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Flat knot 6.326

Min(phi) over symmetries of the knot is: [-3,-3,1,1,2,2,-1,1,2,2,4,1,2,1,3,0,0,0,1,1,-1]
Flat knots (up to 7 crossings) with same phi are :['6.326']
Arrow polynomial of the knot is: 8*K1**3 + 4*K1**2*K2 - 8*K1**2 - 8*K1*K2 - 4*K1*K3 - 2*K1 + 4*K2 + 2*K3 + K4 + 4
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.326']
Outer characteristic polynomial of the knot is: t^7+72t^5+43t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.326']
2-strand cable arrow polynomial of the knot is: -192*K1**4*K2**2 + 384*K1**4*K2 - 1680*K1**4 + 352*K1**3*K2*K3 - 256*K1**3*K3 - 256*K1**2*K2**4 + 608*K1**2*K2**3 + 64*K1**2*K2**2*K4 - 4832*K1**2*K2**2 - 608*K1**2*K2*K4 + 6488*K1**2*K2 - 496*K1**2*K3**2 - 128*K1**2*K3*K5 - 48*K1**2*K4**2 - 32*K1**2*K5**2 - 4488*K1**2 + 864*K1*K2**3*K3 + 224*K1*K2**2*K3*K4 - 864*K1*K2**2*K3 + 64*K1*K2**2*K4*K5 - 512*K1*K2**2*K5 - 384*K1*K2*K3*K4 - 192*K1*K2*K3*K6 + 6392*K1*K2*K3 - 64*K1*K2*K4*K5 - 64*K1*K2*K5*K6 - 32*K1*K3**2*K5 + 1456*K1*K3*K4 + 392*K1*K4*K5 + 88*K1*K5*K6 + 8*K1*K6*K7 - 64*K2**6 - 64*K2**4*K3**2 - 32*K2**4*K4**2 + 128*K2**4*K4 - 1312*K2**4 + 160*K2**3*K3*K5 + 64*K2**3*K4*K6 - 64*K2**3*K6 - 944*K2**2*K3**2 - 32*K2**2*K3*K7 - 320*K2**2*K4**2 + 1824*K2**2*K4 - 144*K2**2*K5**2 - 48*K2**2*K6**2 - 3540*K2**2 + 1152*K2*K3*K5 + 320*K2*K4*K6 + 88*K2*K5*K7 + 16*K2*K6*K8 + 56*K3**2*K6 - 2168*K3**2 + 8*K3*K4*K7 - 892*K4**2 - 352*K5**2 - 92*K6**2 - 16*K7**2 - 2*K8**2 + 4012
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.326']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11229', 'vk6.11310', 'vk6.12490', 'vk6.12603', 'vk6.18228', 'vk6.18564', 'vk6.24694', 'vk6.25110', 'vk6.30899', 'vk6.31024', 'vk6.32083', 'vk6.32204', 'vk6.36816', 'vk6.37276', 'vk6.44056', 'vk6.44396', 'vk6.51975', 'vk6.52072', 'vk6.52856', 'vk6.52905', 'vk6.56033', 'vk6.56308', 'vk6.60580', 'vk6.60918', 'vk6.63631', 'vk6.63678', 'vk6.64059', 'vk6.64106', 'vk6.65692', 'vk6.65985', 'vk6.68737', 'vk6.68946']
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 O1O2O3O4O5U2U5O6U1U4U6U3
R3 orbit {'O1O2O3O4O5U2U5O6U1U4U6U3'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U3U6U2U5O6U1U4
Gauss code of K* O1O2O3O4U1U5U4U2U6O5O6U3
Gauss code of -K* O1O2O3O4U2O5O6U5U3U1U6U4
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 -3 2 1 1 2],[ 3 0 -1 4 2 1 2],[ 3 1 0 3 2 1 1],[-2 -4 -3 0 -1 0 1],[-1 -2 -2 1 0 0 1],[-1 -1 -1 0 0 0 0],[-2 -2 -1 -1 -1 0 0]]
Primitive based matrix [[ 0 2 2 1 1 -3 -3],[-2 0 1 0 -1 -3 -4],[-2 -1 0 0 -1 -1 -2],[-1 0 0 0 0 -1 -1],[-1 1 1 0 0 -2 -2],[ 3 3 1 1 2 0 1],[ 3 4 2 1 2 -1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,-1,-1,3,3,-1,0,1,3,4,0,1,1,2,0,1,1,2,2,-1]
Phi over symmetry [-3,-3,1,1,2,2,-1,1,2,2,4,1,2,1,3,0,0,0,1,1,-1]
Phi of -K [-3,-3,1,1,2,2,-1,2,3,2,4,2,3,1,3,0,0,0,1,1,-1]
Phi of K* [-2,-2,-1,-1,3,3,-1,0,1,3,4,0,1,1,2,0,2,2,3,3,-1]
Phi of -K* [-3,-3,1,1,2,2,-1,1,2,2,4,1,2,1,3,0,0,0,1,1,-1]
Symmetry type of based matrix c
u-polynomial 2t^3-2t^2-2t
Normalized Jones-Krushkal polynomial 3z^2+20z+29
Enhanced Jones-Krushkal polynomial 3w^3z^2+20w^2z+29w
Inner characteristic polynomial t^6+44t^4+11t^2
Outer characteristic polynomial t^7+72t^5+43t^3+4t
Flat arrow polynomial 8*K1**3 + 4*K1**2*K2 - 8*K1**2 - 8*K1*K2 - 4*K1*K3 - 2*K1 + 4*K2 + 2*K3 + K4 + 4
2-strand cable arrow polynomial -192*K1**4*K2**2 + 384*K1**4*K2 - 1680*K1**4 + 352*K1**3*K2*K3 - 256*K1**3*K3 - 256*K1**2*K2**4 + 608*K1**2*K2**3 + 64*K1**2*K2**2*K4 - 4832*K1**2*K2**2 - 608*K1**2*K2*K4 + 6488*K1**2*K2 - 496*K1**2*K3**2 - 128*K1**2*K3*K5 - 48*K1**2*K4**2 - 32*K1**2*K5**2 - 4488*K1**2 + 864*K1*K2**3*K3 + 224*K1*K2**2*K3*K4 - 864*K1*K2**2*K3 + 64*K1*K2**2*K4*K5 - 512*K1*K2**2*K5 - 384*K1*K2*K3*K4 - 192*K1*K2*K3*K6 + 6392*K1*K2*K3 - 64*K1*K2*K4*K5 - 64*K1*K2*K5*K6 - 32*K1*K3**2*K5 + 1456*K1*K3*K4 + 392*K1*K4*K5 + 88*K1*K5*K6 + 8*K1*K6*K7 - 64*K2**6 - 64*K2**4*K3**2 - 32*K2**4*K4**2 + 128*K2**4*K4 - 1312*K2**4 + 160*K2**3*K3*K5 + 64*K2**3*K4*K6 - 64*K2**3*K6 - 944*K2**2*K3**2 - 32*K2**2*K3*K7 - 320*K2**2*K4**2 + 1824*K2**2*K4 - 144*K2**2*K5**2 - 48*K2**2*K6**2 - 3540*K2**2 + 1152*K2*K3*K5 + 320*K2*K4*K6 + 88*K2*K5*K7 + 16*K2*K6*K8 + 56*K3**2*K6 - 2168*K3**2 + 8*K3*K4*K7 - 892*K4**2 - 352*K5**2 - 92*K6**2 - 16*K7**2 - 2*K8**2 + 4012
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {3, 5}, {2, 4}], [{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {3, 5}, {1, 4}], [{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {2, 5}, {1, 4}]]
If K is slice False
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