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Flat knot 6.1066

Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,1,0,2,3,3,-1,1,0,1,0,1,0,0,1,0]
Flat knots (up to 7 crossings) with same phi are :['6.1066']
Arrow polynomial of the knot is: -6K12 - 2K1K2 + K1 + 3K2 + K3 + 4
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.323', '6.380', '6.444', '6.472', '6.523', '6.579', '6.592', '6.595', '6.609', '6.614', '6.620', '6.644', '6.648', '6.669', '6.671', '6.681', '6.693', '6.724', '6.725', '6.757', '6.766', '6.785', '6.786', '6.797', '6.798', '6.816', '6.833', '6.972', '6.978', '6.1056', '6.1064', '6.1066', '6.1087', '6.1094', '6.1273', '6.1277', '6.1282', '6.1295', '6.1300', '6.1313', '6.1344', '6.1353', '6.1354']
Outer characteristic polynomial of the knot is: t^7+44t^5+134t^3+16t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1066']
2-strand cable arrow polynomial of the knot is: -64K14K2*2 + 256K1*4K2 - 416K14 + 128K1*3K2K3 + 32K1*3K3K4 - 288K1*3K3 + 1408K12K2*3 - 5840K1*2K2*2 + 64K1*2K2K32 - 576K1*2K2K4 + 7088K1*2K2 - 256K12K3*2 - 32K1*2K3K5 - 48K1*2K4*2 - 5832K1*2 + 192K1K23K3 - 1248K1K2*2K3 - 64K1K2*2K5 - 224K1K2K3K4 + 6696K1K2K3 + 968K1K3K4 + 112K1K4K5 - 1416K2*4 - 304K2*2K3*2 - 8K2*2K4*2 + 1536K2*2K4 - 3630K22 + 280K2K3K5 + 8K2K4K6 - 1968K3*2 - 562K4*2 - 64K5*2 - 2K6**2 + 4072
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1066']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.73754', 'vk6.73755', 'vk6.73893', 'vk6.73895', 'vk6.75700', 'vk6.75701', 'vk6.75894', 'vk6.75896', 'vk6.78686', 'vk6.78688', 'vk6.78885', 'vk6.78886', 'vk6.80310', 'vk6.80312', 'vk6.80436', 'vk6.80437', 'vk6.81694', 'vk6.81702', 'vk6.81705', 'vk6.81802', 'vk6.82206', 'vk6.82449', 'vk6.82451', 'vk6.82465', 'vk6.82468', 'vk6.84425', 'vk6.84434', 'vk6.84443', 'vk6.87772', 'vk6.88112', 'vk6.88402', 'vk6.89637']
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

Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,1,0,2,3,3,-1,1,0,1,0,1,0,0,1,0]

Flat knots (up to 7 crossings) with same phi are :['6.1066']

Arrow polynomial of the knot is: -6*K1**2 - 2*K1*K2 + K1 + 3*K2 + K3 + 4

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.323', '6.380', '6.444', '6.472', '6.523', '6.579', '6.592', '6.595', '6.609', '6.614', '6.620', '6.644', '6.648', '6.669', '6.671', '6.681', '6.693', '6.724', '6.725', '6.757', '6.766', '6.785', '6.786', '6.797', '6.798', '6.816', '6.833', '6.972', '6.978', '6.1056', '6.1064', '6.1066', '6.1087', '6.1094', '6.1273', '6.1277', '6.1282', '6.1295', '6.1300', '6.1313', '6.1344', '6.1353', '6.1354']

Outer characteristic polynomial of the knot is: t^7+44t^5+134t^3+16t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1066']

2-strand cable arrow polynomial of the knot is: -64*K1**4*K2**2 + 256*K1**4*K2 - 416*K1**4 + 128*K1**3*K2*K3 + 32*K1**3*K3*K4 - 288*K1**3*K3 + 1408*K1**2*K2**3 - 5840*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 576*K1**2*K2*K4 + 7088*K1**2*K2 - 256*K1**2*K3**2 - 32*K1**2*K3*K5 - 48*K1**2*K4**2 - 5832*K1**2 + 192*K1*K2**3*K3 - 1248*K1*K2**2*K3 - 64*K1*K2**2*K5 - 224*K1*K2*K3*K4 + 6696*K1*K2*K3 + 968*K1*K3*K4 + 112*K1*K4*K5 - 1416*K2**4 - 304*K2**2*K3**2 - 8*K2**2*K4**2 + 1536*K2**2*K4 - 3630*K2**2 + 280*K2*K3*K5 + 8*K2*K4*K6 - 1968*K3**2 - 562*K4**2 - 64*K5**2 - 2*K6**2 + 4072

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1066']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.73754', 'vk6.73755', 'vk6.73893', 'vk6.73895', 'vk6.75700', 'vk6.75701', 'vk6.75894', 'vk6.75896', 'vk6.78686', 'vk6.78688', 'vk6.78885', 'vk6.78886', 'vk6.80310', 'vk6.80312', 'vk6.80436', 'vk6.80437', 'vk6.81694', 'vk6.81702', 'vk6.81705', 'vk6.81802', 'vk6.82206', 'vk6.82449', 'vk6.82451', 'vk6.82465', 'vk6.82468', 'vk6.84425', 'vk6.84434', 'vk6.84443', 'vk6.87772', 'vk6.88112', 'vk6.88402', 'vk6.89637']

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	O1O2O3O4U5U3O6U1O5U2U6U4
R3 orbit	{'O1O2O3O4U5U3O6U1O5U2U6U4'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4U1U5U3O6U4O5U2U6
Gauss code of K*	O1O2O3U4U1U5U3O6O5U2O4U6
Gauss code of -K*	O1O2O3U4O5U2O6O4U1U6U3U5
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 -1 0 3 -1 1],[ 2 0 0 0 3 1 1],[ 1 0 0 1 3 0 0],[ 0 0 -1 0 0 0 -1],[-3 -3 -3 0 0 -2 -1],[ 1 -1 0 0 2 0 1],[-1 -1 0 1 1 -1 0]]
Primitive based matrix	[[ 0 3 1 0 -1 -1 -2],[-3 0 -1 0 -2 -3 -3],[-1 1 0 1 -1 0 -1],[ 0 0 -1 0 0 -1 0],[ 1 2 1 0 0 0 -1],[ 1 3 0 1 0 0 0],[ 2 3 1 0 1 0 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-3,-1,0,1,1,2,1,0,2,3,3,-1,1,0,1,0,1,0,0,1,0]
Phi over symmetry	[-3,-1,0,1,1,2,1,0,2,3,3,-1,1,0,1,0,1,0,0,1,0]
Phi of -K	[-2,-1,-1,0,1,3,0,1,2,2,2,0,1,1,2,0,2,1,2,3,1]
Phi of K*	[-3,-1,0,1,1,2,1,3,1,2,2,2,2,1,2,0,1,2,0,1,0]
Phi of -K*	[-2,-1,-1,0,1,3,0,1,0,1,3,0,1,0,3,0,1,2,-1,0,1]
Symmetry type of based matrix	c
u-polynomial	-t^3+t^2+t
Normalized Jones-Krushkal polynomial	3z^2+16z+21
Enhanced Jones-Krushkal polynomial	-2w^4z^2+5w^3z^2-8w^3z+24w^2z+21w
Inner characteristic polynomial	t^6+28t^4+65t^2+1
Outer characteristic polynomial	t^7+44t^5+134t^3+16t
Flat arrow polynomial	-6K12 - 2K1K2 + K1 + 3K2 + K3 + 4
2-strand cable arrow polynomial	-64K14K2*2 + 256K1*4K2 - 416K14 + 128K1*3K2K3 + 32K1*3K3K4 - 288K1*3K3 + 1408K12K2*3 - 5840K1*2K2*2 + 64K1*2K2K32 - 576K1*2K2K4 + 7088K1*2K2 - 256K12K3*2 - 32K1*2K3K5 - 48K1*2K4*2 - 5832K1*2 + 192K1K23K3 - 1248K1K2*2K3 - 64K1K2*2K5 - 224K1K2K3K4 + 6696K1K2K3 + 968K1K3K4 + 112K1K4K5 - 1416K2*4 - 304K2*2K3*2 - 8K2*2K4*2 + 1536K2*2K4 - 3630K22 + 280K2K3K5 + 8K2K4K6 - 1968K3*2 - 562K4*2 - 64K5*2 - 2K6**2 + 4072
Genus of based matrix	2
Fillings of based matrix	[[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {2, 5}, {4}, {3}], [{1, 6}, {3, 5}, {2, 4}], [{1, 6}, {3, 5}, {4}, {2}], [{1, 6}, {4, 5}, {2, 3}], [{1, 6}, {4, 5}, {3}, {2}], [{1, 6}, {5}, {2, 4}, {3}], [{1, 6}, {5}, {3, 4}, {2}], [{1, 6}, {5}, {4}, {2, 3}], [{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {1, 5}, {4}, {3}], [{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {3, 5}, {4}, {1}], [{2, 6}, {4, 5}, {1, 3}], [{2, 6}, {4, 5}, {3}, {1}], [{2, 6}, {5}, {1, 4}, {3}], [{2, 6}, {5}, {3, 4}, {1}], [{2, 6}, {5}, {4}, {1, 3}], [{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {1, 5}, {4}, {2}], [{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {2, 5}, {4}, {1}], [{3, 6}, {4, 5}, {1, 2}], [{3, 6}, {4, 5}, {2}, {1}], [{3, 6}, {5}, {1, 4}, {2}], [{3, 6}, {5}, {2, 4}, {1}], [{3, 6}, {5}, {4}, {1, 2}], [{3, 6}, {5}, {4}, {2}, {1}], [{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {1, 5}, {3}, {2}], [{4, 6}, {2, 5}, {1, 3}], [{4, 6}, {2, 5}, {3}, {1}], [{4, 6}, {3, 5}, {1, 2}], [{4, 6}, {3, 5}, {2}, {1}], [{4, 6}, {5}, {1, 3}, {2}], [{4, 6}, {5}, {2, 3}, {1}], [{4, 6}, {5}, {3}, {1, 2}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {1, 4}, {3}, {2}], [{5, 6}, {2, 4}, {1, 3}], [{5, 6}, {2, 4}, {3}, {1}], [{5, 6}, {3, 4}, {1, 2}], [{5, 6}, {3, 4}, {2}, {1}], [{5, 6}, {4}, {1, 3}, {2}], [{5, 6}, {4}, {2, 3}, {1}], [{5, 6}, {4}, {3}, {1, 2}], [{6}, {1, 5}, {2, 4}, {3}], [{6}, {1, 5}, {3, 4}, {2}], [{6}, {1, 5}, {4}, {2, 3}], [{6}, {2, 5}, {1, 4}, {3}], [{6}, {2, 5}, {3, 4}, {1}], [{6}, {2, 5}, {4}, {1, 3}], [{6}, {3, 5}, {1, 4}, {2}], [{6}, {3, 5}, {2, 4}, {1}], [{6}, {3, 5}, {4}, {1, 2}], [{6}, {4, 5}, {1, 3}, {2}], [{6}, {4, 5}, {2, 3}, {1}], [{6}, {4, 5}, {3}, {1, 2}], [{6}, {4, 5}, {3}, {2}, {1}], [{6}, {5}, {1, 4}, {2, 3}], [{6}, {5}, {2, 4}, {1, 3}], [{6}, {5}, {3, 4}, {1, 2}]]
If K is slice	False

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