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Flat knot 6.316

Min(phi) over symmetries of the knot is: [-4,-2,0,0,3,3,0,1,3,3,5,0,1,1,2,0,1,1,2,2,-1]
Flat knots (up to 7 crossings) with same phi are :['6.316']
Arrow polynomial of the knot is: 12*K1**3 - 8*K1**2 - 8*K1*K2 - 2*K1*K3 - 5*K1 + 5*K2 + K3 + K4 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.316']
Outer characteristic polynomial of the knot is: t^7+99t^5+51t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.316']
2-strand cable arrow polynomial of the knot is: -768*K1**4 + 320*K1**3*K2*K3 + 32*K1**3*K3*K4 - 384*K1**3*K3 + 416*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 - 4784*K1**2*K2**2 + 224*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 640*K1**2*K2*K4 + 6928*K1**2*K2 - 656*K1**2*K3**2 - 32*K1**2*K3*K5 - 64*K1**2*K4**2 - 5676*K1**2 + 992*K1*K2**3*K3 + 64*K1*K2**2*K3*K4 - 1120*K1*K2**2*K3 - 288*K1*K2**2*K5 + 96*K1*K2*K3**3 + 32*K1*K2*K3*K4**2 - 384*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 7088*K1*K2*K3 - 64*K1*K2*K4*K5 - 32*K1*K2*K4*K7 + 32*K1*K3**3*K4 + 1352*K1*K3*K4 + 256*K1*K4*K5 + 40*K1*K5*K6 + 16*K1*K6*K7 - 96*K2**6 + 128*K2**4*K4 - 1616*K2**4 + 32*K2**3*K3*K5 - 1184*K2**2*K3**2 - 144*K2**2*K4**2 + 1776*K2**2*K4 - 32*K2**2*K5**2 - 8*K2**2*K6**2 - 3666*K2**2 - 64*K2*K3**2*K4 + 784*K2*K3*K5 + 144*K2*K4*K6 + 24*K2*K5*K7 + 8*K2*K6*K8 - 48*K3**4 - 32*K3**2*K4**2 + 48*K3**2*K6 - 2172*K3**2 + 16*K3*K4*K7 - 750*K4**2 - 192*K5**2 - 54*K6**2 - 16*K7**2 - 2*K8**2 + 4278
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.316']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.71354', 'vk6.71406', 'vk6.71418', 'vk6.71865', 'vk6.71883', 'vk6.71928', 'vk6.71944', 'vk6.74316', 'vk6.74326', 'vk6.74961', 'vk6.74971', 'vk6.76532', 'vk6.76538', 'vk6.76939', 'vk6.77000', 'vk6.77014', 'vk6.77061', 'vk6.77071', 'vk6.77388', 'vk6.79370', 'vk6.79794', 'vk6.79804', 'vk6.80831', 'vk6.80835', 'vk6.81268', 'vk6.81467', 'vk6.81485', 'vk6.83842', 'vk6.87060', 'vk6.87078', 'vk6.88043', 'vk6.89561']
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 O1O2O3O4O5U2U4O6U1U3U6U5
R3 orbit {'O1O2O3O4O5U2U4O6U1U3U6U5'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U1U6U3U5O6U2U4
Gauss code of K* O1O2O3O4U1U5U2U6U4O5O6U3
Gauss code of -K* O1O2O3O4U2O5O6U1U5U3U6U4
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 -3 0 0 4 2],[ 3 0 -1 2 1 5 2],[ 3 1 0 2 1 3 1],[ 0 -2 -2 0 0 3 1],[ 0 -1 -1 0 0 1 0],[-4 -5 -3 -3 -1 0 0],[-2 -2 -1 -1 0 0 0]]
Primitive based matrix [[ 0 4 2 0 0 -3 -3],[-4 0 0 -1 -3 -3 -5],[-2 0 0 0 -1 -1 -2],[ 0 1 0 0 0 -1 -1],[ 0 3 1 0 0 -2 -2],[ 3 3 1 1 2 0 1],[ 3 5 2 1 2 -1 0]]
If based matrix primitive True
Phi of primitive based matrix [-4,-2,0,0,3,3,0,1,3,3,5,0,1,1,2,0,1,1,2,2,-1]
Phi over symmetry [-4,-2,0,0,3,3,0,1,3,3,5,0,1,1,2,0,1,1,2,2,-1]
Phi of -K [-3,-3,0,0,2,4,-1,1,2,4,4,1,2,3,2,0,1,1,2,3,2]
Phi of K* [-4,-2,0,0,3,3,2,1,3,2,4,1,2,3,4,0,1,1,2,2,-1]
Phi of -K* [-3,-3,0,0,2,4,-1,1,2,2,5,1,2,1,3,0,0,1,1,3,0]
Symmetry type of based matrix c
u-polynomial -t^4+2t^3-t^2
Normalized Jones-Krushkal polynomial 3z^2+20z+29
Enhanced Jones-Krushkal polynomial 3w^3z^2+20w^2z+29w
Inner characteristic polynomial t^6+61t^4+16t^2
Outer characteristic polynomial t^7+99t^5+51t^3+4t
Flat arrow polynomial 12*K1**3 - 8*K1**2 - 8*K1*K2 - 2*K1*K3 - 5*K1 + 5*K2 + K3 + K4 + 5
2-strand cable arrow polynomial -768*K1**4 + 320*K1**3*K2*K3 + 32*K1**3*K3*K4 - 384*K1**3*K3 + 416*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 - 4784*K1**2*K2**2 + 224*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 640*K1**2*K2*K4 + 6928*K1**2*K2 - 656*K1**2*K3**2 - 32*K1**2*K3*K5 - 64*K1**2*K4**2 - 5676*K1**2 + 992*K1*K2**3*K3 + 64*K1*K2**2*K3*K4 - 1120*K1*K2**2*K3 - 288*K1*K2**2*K5 + 96*K1*K2*K3**3 + 32*K1*K2*K3*K4**2 - 384*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 7088*K1*K2*K3 - 64*K1*K2*K4*K5 - 32*K1*K2*K4*K7 + 32*K1*K3**3*K4 + 1352*K1*K3*K4 + 256*K1*K4*K5 + 40*K1*K5*K6 + 16*K1*K6*K7 - 96*K2**6 + 128*K2**4*K4 - 1616*K2**4 + 32*K2**3*K3*K5 - 1184*K2**2*K3**2 - 144*K2**2*K4**2 + 1776*K2**2*K4 - 32*K2**2*K5**2 - 8*K2**2*K6**2 - 3666*K2**2 - 64*K2*K3**2*K4 + 784*K2*K3*K5 + 144*K2*K4*K6 + 24*K2*K5*K7 + 8*K2*K6*K8 - 48*K3**4 - 32*K3**2*K4**2 + 48*K3**2*K6 - 2172*K3**2 + 16*K3*K4*K7 - 750*K4**2 - 192*K5**2 - 54*K6**2 - 16*K7**2 - 2*K8**2 + 4278
Genus of based matrix 1
Fillings of based matrix [[{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {1, 4}, {3}, {2}]]
If K is slice False
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