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Flat knot 6.1174

Min(phi) over symmetries of the knot is: [-3,-1,-1,1,2,2,-1,1,1,2,3,0,1,0,0,0,1,2,0,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.1174', '7.24085']
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+82t^5+174t^3+13t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1174', '7.24085']
2-strand cable arrow polynomial of the knot is: 256*K1**4*K2 - 1728*K1**4 - 256*K1**3*K2**2*K3 + 1920*K1**3*K2*K3 + 128*K1**3*K3*K4 - 1248*K1**3*K3 - 512*K1**2*K2**4 + 832*K1**2*K2**3 - 256*K1**2*K2**2*K3**2 + 256*K1**2*K2**2*K4 - 4912*K1**2*K2**2 + 256*K1**2*K2*K3**2 + 32*K1**2*K2*K4**2 - 736*K1**2*K2*K4 + 4704*K1**2*K2 - 2912*K1**2*K3**2 - 64*K1**2*K3*K5 - 240*K1**2*K4**2 - 32*K1**2*K4*K6 - 2188*K1**2 + 256*K1*K2**5*K3 - 256*K1*K2**3*K3*K4 + 2624*K1*K2**3*K3 + 160*K1*K2**2*K3*K4 - 896*K1*K2**2*K3 - 416*K1*K2**2*K5 + 384*K1*K2*K3**3 + 96*K1*K2*K3*K4**2 - 960*K1*K2*K3*K4 + 6472*K1*K2*K3 - 32*K1*K2*K4*K5 - 64*K1*K3**2*K5 - 32*K1*K3*K4*K6 + 2016*K1*K3*K4 + 256*K1*K4*K5 + 8*K1*K5*K6 + 8*K1*K6*K7 - 128*K2**6 - 256*K2**4*K3**2 - 32*K2**4*K4**2 + 160*K2**4*K4 - 1144*K2**4 + 64*K2**3*K3*K5 + 32*K2**3*K4*K6 + 256*K2**2*K3**2*K4 - 2352*K2**2*K3**2 + 32*K2**2*K4**3 - 232*K2**2*K4**2 + 872*K2**2*K4 - 8*K2**2*K6**2 - 1330*K2**2 - 96*K2*K3**2*K4 - 64*K2*K3*K4*K5 + 1128*K2*K3*K5 - 32*K2*K4**2*K6 + 136*K2*K4*K6 + 8*K2*K6*K8 - 128*K3**4 - 80*K3**2*K4**2 + 32*K3**2*K6 - 1392*K3**2 + 32*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 380*K4**2 - 64*K5**2 - 14*K6**2 - 4*K7**2 - 2*K8**2 + 2060
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1174']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.474', 'vk6.536', 'vk6.539', 'vk6.937', 'vk6.1032', 'vk6.1037', 'vk6.1624', 'vk6.1723', 'vk6.1732', 'vk6.2119', 'vk6.2220', 'vk6.2225', 'vk6.2532', 'vk6.2830', 'vk6.3033', 'vk6.3166', 'vk6.20387', 'vk6.20389', 'vk6.21728', 'vk6.21731', 'vk6.27713', 'vk6.27717', 'vk6.29257', 'vk6.29262', 'vk6.39157', 'vk6.39164', 'vk6.45885', 'vk6.45888', 'vk6.57251', 'vk6.57255', 'vk6.61886', 'vk6.61894']
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 O1O2O3O4U1U5U6O5O6U2U4U3
R3 orbit {'O1O2O3O4U1U5U6O5O6U2U4U3'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U2U1U3O5O6U5U6U4
Gauss code of K* O1O2O3U2U3O4O5O6U1U4U6U5
Gauss code of -K* O1O2O3U4U5O4O5O6U2U1U3U6
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 -1 2 2 -1 1],[ 3 0 1 3 2 3 3],[ 1 -1 0 2 1 0 2],[-2 -3 -2 0 0 -3 -1],[-2 -2 -1 0 0 -3 -1],[ 1 -3 0 3 3 0 1],[-1 -3 -2 1 1 -1 0]]
Primitive based matrix [[ 0 2 2 1 -1 -1 -3],[-2 0 0 -1 -1 -3 -2],[-2 0 0 -1 -2 -3 -3],[-1 1 1 0 -2 -1 -3],[ 1 1 2 2 0 0 -1],[ 1 3 3 1 0 0 -3],[ 3 2 3 3 1 3 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,-1,1,1,3,0,1,1,3,2,1,2,3,3,2,1,3,0,1,3]
Phi over symmetry [-3,-1,-1,1,2,2,-1,1,1,2,3,0,1,0,0,0,1,2,0,0,0]
Phi of -K [-3,-1,-1,1,2,2,-1,1,1,2,3,0,1,0,0,0,1,2,0,0,0]
Phi of K* [-2,-2,-1,1,1,3,0,0,0,1,2,0,0,2,3,1,0,1,0,-1,1]
Phi of -K* [-3,-1,-1,1,2,2,1,3,3,2,3,0,2,1,2,1,3,3,1,1,0]
Symmetry type of based matrix c
u-polynomial t^3-2t^2+t
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+62t^4+126t^2+1
Outer characteristic polynomial t^7+82t^5+174t^3+13t
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 256*K1**4*K2 - 1728*K1**4 - 256*K1**3*K2**2*K3 + 1920*K1**3*K2*K3 + 128*K1**3*K3*K4 - 1248*K1**3*K3 - 512*K1**2*K2**4 + 832*K1**2*K2**3 - 256*K1**2*K2**2*K3**2 + 256*K1**2*K2**2*K4 - 4912*K1**2*K2**2 + 256*K1**2*K2*K3**2 + 32*K1**2*K2*K4**2 - 736*K1**2*K2*K4 + 4704*K1**2*K2 - 2912*K1**2*K3**2 - 64*K1**2*K3*K5 - 240*K1**2*K4**2 - 32*K1**2*K4*K6 - 2188*K1**2 + 256*K1*K2**5*K3 - 256*K1*K2**3*K3*K4 + 2624*K1*K2**3*K3 + 160*K1*K2**2*K3*K4 - 896*K1*K2**2*K3 - 416*K1*K2**2*K5 + 384*K1*K2*K3**3 + 96*K1*K2*K3*K4**2 - 960*K1*K2*K3*K4 + 6472*K1*K2*K3 - 32*K1*K2*K4*K5 - 64*K1*K3**2*K5 - 32*K1*K3*K4*K6 + 2016*K1*K3*K4 + 256*K1*K4*K5 + 8*K1*K5*K6 + 8*K1*K6*K7 - 128*K2**6 - 256*K2**4*K3**2 - 32*K2**4*K4**2 + 160*K2**4*K4 - 1144*K2**4 + 64*K2**3*K3*K5 + 32*K2**3*K4*K6 + 256*K2**2*K3**2*K4 - 2352*K2**2*K3**2 + 32*K2**2*K4**3 - 232*K2**2*K4**2 + 872*K2**2*K4 - 8*K2**2*K6**2 - 1330*K2**2 - 96*K2*K3**2*K4 - 64*K2*K3*K4*K5 + 1128*K2*K3*K5 - 32*K2*K4**2*K6 + 136*K2*K4*K6 + 8*K2*K6*K8 - 128*K3**4 - 80*K3**2*K4**2 + 32*K3**2*K6 - 1392*K3**2 + 32*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 380*K4**2 - 64*K5**2 - 14*K6**2 - 4*K7**2 - 2*K8**2 + 2060
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {4, 5}, {1, 3}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {2, 4}, {1, 3}], [{5, 6}, {3, 4}, {1, 2}], [{5, 6}, {3, 4}, {2}, {1}]]
If K is slice False
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