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Flat knot 6.453

Min(phi) over symmetries of the knot is: [-4,-2,1,1,2,2,1,1,4,2,3,1,3,1,2,-1,0,0,0,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.453']
Arrow polynomial of the knot is: 4*K1**3 - 4*K1*K2 - 2*K1*K3 - K1 + K2 + K3 + K4 + 1
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.71', '6.148', '6.170', '6.253', '6.259', '6.298', '6.439', '6.453', '6.499', '6.503']
Outer characteristic polynomial of the knot is: t^7+95t^5+290t^3+19t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.453']
2-strand cable arrow polynomial of the knot is: -256*K1**4*K2**2 + 224*K1**4*K2 - 272*K1**4 + 384*K1**3*K2*K3 - 224*K1**3*K3 - 128*K1**2*K2**4 + 608*K1**2*K2**3 - 4128*K1**2*K2**2 - 352*K1**2*K2*K4 + 3752*K1**2*K2 - 144*K1**2*K3**2 - 2824*K1**2 + 1120*K1*K2**3*K3 + 288*K1*K2**2*K3*K4 - 960*K1*K2**2*K3 + 64*K1*K2**2*K4*K5 - 416*K1*K2**2*K5 - 192*K1*K2*K3*K4 + 4760*K1*K2*K3 - 32*K1*K2*K4*K5 + 528*K1*K3*K4 + 104*K1*K4*K5 + 8*K1*K5*K6 - 288*K2**6 + 832*K2**4*K4 - 3088*K2**4 + 128*K2**3*K3*K5 - 192*K2**3*K6 - 1088*K2**2*K3**2 - 32*K2**2*K3*K7 - 560*K2**2*K4**2 + 2928*K2**2*K4 - 208*K2**2*K5**2 - 8*K2**2*K6**2 - 1770*K2**2 + 928*K2*K3*K5 + 144*K2*K4*K6 + 80*K2*K5*K7 + 8*K2*K6*K8 + 8*K3**2*K6 - 1484*K3**2 - 766*K4**2 - 240*K5**2 - 14*K6**2 - 4*K7**2 - 2*K8**2 + 2774
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.453']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11422', 'vk6.11715', 'vk6.12732', 'vk6.13079', 'vk6.20609', 'vk6.22026', 'vk6.28082', 'vk6.29532', 'vk6.31167', 'vk6.31506', 'vk6.32329', 'vk6.32753', 'vk6.39489', 'vk6.41699', 'vk6.46085', 'vk6.47743', 'vk6.52182', 'vk6.52434', 'vk6.53010', 'vk6.53326', 'vk6.57485', 'vk6.58654', 'vk6.62165', 'vk6.63120', 'vk6.63751', 'vk6.63861', 'vk6.64177', 'vk6.64365', 'vk6.67009', 'vk6.67879', 'vk6.69632', 'vk6.70319']
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 O1O2O3O4O5U6U3O6U2U1U4U5
R3 orbit {'O1O2O3O4O5U6U3O6U2U1U4U5'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U1U2U5U4O6U3U6
Gauss code of K* O1O2O3O4U2U1U5U3U4O6O5U6
Gauss code of -K* O1O2O3O4U5O6O5U1U2U6U4U3
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 -2 -2 -1 2 4 -1],[ 2 0 0 1 3 4 1],[ 2 0 0 1 2 3 1],[ 1 -1 -1 0 0 1 1],[-2 -3 -2 0 0 1 -2],[-4 -4 -3 -1 -1 0 -4],[ 1 -1 -1 -1 2 4 0]]
Primitive based matrix [[ 0 4 2 -1 -1 -2 -2],[-4 0 -1 -1 -4 -3 -4],[-2 1 0 0 -2 -2 -3],[ 1 1 0 0 1 -1 -1],[ 1 4 2 -1 0 -1 -1],[ 2 3 2 1 1 0 0],[ 2 4 3 1 1 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-4,-2,1,1,2,2,1,1,4,3,4,0,2,2,3,-1,1,1,1,1,0]
Phi over symmetry [-4,-2,1,1,2,2,1,1,4,2,3,1,3,1,2,-1,0,0,0,0,0]
Phi of -K [-2,-2,-1,-1,2,4,0,0,0,1,2,0,0,2,3,-1,3,4,1,1,1]
Phi of K* [-4,-2,1,1,2,2,1,1,4,2,3,1,3,1,2,-1,0,0,0,0,0]
Phi of -K* [-2,-2,-1,-1,2,4,0,1,1,2,3,1,1,3,4,-1,2,4,0,1,1]
Symmetry type of based matrix c
u-polynomial -t^4+t^2+2t
Normalized Jones-Krushkal polynomial 5z^2+18z+17
Enhanced Jones-Krushkal polynomial -4w^4z^2+9w^3z^2-8w^3z+26w^2z+17w
Inner characteristic polynomial t^6+65t^4+155t^2+4
Outer characteristic polynomial t^7+95t^5+290t^3+19t
Flat arrow polynomial 4*K1**3 - 4*K1*K2 - 2*K1*K3 - K1 + K2 + K3 + K4 + 1
2-strand cable arrow polynomial -256*K1**4*K2**2 + 224*K1**4*K2 - 272*K1**4 + 384*K1**3*K2*K3 - 224*K1**3*K3 - 128*K1**2*K2**4 + 608*K1**2*K2**3 - 4128*K1**2*K2**2 - 352*K1**2*K2*K4 + 3752*K1**2*K2 - 144*K1**2*K3**2 - 2824*K1**2 + 1120*K1*K2**3*K3 + 288*K1*K2**2*K3*K4 - 960*K1*K2**2*K3 + 64*K1*K2**2*K4*K5 - 416*K1*K2**2*K5 - 192*K1*K2*K3*K4 + 4760*K1*K2*K3 - 32*K1*K2*K4*K5 + 528*K1*K3*K4 + 104*K1*K4*K5 + 8*K1*K5*K6 - 288*K2**6 + 832*K2**4*K4 - 3088*K2**4 + 128*K2**3*K3*K5 - 192*K2**3*K6 - 1088*K2**2*K3**2 - 32*K2**2*K3*K7 - 560*K2**2*K4**2 + 2928*K2**2*K4 - 208*K2**2*K5**2 - 8*K2**2*K6**2 - 1770*K2**2 + 928*K2*K3*K5 + 144*K2*K4*K6 + 80*K2*K5*K7 + 8*K2*K6*K8 + 8*K3**2*K6 - 1484*K3**2 - 766*K4**2 - 240*K5**2 - 14*K6**2 - 4*K7**2 - 2*K8**2 + 2774
Genus of based matrix 1
Fillings of based matrix [[{4, 6}, {2, 5}, {1, 3}]]
If K is slice False
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