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Flat knot 6.282

Min(phi) over symmetries of the knot is: [-4,-2,1,1,1,3,0,2,4,4,3,1,2,3,2,0,0,1,0,2,2]
Flat knots (up to 7 crossings) with same phi are :['6.282']
Arrow polynomial of the knot is: 8*K1**3 + 4*K1**2*K2 - 8*K1**2 - 6*K1*K2 - 2*K1*K3 - 3*K1 + 3*K2 + K3 + 4
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.143', '6.158', '6.264', '6.282', '6.501']
Outer characteristic polynomial of the knot is: t^7+78t^5+69t^3+8t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.282']
2-strand cable arrow polynomial of the knot is: -128*K1**6 + 512*K1**4*K2**3 - 1152*K1**4*K2**2 + 1344*K1**4*K2 - 2048*K1**4 - 384*K1**3*K2**2*K3 + 800*K1**3*K2*K3 - 864*K1**3*K3 + 640*K1**2*K2**5 - 2240*K1**2*K2**4 + 256*K1**2*K2**3*K3**2 + 4288*K1**2*K2**3 - 384*K1**2*K2**2*K3**2 + 64*K1**2*K2**2*K4 - 7376*K1**2*K2**2 + 192*K1**2*K2*K3**2 - 736*K1**2*K2*K4 + 7000*K1**2*K2 - 288*K1**2*K3**2 - 32*K1**2*K4**2 - 4308*K1**2 - 1024*K1*K2**4*K3 - 128*K1*K2**3*K3*K4 + 2752*K1*K2**3*K3 + 512*K1*K2**2*K3*K4 - 1120*K1*K2**2*K3 - 96*K1*K2**2*K5 - 288*K1*K2*K3*K4 + 5768*K1*K2*K3 - 32*K1*K3**2*K5 + 808*K1*K3*K4 + 152*K1*K4*K5 + 24*K1*K5*K6 - 576*K2**6 - 320*K2**4*K3**2 - 32*K2**4*K4**2 + 672*K2**4*K4 - 2192*K2**4 + 224*K2**3*K3*K5 + 32*K2**3*K4*K6 - 1040*K2**2*K3**2 - 248*K2**2*K4**2 + 1168*K2**2*K4 - 48*K2**2*K5**2 - 8*K2**2*K6**2 - 1802*K2**2 - 32*K2*K3**2*K4 + 480*K2*K3*K5 + 56*K2*K4*K6 + 8*K2*K5*K7 + 32*K3**2*K6 - 1620*K3**2 - 506*K4**2 - 168*K5**2 - 30*K6**2 + 3464
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.282']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.14087', 'vk6.14097', 'vk6.14278', 'vk6.14298', 'vk6.15507', 'vk6.15527', 'vk6.16009', 'vk6.16019', 'vk6.16267', 'vk6.16274', 'vk6.16280', 'vk6.22577', 'vk6.22587', 'vk6.34047', 'vk6.34090', 'vk6.34101', 'vk6.34487', 'vk6.34529', 'vk6.34539', 'vk6.34556', 'vk6.34570', 'vk6.42257', 'vk6.54062', 'vk6.54070', 'vk6.54283', 'vk6.54514', 'vk6.54551', 'vk6.54562', 'vk6.59007', 'vk6.64518', 'vk6.64524', 'vk6.64622']
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 O1O2O3O4O5U1U5O6U2U6U3U4
R3 orbit {'O1O2O3O4O5U1U5O6U2U6U3U4'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U2U3U6U4O6U1U5
Gauss code of K* O1O2O3O4U5U1U3U4U6O5O6U2
Gauss code of -K* O1O2O3O4U3O5O6U5U1U2U4U6
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 3 1 1],[ 4 0 2 3 4 1 1],[ 2 -2 0 2 3 0 1],[-1 -3 -2 0 1 0 0],[-3 -4 -3 -1 0 0 0],[-1 -1 0 0 0 0 0],[-1 -1 -1 0 0 0 0]]
Primitive based matrix [[ 0 3 1 1 1 -2 -4],[-3 0 0 0 -1 -3 -4],[-1 0 0 0 0 0 -1],[-1 0 0 0 0 -1 -1],[-1 1 0 0 0 -2 -3],[ 2 3 0 1 2 0 -2],[ 4 4 1 1 3 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-1,-1,-1,2,4,0,0,1,3,4,0,0,0,1,0,1,1,2,3,2]
Phi over symmetry [-4,-2,1,1,1,3,0,2,4,4,3,1,2,3,2,0,0,1,0,2,2]
Phi of -K [-4,-2,1,1,1,3,0,2,4,4,3,1,2,3,2,0,0,1,0,2,2]
Phi of K* [-3,-1,-1,-1,2,4,1,2,2,2,3,0,0,1,2,0,2,4,3,4,0]
Phi of -K* [-4,-2,1,1,1,3,2,1,1,3,4,0,1,2,3,0,0,0,0,0,1]
Symmetry type of based matrix c
u-polynomial t^4-t^3+t^2-3t
Normalized Jones-Krushkal polynomial 2z^2+15z+23
Enhanced Jones-Krushkal polynomial -2w^4z^2+4w^3z^2-6w^3z+21w^2z+23w
Inner characteristic polynomial t^6+46t^4+20t^2+1
Outer characteristic polynomial t^7+78t^5+69t^3+8t
Flat arrow polynomial 8*K1**3 + 4*K1**2*K2 - 8*K1**2 - 6*K1*K2 - 2*K1*K3 - 3*K1 + 3*K2 + K3 + 4
2-strand cable arrow polynomial -128*K1**6 + 512*K1**4*K2**3 - 1152*K1**4*K2**2 + 1344*K1**4*K2 - 2048*K1**4 - 384*K1**3*K2**2*K3 + 800*K1**3*K2*K3 - 864*K1**3*K3 + 640*K1**2*K2**5 - 2240*K1**2*K2**4 + 256*K1**2*K2**3*K3**2 + 4288*K1**2*K2**3 - 384*K1**2*K2**2*K3**2 + 64*K1**2*K2**2*K4 - 7376*K1**2*K2**2 + 192*K1**2*K2*K3**2 - 736*K1**2*K2*K4 + 7000*K1**2*K2 - 288*K1**2*K3**2 - 32*K1**2*K4**2 - 4308*K1**2 - 1024*K1*K2**4*K3 - 128*K1*K2**3*K3*K4 + 2752*K1*K2**3*K3 + 512*K1*K2**2*K3*K4 - 1120*K1*K2**2*K3 - 96*K1*K2**2*K5 - 288*K1*K2*K3*K4 + 5768*K1*K2*K3 - 32*K1*K3**2*K5 + 808*K1*K3*K4 + 152*K1*K4*K5 + 24*K1*K5*K6 - 576*K2**6 - 320*K2**4*K3**2 - 32*K2**4*K4**2 + 672*K2**4*K4 - 2192*K2**4 + 224*K2**3*K3*K5 + 32*K2**3*K4*K6 - 1040*K2**2*K3**2 - 248*K2**2*K4**2 + 1168*K2**2*K4 - 48*K2**2*K5**2 - 8*K2**2*K6**2 - 1802*K2**2 - 32*K2*K3**2*K4 + 480*K2*K3*K5 + 56*K2*K4*K6 + 8*K2*K5*K7 + 32*K3**2*K6 - 1620*K3**2 - 506*K4**2 - 168*K5**2 - 30*K6**2 + 3464
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{3, 6}, {2, 5}, {1, 4}], [{4, 6}, {3, 5}, {1, 2}]]
If K is slice False
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