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Flat knot 6.375

Min(phi) over symmetries of the knot is: [-4,-2,0,1,2,3,0,3,1,4,4,1,0,2,2,0,1,2,0,0,1]
Flat knots (up to 7 crossings) with same phi are :['6.375']
Arrow polynomial of the knot is: 12K13 + 4K1*2K2 - 6K12 - 10K1K2 - 2K1K3 - 4K1 + 2K2 + 2K3 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.375']
Outer characteristic polynomial of the knot is: t^7+91t^5+69t^3+7t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.375']
2-strand cable arrow polynomial of the knot is: 256K14K2*3 - 1536K1*4K2*2 + 2688K1*4K2 - 2528K14 - 128K1*3K2*2K3 + 640K13K2K3 - 1024K1*3K3 - 1216K12K2*4 + 4960K1*2K2*3 - 11648K1*2K2*2 - 1024K1*2K2K4 + 9304K1*2K2 - 288K12K3*2 - 4956K1*2 + 2752K1K23K3 + 224K1K2*2K3K4 - 3136K1K22K3 - 512K1K2*2K5 - 384K1K2K3K4 - 32K1K2K3K6 + 9584K1K2K3 + 1016K1K3K4 + 120K1K4K5 + 8K1K5K6 - 224K26 - 128K2*4K3*2 - 32K2*4K4*2 + 736K2*4K4 - 4184K24 + 288K2*3K3K5 + 32K2*3K4K6 - 128K2*3K6 - 1936K22K3*2 - 32K2*2K3K7 - 536K2*2K4*2 + 3336K2*2K4 - 96K22K5*2 - 8K2*2K6*2 - 2612K2*2 + 984K2K3K5 + 168K2K4K6 + 8K2K5K7 - 2076K32 - 720K4*2 - 152K5*2 - 20K6**2 + 4190
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.375']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16918', 'vk6.17162', 'vk6.20513', 'vk6.21897', 'vk6.23304', 'vk6.23605', 'vk6.27952', 'vk6.29431', 'vk6.35332', 'vk6.35766', 'vk6.39366', 'vk6.41546', 'vk6.42826', 'vk6.43110', 'vk6.45931', 'vk6.47618', 'vk6.55069', 'vk6.55320', 'vk6.57378', 'vk6.58537', 'vk6.59460', 'vk6.59753', 'vk6.62029', 'vk6.63025', 'vk6.64906', 'vk6.65121', 'vk6.66927', 'vk6.67778', 'vk6.68211', 'vk6.68357', 'vk6.69529', 'vk6.70235']
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: [-4,-2,0,1,2,3,0,3,1,4,4,1,0,2,2,0,1,2,0,0,1]

Flat knots (up to 7 crossings) with same phi are :['6.375']

Arrow polynomial of the knot is: 12*K1**3 + 4*K1**2*K2 - 6*K1**2 - 10*K1*K2 - 2*K1*K3 - 4*K1 + 2*K2 + 2*K3 + 3

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.375']

Outer characteristic polynomial of the knot is: t^7+91t^5+69t^3+7t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.375']

2-strand cable arrow polynomial of the knot is: 256*K1**4*K2**3 - 1536*K1**4*K2**2 + 2688*K1**4*K2 - 2528*K1**4 - 128*K1**3*K2**2*K3 + 640*K1**3*K2*K3 - 1024*K1**3*K3 - 1216*K1**2*K2**4 + 4960*K1**2*K2**3 - 11648*K1**2*K2**2 - 1024*K1**2*K2*K4 + 9304*K1**2*K2 - 288*K1**2*K3**2 - 4956*K1**2 + 2752*K1*K2**3*K3 + 224*K1*K2**2*K3*K4 - 3136*K1*K2**2*K3 - 512*K1*K2**2*K5 - 384*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 9584*K1*K2*K3 + 1016*K1*K3*K4 + 120*K1*K4*K5 + 8*K1*K5*K6 - 224*K2**6 - 128*K2**4*K3**2 - 32*K2**4*K4**2 + 736*K2**4*K4 - 4184*K2**4 + 288*K2**3*K3*K5 + 32*K2**3*K4*K6 - 128*K2**3*K6 - 1936*K2**2*K3**2 - 32*K2**2*K3*K7 - 536*K2**2*K4**2 + 3336*K2**2*K4 - 96*K2**2*K5**2 - 8*K2**2*K6**2 - 2612*K2**2 + 984*K2*K3*K5 + 168*K2*K4*K6 + 8*K2*K5*K7 - 2076*K3**2 - 720*K4**2 - 152*K5**2 - 20*K6**2 + 4190

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.375']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16918', 'vk6.17162', 'vk6.20513', 'vk6.21897', 'vk6.23304', 'vk6.23605', 'vk6.27952', 'vk6.29431', 'vk6.35332', 'vk6.35766', 'vk6.39366', 'vk6.41546', 'vk6.42826', 'vk6.43110', 'vk6.45931', 'vk6.47618', 'vk6.55069', 'vk6.55320', 'vk6.57378', 'vk6.58537', 'vk6.59460', 'vk6.59753', 'vk6.62029', 'vk6.63025', 'vk6.64906', 'vk6.65121', 'vk6.66927', 'vk6.67778', 'vk6.68211', 'vk6.68357', 'vk6.69529', 'vk6.70235']

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	O1O2O3O4O5U4U1O6U2U3U6U5
R3 orbit	{'O1O2O3O4O5U4U1O6U2U3U6U5'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4O5U1U6U3U4O6U5U2
Gauss code of K*	O1O2O3O4U5U1U2U6U4O6O5U3
Gauss code of -K*	O1O2O3O4U2O5O6U1U6U3U4U5
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 -2 0 -1 4 2],[ 3 0 1 2 0 4 2],[ 2 -1 0 1 0 4 2],[ 0 -2 -1 0 0 3 1],[ 1 0 0 0 0 1 0],[-4 -4 -4 -3 -1 0 0],[-2 -2 -2 -1 0 0 0]]
Primitive based matrix	[[ 0 4 2 0 -1 -2 -3],[-4 0 0 -3 -1 -4 -4],[-2 0 0 -1 0 -2 -2],[ 0 3 1 0 0 -1 -2],[ 1 1 0 0 0 0 0],[ 2 4 2 1 0 0 -1],[ 3 4 2 2 0 1 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-4,-2,0,1,2,3,0,3,1,4,4,1,0,2,2,0,1,2,0,0,1]
Phi over symmetry	[-4,-2,0,1,2,3,0,3,1,4,4,1,0,2,2,0,1,2,0,0,1]
Phi of -K	[-3,-2,-1,0,2,4,0,2,1,3,3,1,1,2,2,1,3,4,1,1,2]
Phi of K*	[-4,-2,0,1,2,3,2,1,4,2,3,1,3,2,3,1,1,1,1,2,0]
Phi of -K*	[-3,-2,-1,0,2,4,1,0,2,2,4,0,1,2,4,0,0,1,1,3,0]
Symmetry type of based matrix	c
u-polynomial	-t^4+t^3+t
Normalized Jones-Krushkal polynomial	8z^2+29z+27
Enhanced Jones-Krushkal polynomial	8w^3z^2+29w^2z+27w
Inner characteristic polynomial	t^6+57t^4+25t^2+1
Outer characteristic polynomial	t^7+91t^5+69t^3+7t
Flat arrow polynomial	12K13 + 4K1*2K2 - 6K12 - 10K1K2 - 2K1K3 - 4K1 + 2K2 + 2K3 + 3
2-strand cable arrow polynomial	256K14K2*3 - 1536K1*4K2*2 + 2688K1*4K2 - 2528K14 - 128K1*3K2*2K3 + 640K13K2K3 - 1024K1*3K3 - 1216K12K2*4 + 4960K1*2K2*3 - 11648K1*2K2*2 - 1024K1*2K2K4 + 9304K1*2K2 - 288K12K3*2 - 4956K1*2 + 2752K1K23K3 + 224K1K2*2K3K4 - 3136K1K22K3 - 512K1K2*2K5 - 384K1K2K3K4 - 32K1K2K3K6 + 9584K1K2K3 + 1016K1K3K4 + 120K1K4K5 + 8K1K5K6 - 224K26 - 128K2*4K3*2 - 32K2*4K4*2 + 736K2*4K4 - 4184K24 + 288K2*3K3K5 + 32K2*3K4K6 - 128K2*3K6 - 1936K22K3*2 - 32K2*2K3K7 - 536K2*2K4*2 + 3336K2*2K4 - 96K22K5*2 - 8K2*2K6*2 - 2612K2*2 + 984K2K3K5 + 168K2K4K6 + 8K2K5K7 - 2076K32 - 720K4*2 - 152K5*2 - 20K6**2 + 4190
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}], [{4, 6}, {5}, {3}, {2}, {1}], [{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}, {5}, {1, 4}, {2, 3}], [{6}, {5}, {2, 4}, {1, 3}], [{6}, {5}, {3, 4}, {1, 2}], [{6}, {5}, {3, 4}, {2}, {1}]]
If K is slice	False

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