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Flat knot 6.253

Min(phi) over symmetries of the knot is: [-4,-2,0,0,3,3,1,1,2,3,4,0,1,2,2,0,1,0,2,1,-1]
Flat knots (up to 7 crossings) with same phi are :['6.253']
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+119t^5+95t^3+10t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.253']
2-strand cable arrow polynomial of the knot is: -768*K1**4 + 352*K1**3*K2*K3 - 864*K1**3*K3 + 192*K1**2*K2**3 - 256*K1**2*K2**2*K3**2 - 3200*K1**2*K2**2 + 384*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 224*K1**2*K2*K4 + 4752*K1**2*K2 - 1760*K1**2*K3**2 - 96*K1**2*K3*K5 - 4028*K1**2 + 1632*K1*K2**3*K3 + 160*K1*K2**2*K3*K4 - 1088*K1*K2**2*K3 - 288*K1*K2**2*K5 + 256*K1*K2*K3**3 - 768*K1*K2*K3*K4 - 128*K1*K2*K3*K6 + 7392*K1*K2*K3 - 32*K1*K3**2*K5 + 1728*K1*K3*K4 + 64*K1*K4*K5 + 24*K1*K5*K6 + 8*K1*K6*K7 - 32*K2**6 + 32*K2**4*K4 - 1296*K2**4 + 64*K2**3*K3*K5 - 2480*K2**2*K3**2 - 32*K2**2*K3*K7 - 48*K2**2*K4**2 + 1176*K2**2*K4 - 32*K2**2*K5**2 - 8*K2**2*K6**2 - 2706*K2**2 - 64*K2*K3**2*K4 + 1624*K2*K3*K5 + 56*K2*K4*K6 + 16*K2*K5*K7 + 8*K2*K6*K8 - 64*K3**4 + 56*K3**2*K6 - 2376*K3**2 + 8*K3*K4*K7 - 442*K4**2 - 188*K5**2 - 30*K6**2 - 8*K7**2 - 2*K8**2 + 3290
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.253']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.81840', 'vk6.81894', 'vk6.82060', 'vk6.82086', 'vk6.82558', 'vk6.82616', 'vk6.82775', 'vk6.82788', 'vk6.82828', 'vk6.82842', 'vk6.82950', 'vk6.83058', 'vk6.83064', 'vk6.83271', 'vk6.83328', 'vk6.83362', 'vk6.83526', 'vk6.84536', 'vk6.84643', 'vk6.84906', 'vk6.84961', 'vk6.85828', 'vk6.86105', 'vk6.86118', 'vk6.86155', 'vk6.86840', 'vk6.88453', 'vk6.88890', 'vk6.89037', 'vk6.89700', 'vk6.89926', 'vk6.90016']
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 O1O2O3O4O5U1U2O6U4U3U5U6
R3 orbit {'O1O2O3O4O5U1U2O6U4U3U5U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U6U1U3U2O6U4U5
Gauss code of K* O1O2O3O4U5U6U2U1U3O5O6U4
Gauss code of -K* O1O2O3O4U1O5O6U2U4U3U5U6
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 -4 -2 0 0 3 3],[ 4 0 1 3 2 4 3],[ 2 -1 0 2 1 3 3],[ 0 -3 -2 0 0 2 3],[ 0 -2 -1 0 0 1 2],[-3 -4 -3 -2 -1 0 1],[-3 -3 -3 -3 -2 -1 0]]
Primitive based matrix [[ 0 3 3 0 0 -2 -4],[-3 0 1 -1 -2 -3 -4],[-3 -1 0 -2 -3 -3 -3],[ 0 1 2 0 0 -1 -2],[ 0 2 3 0 0 -2 -3],[ 2 3 3 1 2 0 -1],[ 4 4 3 2 3 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-3,0,0,2,4,-1,1,2,3,4,2,3,3,3,0,1,2,2,3,1]
Phi over symmetry [-4,-2,0,0,3,3,1,1,2,3,4,0,1,2,2,0,1,0,2,1,-1]
Phi of -K [-4,-2,0,0,3,3,1,1,2,3,4,0,1,2,2,0,1,0,2,1,-1]
Phi of K* [-3,-3,0,0,2,4,-1,0,1,2,4,1,2,2,3,0,0,1,1,2,1]
Phi of -K* [-4,-2,0,0,3,3,1,2,3,3,4,1,2,3,3,0,2,1,3,2,-1]
Symmetry type of based matrix c
u-polynomial t^4-2t^3+t^2
Normalized Jones-Krushkal polynomial 7z^2+26z+25
Enhanced Jones-Krushkal polynomial -2w^4z^2+9w^3z^2+26w^2z+25w
Inner characteristic polynomial t^6+81t^4+34t^2+1
Outer characteristic polynomial t^7+119t^5+95t^3+10t
Flat arrow polynomial 4*K1**3 - 4*K1*K2 - 2*K1*K3 - K1 + K2 + K3 + K4 + 1
2-strand cable arrow polynomial -768*K1**4 + 352*K1**3*K2*K3 - 864*K1**3*K3 + 192*K1**2*K2**3 - 256*K1**2*K2**2*K3**2 - 3200*K1**2*K2**2 + 384*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 224*K1**2*K2*K4 + 4752*K1**2*K2 - 1760*K1**2*K3**2 - 96*K1**2*K3*K5 - 4028*K1**2 + 1632*K1*K2**3*K3 + 160*K1*K2**2*K3*K4 - 1088*K1*K2**2*K3 - 288*K1*K2**2*K5 + 256*K1*K2*K3**3 - 768*K1*K2*K3*K4 - 128*K1*K2*K3*K6 + 7392*K1*K2*K3 - 32*K1*K3**2*K5 + 1728*K1*K3*K4 + 64*K1*K4*K5 + 24*K1*K5*K6 + 8*K1*K6*K7 - 32*K2**6 + 32*K2**4*K4 - 1296*K2**4 + 64*K2**3*K3*K5 - 2480*K2**2*K3**2 - 32*K2**2*K3*K7 - 48*K2**2*K4**2 + 1176*K2**2*K4 - 32*K2**2*K5**2 - 8*K2**2*K6**2 - 2706*K2**2 - 64*K2*K3**2*K4 + 1624*K2*K3*K5 + 56*K2*K4*K6 + 16*K2*K5*K7 + 8*K2*K6*K8 - 64*K3**4 + 56*K3**2*K6 - 2376*K3**2 + 8*K3*K4*K7 - 442*K4**2 - 188*K5**2 - 30*K6**2 - 8*K7**2 - 2*K8**2 + 3290
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {2, 5}, {4}, {3}], [{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {1, 5}, {4}, {3}], [{3, 6}, {1, 5}, {2, 4}], [{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {1, 5}, {3}, {2}], [{5, 6}, {3, 4}, {1, 2}], [{5, 6}, {4}, {3}, {1, 2}], [{6}, {1, 5}, {2, 4}, {3}]]
If K is slice False
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