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Flat knot 6.363

Min(phi) over symmetries of the knot is: [-3,-2,-1,1,2,3,-1,1,1,2,4,1,1,0,2,1,1,3,0,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.363']
Arrow polynomial of the knot is: 4*K1**3 + 4*K1**2*K2 - 4*K1**2 - 4*K1*K2 - 4*K1*K3 - K1 + 2*K2 + K3 + K4 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.114', '6.224', '6.363']
Outer characteristic polynomial of the knot is: t^7+68t^5+91t^3+8t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.363']
2-strand cable arrow polynomial of the knot is: 128*K1**4*K2**3 - 448*K1**4*K2**2 + 352*K1**4*K2 - 464*K1**4 - 128*K1**3*K2**2*K3 + 608*K1**3*K2*K3 - 544*K1**3*K3 - 256*K1**2*K2**4 + 2912*K1**2*K2**3 - 6656*K1**2*K2**2 + 32*K1**2*K2*K3**2 - 832*K1**2*K2*K4 + 5936*K1**2*K2 - 240*K1**2*K3**2 - 4604*K1**2 - 640*K1*K2**4*K3 - 256*K1*K2**3*K3*K4 + 2016*K1*K2**3*K3 + 448*K1*K2**2*K3*K4 - 1504*K1*K2**2*K3 + 96*K1*K2**2*K4*K5 - 128*K1*K2**2*K5 - 416*K1*K2*K3*K4 + 6416*K1*K2*K3 + 888*K1*K3*K4 + 152*K1*K4*K5 + 40*K1*K5*K6 - 32*K2**6 - 128*K2**4*K3**2 - 32*K2**4*K4**2 + 800*K2**4*K4 - 3312*K2**4 + 352*K2**3*K3*K5 + 64*K2**3*K4*K6 - 32*K2**3*K6 - 1904*K2**2*K3**2 - 720*K2**2*K4**2 + 2224*K2**2*K4 - 208*K2**2*K5**2 - 48*K2**2*K6**2 - 2022*K2**2 + 1088*K2*K3*K5 + 272*K2*K4*K6 + 48*K2*K5*K7 + 16*K2*K6*K8 + 24*K3**2*K6 - 1900*K3**2 - 740*K4**2 - 276*K5**2 - 90*K6**2 - 4*K7**2 - 2*K8**2 + 3652
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.363']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4189', 'vk6.4270', 'vk6.5435', 'vk6.5553', 'vk6.7546', 'vk6.7629', 'vk6.9054', 'vk6.9135', 'vk6.18246', 'vk6.18581', 'vk6.24722', 'vk6.25135', 'vk6.36844', 'vk6.37307', 'vk6.44077', 'vk6.44416', 'vk6.48501', 'vk6.48582', 'vk6.49185', 'vk6.49297', 'vk6.50282', 'vk6.50354', 'vk6.51051', 'vk6.51084', 'vk6.56049', 'vk6.56323', 'vk6.60606', 'vk6.60949', 'vk6.65715', 'vk6.66009', 'vk6.68756', 'vk6.68964']
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 O1O2O3O4O5U3U5O6U1U2U6U4
R3 orbit {'O1O2O3O4O5U3U5O6U1U2U6U4'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U2U6U4U5O6U1U3
Gauss code of K* O1O2O3O4U1U2U5U4U6O5O6U3
Gauss code of -K* O1O2O3O4U2O5O6U5U1U6U3U4
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 -2 3 1 2],[ 3 0 1 -1 4 1 2],[ 1 -1 0 -1 3 1 1],[ 2 1 1 0 2 1 0],[-3 -4 -3 -2 0 0 0],[-1 -1 -1 -1 0 0 0],[-2 -2 -1 0 0 0 0]]
Primitive based matrix [[ 0 3 2 1 -1 -2 -3],[-3 0 0 0 -3 -2 -4],[-2 0 0 0 -1 0 -2],[-1 0 0 0 -1 -1 -1],[ 1 3 1 1 0 -1 -1],[ 2 2 0 1 1 0 1],[ 3 4 2 1 1 -1 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-2,-1,1,2,3,0,0,3,2,4,0,1,0,2,1,1,1,1,1,-1]
Phi over symmetry [-3,-2,-1,1,2,3,-1,1,1,2,4,1,1,0,2,1,1,3,0,0,0]
Phi of -K [-3,-2,-1,1,2,3,2,1,3,3,2,0,2,4,3,1,2,1,1,2,1]
Phi of K* [-3,-2,-1,1,2,3,1,2,1,3,2,1,2,4,3,1,2,3,0,1,2]
Phi of -K* [-3,-2,-1,1,2,3,-1,1,1,2,4,1,1,0,2,1,1,3,0,0,0]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 6z^2+23z+23
Enhanced Jones-Krushkal polynomial 6w^3z^2-4w^3z+27w^2z+23w
Inner characteristic polynomial t^6+40t^4+39t^2
Outer characteristic polynomial t^7+68t^5+91t^3+8t
Flat arrow polynomial 4*K1**3 + 4*K1**2*K2 - 4*K1**2 - 4*K1*K2 - 4*K1*K3 - K1 + 2*K2 + K3 + K4 + 2
2-strand cable arrow polynomial 128*K1**4*K2**3 - 448*K1**4*K2**2 + 352*K1**4*K2 - 464*K1**4 - 128*K1**3*K2**2*K3 + 608*K1**3*K2*K3 - 544*K1**3*K3 - 256*K1**2*K2**4 + 2912*K1**2*K2**3 - 6656*K1**2*K2**2 + 32*K1**2*K2*K3**2 - 832*K1**2*K2*K4 + 5936*K1**2*K2 - 240*K1**2*K3**2 - 4604*K1**2 - 640*K1*K2**4*K3 - 256*K1*K2**3*K3*K4 + 2016*K1*K2**3*K3 + 448*K1*K2**2*K3*K4 - 1504*K1*K2**2*K3 + 96*K1*K2**2*K4*K5 - 128*K1*K2**2*K5 - 416*K1*K2*K3*K4 + 6416*K1*K2*K3 + 888*K1*K3*K4 + 152*K1*K4*K5 + 40*K1*K5*K6 - 32*K2**6 - 128*K2**4*K3**2 - 32*K2**4*K4**2 + 800*K2**4*K4 - 3312*K2**4 + 352*K2**3*K3*K5 + 64*K2**3*K4*K6 - 32*K2**3*K6 - 1904*K2**2*K3**2 - 720*K2**2*K4**2 + 2224*K2**2*K4 - 208*K2**2*K5**2 - 48*K2**2*K6**2 - 2022*K2**2 + 1088*K2*K3*K5 + 272*K2*K4*K6 + 48*K2*K5*K7 + 16*K2*K6*K8 + 24*K3**2*K6 - 1900*K3**2 - 740*K4**2 - 276*K5**2 - 90*K6**2 - 4*K7**2 - 2*K8**2 + 3652
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{3, 6}, {2, 5}, {1, 4}], [{4, 6}, {2, 5}, {1, 3}]]
If K is slice False
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