*K4 + 808*K2*K3*K5 + 272*K2*K4*K6 + 88*K2*K5*K7 + 32*K2*K6*K8 + 8*K2*K7*K9 + 64*K3**2*K6 - 1988*K3**2 + 8*K3*K4*K7 - 858*K4**2 - 320*K5**2 - 128*K6**2 - 28*K7**2 - 6*K8**2 + 3190
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {4, 5}, {2, 3}], [{4, 6}, {1, 5}, {2, 3}], [{5, 6}, {1, 4}, {2, 3}]]
If K is slice False
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