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Flat knot 6.365

Min(phi) over symmetries of the knot is: [-3,-2,0,1,1,3,-1,2,1,1,4,1,0,1,2,0,1,2,0,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.365']
Arrow polynomial of the knot is: 4*K1**3 + 4*K1**2*K2 - 8*K1**2 - 4*K1*K2 - 2*K1*K3 - K1 - 2*K2**2 + 3*K2 + K3 + K4 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.365', '6.451']
Outer characteristic polynomial of the knot is: t^7+58t^5+56t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.365']
2-strand cable arrow polynomial of the knot is: -64*K1**6 + 448*K1**4*K2 - 1776*K1**4 + 256*K1**3*K2*K3 - 1184*K1**3*K3 - 128*K1**2*K2**4 + 192*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 2464*K1**2*K2**2 + 128*K1**2*K2*K3**2 + 64*K1**2*K2*K4**2 - 384*K1**2*K2*K4 + 5664*K1**2*K2 - 1136*K1**2*K3**2 - 96*K1**2*K3*K5 - 240*K1**2*K4**2 - 96*K1**2*K4*K6 - 4276*K1**2 + 384*K1*K2**3*K3 - 512*K1*K2**2*K3 - 64*K1*K2**2*K5 + 96*K1*K2*K3**3 + 32*K1*K2*K3*K4**2 - 672*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 5976*K1*K2*K3 - 128*K1*K3**2*K5 - 32*K1*K3*K4*K6 + 1784*K1*K3*K4 + 368*K1*K4*K5 + 56*K1*K5*K6 - 32*K2**6 - 64*K2**4*K3**2 - 32*K2**4*K4**2 + 96*K2**4*K4 - 432*K2**4 + 32*K2**3*K3*K5 + 32*K2**3*K4*K6 + 64*K2**2*K3**2*K4 - 640*K2**2*K3**2 + 32*K2**2*K4**3 - 240*K2**2*K4**2 + 976*K2**2*K4 - 8*K2**2*K6**2 - 3346*K2**2 - 32*K2*K3*K4*K5 + 768*K2*K3*K5 - 32*K2*K4**2*K6 + 176*K2*K4*K6 + 8*K2*K6*K8 - 192*K3**4 - 80*K3**2*K4**2 + 192*K3**2*K6 - 2208*K3**2 + 64*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 742*K4**2 - 188*K5**2 - 70*K6**2 - 8*K7**2 - 2*K8**2 + 3614
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.365']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4069', 'vk6.4102', 'vk6.5307', 'vk6.5340', 'vk6.7435', 'vk6.7462', 'vk6.8930', 'vk6.8963', 'vk6.10125', 'vk6.10294', 'vk6.10319', 'vk6.14554', 'vk6.15265', 'vk6.15392', 'vk6.15774', 'vk6.16189', 'vk6.29865', 'vk6.29898', 'vk6.33911', 'vk6.33994', 'vk6.34221', 'vk6.34375', 'vk6.48456', 'vk6.49155', 'vk6.50203', 'vk6.50228', 'vk6.51601', 'vk6.53962', 'vk6.54025', 'vk6.54190', 'vk6.54463', 'vk6.63312']
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 O1O2O3O4O5U3U5O6U1U6U2U4
R3 orbit {'O1O2O3O4O5U3U5O6U1U6U2U4'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U2U4U6U5O6U1U3
Gauss code of K* O1O2O3O4U1U3U5U4U6O5O6U2
Gauss code of -K* O1O2O3O4U3O5O6U5U1U6U2U4
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 0 -2 3 1 1],[ 3 0 2 -1 4 1 1],[ 0 -2 0 -1 2 1 0],[ 2 1 1 0 2 1 0],[-3 -4 -2 -2 0 0 0],[-1 -1 -1 -1 0 0 0],[-1 -1 0 0 0 0 0]]
Primitive based matrix [[ 0 3 1 1 0 -2 -3],[-3 0 0 0 -2 -2 -4],[-1 0 0 0 0 0 -1],[-1 0 0 0 -1 -1 -1],[ 0 2 0 1 0 -1 -2],[ 2 2 0 1 1 0 1],[ 3 4 1 1 2 -1 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-1,-1,0,2,3,0,0,2,2,4,0,0,0,1,1,1,1,1,2,-1]
Phi over symmetry [-3,-2,0,1,1,3,-1,2,1,1,4,1,0,1,2,0,1,2,0,0,0]
Phi of -K [-3,-2,0,1,1,3,2,1,3,3,2,1,2,3,3,0,1,1,0,2,2]
Phi of K* [-3,-1,-1,0,2,3,2,2,1,3,2,0,0,2,3,1,3,3,1,1,2]
Phi of -K* [-3,-2,0,1,1,3,-1,2,1,1,4,1,0,1,2,0,1,2,0,0,0]
Symmetry type of based matrix c
u-polynomial t^2-2t
Normalized Jones-Krushkal polynomial 2z^2+19z+31
Enhanced Jones-Krushkal polynomial 2w^3z^2+19w^2z+31w
Inner characteristic polynomial t^6+34t^4+27t^2
Outer characteristic polynomial t^7+58t^5+56t^3+4t
Flat arrow polynomial 4*K1**3 + 4*K1**2*K2 - 8*K1**2 - 4*K1*K2 - 2*K1*K3 - K1 - 2*K2**2 + 3*K2 + K3 + K4 + 5
2-strand cable arrow polynomial -64*K1**6 + 448*K1**4*K2 - 1776*K1**4 + 256*K1**3*K2*K3 - 1184*K1**3*K3 - 128*K1**2*K2**4 + 192*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 2464*K1**2*K2**2 + 128*K1**2*K2*K3**2 + 64*K1**2*K2*K4**2 - 384*K1**2*K2*K4 + 5664*K1**2*K2 - 1136*K1**2*K3**2 - 96*K1**2*K3*K5 - 240*K1**2*K4**2 - 96*K1**2*K4*K6 - 4276*K1**2 + 384*K1*K2**3*K3 - 512*K1*K2**2*K3 - 64*K1*K2**2*K5 + 96*K1*K2*K3**3 + 32*K1*K2*K3*K4**2 - 672*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 5976*K1*K2*K3 - 128*K1*K3**2*K5 - 32*K1*K3*K4*K6 + 1784*K1*K3*K4 + 368*K1*K4*K5 + 56*K1*K5*K6 - 32*K2**6 - 64*K2**4*K3**2 - 32*K2**4*K4**2 + 96*K2**4*K4 - 432*K2**4 + 32*K2**3*K3*K5 + 32*K2**3*K4*K6 + 64*K2**2*K3**2*K4 - 640*K2**2*K3**2 + 32*K2**2*K4**3 - 240*K2**2*K4**2 + 976*K2**2*K4 - 8*K2**2*K6**2 - 3346*K2**2 - 32*K2*K3*K4*K5 + 768*K2*K3*K5 - 32*K2*K4**2*K6 + 176*K2*K4*K6 + 8*K2*K6*K8 - 192*K3**4 - 80*K3**2*K4**2 + 192*K3**2*K6 - 2208*K3**2 + 64*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 742*K4**2 - 188*K5**2 - 70*K6**2 - 8*K7**2 - 2*K8**2 + 3614
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {5}, {3, 4}, {2}]]
If K is slice False
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