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Flat knot 6.260

Min(phi) over symmetries of the knot is: [-4,-2,-1,1,3,3,0,2,2,3,4,1,1,2,2,1,2,2,1,0,-1]
Flat knots (up to 7 crossings) with same phi are :['6.260']
Arrow polynomial of the knot is: 4*K1**3 + 4*K1**2*K2 - 12*K1**2 - 4*K1*K2 - 2*K1*K3 - K1 + 5*K2 + K3 + 6
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.260']
Outer characteristic polynomial of the knot is: t^7+116t^5+45t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.260']
2-strand cable arrow polynomial of the knot is: -1648*K1**4 + 448*K1**3*K2*K3 + 32*K1**3*K3*K4 - 704*K1**3*K3 - 128*K1**2*K2**4 + 704*K1**2*K2**3 - 256*K1**2*K2**2*K3**2 - 5472*K1**2*K2**2 + 224*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 640*K1**2*K2*K4 + 7664*K1**2*K2 - 816*K1**2*K3**2 - 64*K1**2*K3*K5 - 144*K1**2*K4**2 - 32*K1**2*K4*K6 - 5060*K1**2 + 1568*K1*K2**3*K3 + 256*K1*K2**2*K3*K4 - 1056*K1*K2**2*K3 + 64*K1*K2**2*K4*K5 - 224*K1*K2**2*K5 + 64*K1*K2*K3**3 - 352*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 6880*K1*K2*K3 - 96*K1*K2*K4*K5 - 32*K1*K2*K4*K7 + 1264*K1*K3*K4 + 272*K1*K4*K5 + 32*K1*K5*K6 - 32*K2**6 - 128*K2**4*K3**2 - 32*K2**4*K4**2 + 128*K2**4*K4 - 1616*K2**4 + 64*K2**3*K3*K5 + 32*K2**3*K4*K6 - 1200*K2**2*K3**2 - 240*K2**2*K4**2 + 1408*K2**2*K4 - 32*K2**2*K5**2 - 8*K2**2*K6**2 - 3050*K2**2 + 584*K2*K3*K5 + 136*K2*K4*K6 + 16*K2*K5*K7 + 8*K3**2*K6 - 1908*K3**2 - 566*K4**2 - 120*K5**2 - 22*K6**2 + 3860
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.260']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.81561', 'vk6.81637', 'vk6.81643', 'vk6.81824', 'vk6.81829', 'vk6.82045', 'vk6.82219', 'vk6.82229', 'vk6.82331', 'vk6.82341', 'vk6.82534', 'vk6.82539', 'vk6.82988', 'vk6.83115', 'vk6.83126', 'vk6.83563', 'vk6.83570', 'vk6.83923', 'vk6.84075', 'vk6.84089', 'vk6.84525', 'vk6.84887', 'vk6.84890', 'vk6.85893', 'vk6.85900', 'vk6.86396', 'vk6.86417', 'vk6.86457', 'vk6.86464', 'vk6.88826', 'vk6.89758', 'vk6.89882']
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 O1O2O3O4O5U1U3O6U2U4U5U6
R3 orbit {'O1O2O3O4O5U1U3O6U2U4U5U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U6U1U2U4O6U3U5
Gauss code of K* O1O2O3O4U5U1U6U2U3O5O6U4
Gauss code of -K* O1O2O3O4U1O5O6U2U3U5U4U6
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 -4 -2 -1 1 3 3],[ 4 0 2 1 3 4 3],[ 2 -2 0 0 2 3 3],[ 1 -1 0 0 1 2 2],[-1 -3 -2 -1 0 1 2],[-3 -4 -3 -2 -1 0 1],[-3 -3 -3 -2 -2 -1 0]]
Primitive based matrix [[ 0 3 3 1 -1 -2 -4],[-3 0 1 -1 -2 -3 -4],[-3 -1 0 -2 -2 -3 -3],[-1 1 2 0 -1 -2 -3],[ 1 2 2 1 0 0 -1],[ 2 3 3 2 0 0 -2],[ 4 4 3 3 1 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-3,-1,1,2,4,-1,1,2,3,4,2,2,3,3,1,2,3,0,1,2]
Phi over symmetry [-4,-2,-1,1,3,3,0,2,2,3,4,1,1,2,2,1,2,2,1,0,-1]
Phi of -K [-4,-2,-1,1,3,3,0,2,2,3,4,1,1,2,2,1,2,2,1,0,-1]
Phi of K* [-3,-3,-1,1,2,4,-1,0,2,2,4,1,2,2,3,1,1,2,1,2,0]
Phi of -K* [-4,-2,-1,1,3,3,2,1,3,3,4,0,2,3,3,1,2,2,2,1,-1]
Symmetry type of based matrix c
u-polynomial t^4-2t^3+t^2
Normalized Jones-Krushkal polynomial 3z^2+20z+29
Enhanced Jones-Krushkal polynomial 3w^3z^2+20w^2z+29w
Inner characteristic polynomial t^6+76t^4+16t^2
Outer characteristic polynomial t^7+116t^5+45t^3+4t
Flat arrow polynomial 4*K1**3 + 4*K1**2*K2 - 12*K1**2 - 4*K1*K2 - 2*K1*K3 - K1 + 5*K2 + K3 + 6
2-strand cable arrow polynomial -1648*K1**4 + 448*K1**3*K2*K3 + 32*K1**3*K3*K4 - 704*K1**3*K3 - 128*K1**2*K2**4 + 704*K1**2*K2**3 - 256*K1**2*K2**2*K3**2 - 5472*K1**2*K2**2 + 224*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 640*K1**2*K2*K4 + 7664*K1**2*K2 - 816*K1**2*K3**2 - 64*K1**2*K3*K5 - 144*K1**2*K4**2 - 32*K1**2*K4*K6 - 5060*K1**2 + 1568*K1*K2**3*K3 + 256*K1*K2**2*K3*K4 - 1056*K1*K2**2*K3 + 64*K1*K2**2*K4*K5 - 224*K1*K2**2*K5 + 64*K1*K2*K3**3 - 352*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 6880*K1*K2*K3 - 96*K1*K2*K4*K5 - 32*K1*K2*K4*K7 + 1264*K1*K3*K4 + 272*K1*K4*K5 + 32*K1*K5*K6 - 32*K2**6 - 128*K2**4*K3**2 - 32*K2**4*K4**2 + 128*K2**4*K4 - 1616*K2**4 + 64*K2**3*K3*K5 + 32*K2**3*K4*K6 - 1200*K2**2*K3**2 - 240*K2**2*K4**2 + 1408*K2**2*K4 - 32*K2**2*K5**2 - 8*K2**2*K6**2 - 3050*K2**2 + 584*K2*K3*K5 + 136*K2*K4*K6 + 16*K2*K5*K7 + 8*K3**2*K6 - 1908*K3**2 - 566*K4**2 - 120*K5**2 - 22*K6**2 + 3860
Genus of based matrix 1
Fillings of based matrix [[{5, 6}, {1, 4}, {2, 3}]]
If K is slice False
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