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Flat knot 6.917

Min(phi) over symmetries of the knot is: [-3,-1,1,1,1,1,-1,1,1,2,3,0,1,0,1,0,0,0,0,-1,0]
Flat knots (up to 7 crossings) with same phi are :['6.917', '7.28398']
Arrow polynomial of the knot is: 4*K1**2*K2 - 2*K1**2 - 2*K1*K2 - 2*K1*K3 + K1 - 2*K2**2 + K3 + K4 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.128', '6.408', '6.452', '6.532', '6.867', '6.917', '6.938', '6.1164', '6.1173', '6.1174']
Outer characteristic polynomial of the knot is: t^7+33t^5+50t^3+8t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.917', '7.28398']
2-strand cable arrow polynomial of the knot is: -2624*K1**4 + 896*K1**3*K2*K3 + 256*K1**3*K3*K4 - 1024*K1**3*K3 + 384*K1**2*K2**2*K4 - 4080*K1**2*K2**2 + 32*K1**2*K2*K4**2 - 928*K1**2*K2*K4 + 5792*K1**2*K2 - 1792*K1**2*K3**2 - 64*K1**2*K3*K5 - 720*K1**2*K4**2 - 32*K1**2*K4*K6 - 2244*K1**2 + 1056*K1*K2**3*K3 + 384*K1*K2**2*K3*K4 - 288*K1*K2**2*K3 - 224*K1*K2**2*K5 + 32*K1*K2*K3*K4**2 - 1216*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 5672*K1*K2*K3 - 64*K1*K2*K4*K5 - 32*K1*K3*K4*K6 + 2264*K1*K3*K4 + 688*K1*K4*K5 + 16*K1*K5*K6 + 8*K1*K6*K7 - 32*K2**4*K4**2 + 448*K2**4*K4 - 1336*K2**4 + 32*K2**3*K4*K6 - 64*K2**3*K6 - 1040*K2**2*K3**2 + 32*K2**2*K4**3 - 936*K2**2*K4**2 + 1744*K2**2*K4 - 8*K2**2*K6**2 - 1930*K2**2 - 160*K2*K3**2*K4 + 736*K2*K3*K5 - 32*K2*K4**2*K6 + 496*K2*K4*K6 + 8*K2*K6*K8 - 16*K3**2*K4**2 - 1488*K3**2 + 16*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 924*K4**2 - 128*K5**2 - 38*K6**2 - 4*K7**2 - 2*K8**2 + 2540
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.917']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.54', 'vk6.109', 'vk6.206', 'vk6.255', 'vk6.382', 'vk6.791', 'vk6.798', 'vk6.1253', 'vk6.1344', 'vk6.1395', 'vk6.1539', 'vk6.2016', 'vk6.2409', 'vk6.2430', 'vk6.2675', 'vk6.2974', 'vk6.10446', 'vk6.10461', 'vk6.10689', 'vk6.10878', 'vk6.14657', 'vk6.16264', 'vk6.19172', 'vk6.25736', 'vk6.25886', 'vk6.30137', 'vk6.30372', 'vk6.30501', 'vk6.33423', 'vk6.33587', 'vk6.53791', 'vk6.63409']
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 O1O2O3O4U3U2O5O6U5U6U1U4
R3 orbit {'O1O2O3O4U3U2O5O6U5U6U1U4'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U1U4U5U6O5O6U3U2
Gauss code of K* O1O2O3O4U3U5U6U4O6O5U1U2
Gauss code of -K* O1O2O3O4U3U4O5O6U1U6U5U2
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 -1 -1 -1 3 -1 1],[ 1 0 0 0 3 -1 1],[ 1 0 0 0 2 0 0],[ 1 0 0 0 1 0 0],[-3 -3 -2 -1 0 -1 1],[ 1 1 0 0 1 0 1],[-1 -1 0 0 -1 -1 0]]
Primitive based matrix [[ 0 3 1 -1 -1 -1 -1],[-3 0 1 -1 -1 -2 -3],[-1 -1 0 0 -1 0 -1],[ 1 1 0 0 0 0 0],[ 1 1 1 0 0 0 1],[ 1 2 0 0 0 0 0],[ 1 3 1 0 -1 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-1,1,1,1,1,-1,1,1,2,3,0,1,0,1,0,0,0,0,-1,0]
Phi over symmetry [-3,-1,1,1,1,1,-1,1,1,2,3,0,1,0,1,0,0,0,0,-1,0]
Phi of -K [-1,-1,-1,-1,1,3,-1,0,0,1,3,0,0,1,1,0,2,2,2,3,3]
Phi of K* [-3,-1,1,1,1,1,3,1,2,3,3,1,2,1,2,0,-1,0,0,0,0]
Phi of -K* [-1,-1,-1,-1,1,3,-1,0,0,1,3,0,0,1,1,0,0,1,0,2,-1]
Symmetry type of based matrix c
u-polynomial -t^3+3t
Normalized Jones-Krushkal polynomial 7z^2+24z+21
Enhanced Jones-Krushkal polynomial -4w^4z^2+11w^3z^2+24w^2z+21w
Inner characteristic polynomial t^6+19t^4+24t^2
Outer characteristic polynomial t^7+33t^5+50t^3+8t
Flat arrow polynomial 4*K1**2*K2 - 2*K1**2 - 2*K1*K2 - 2*K1*K3 + K1 - 2*K2**2 + K3 + K4 + 2
2-strand cable arrow polynomial -2624*K1**4 + 896*K1**3*K2*K3 + 256*K1**3*K3*K4 - 1024*K1**3*K3 + 384*K1**2*K2**2*K4 - 4080*K1**2*K2**2 + 32*K1**2*K2*K4**2 - 928*K1**2*K2*K4 + 5792*K1**2*K2 - 1792*K1**2*K3**2 - 64*K1**2*K3*K5 - 720*K1**2*K4**2 - 32*K1**2*K4*K6 - 2244*K1**2 + 1056*K1*K2**3*K3 + 384*K1*K2**2*K3*K4 - 288*K1*K2**2*K3 - 224*K1*K2**2*K5 + 32*K1*K2*K3*K4**2 - 1216*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 5672*K1*K2*K3 - 64*K1*K2*K4*K5 - 32*K1*K3*K4*K6 + 2264*K1*K3*K4 + 688*K1*K4*K5 + 16*K1*K5*K6 + 8*K1*K6*K7 - 32*K2**4*K4**2 + 448*K2**4*K4 - 1336*K2**4 + 32*K2**3*K4*K6 - 64*K2**3*K6 - 1040*K2**2*K3**2 + 32*K2**2*K4**3 - 936*K2**2*K4**2 + 1744*K2**2*K4 - 8*K2**2*K6**2 - 1930*K2**2 - 160*K2*K3**2*K4 + 736*K2*K3*K5 - 32*K2*K4**2*K6 + 496*K2*K4*K6 + 8*K2*K6*K8 - 16*K3**2*K4**2 - 1488*K3**2 + 16*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 924*K4**2 - 128*K5**2 - 38*K6**2 - 4*K7**2 - 2*K8**2 + 2540
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {3, 5}, {2, 4}], [{1, 6}, {4, 5}, {2, 3}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {2, 4}, {1, 3}], [{5, 6}, {3, 4}, {1, 2}], [{5, 6}, {4}, {1, 3}, {2}]]
If K is slice False
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