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Flat knot 6.277

Min(phi) over symmetries of the knot is: [-4,0,0,1,1,2,1,3,1,3,4,1,0,0,0,0,0,1,0,-1,0]
Flat knots (up to 7 crossings) with same phi are :['6.277']
Arrow polynomial of the knot is: 8K13 - 8K1*2 - 8K1K2 - 2K1K3 - 2K1 + 5K2 + 2K3 + K4 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.277', '6.387']
Outer characteristic polynomial of the knot is: t^7+71t^5+83t^3+9t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.277']
2-strand cable arrow polynomial of the knot is: -64K16 - 192K1*4K2*2 + 1152K1*4K2 - 3072K14 + 288K1*3K2K3 + 32K1*3K3K4 - 928K1*3K3 - 256K12K2*4 + 1248K1*2K2*3 - 6512K1*2K2*2 - 608K1*2K2K4 + 10272K1*2K2 - 784K12K3*2 - 96K1*2K3K5 - 128K1*2K4*2 - 7400K1*2 + 864K1K23K3 - 1216K1K2*2K3 - 352K1K2*2K5 + 64K1K2K33 - 256K1K2K3K4 - 32K1K2K3K6 + 9032K1K2K3 - 96K1K2K4K5 - 32K1K3*2K5 + 1944K1K3K4 + 520K1K4K5 + 72K1K5K6 - 192K2*6 + 320K2*4K4 - 1584K24 + 32K2*3K3K5 - 64K2*3K6 - 928K22K3*2 - 32K2*2K3K7 - 128K2*2K4*2 + 1848K2*2K4 - 32K22K5*2 - 8K2*2K6*2 - 5356K2*2 - 32K2K3K4K5 + 1088K2K3K5 + 120K2K4K6 + 40K2K5K7 + 8K2K6K8 - 48K3*4 + 56K3*2K6 - 3160K32 + 16K3K4K7 - 1118K42 - 456K5*2 - 60K6*2 - 8K7*2 - 2K8**2 + 6214
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.277']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16582', 'vk6.16673', 'vk6.18139', 'vk6.18473', 'vk6.22981', 'vk6.23100', 'vk6.24594', 'vk6.25005', 'vk6.34974', 'vk6.35093', 'vk6.36729', 'vk6.37146', 'vk6.42543', 'vk6.42652', 'vk6.43997', 'vk6.44307', 'vk6.54813', 'vk6.54894', 'vk6.55941', 'vk6.56235', 'vk6.59241', 'vk6.59317', 'vk6.60475', 'vk6.60835', 'vk6.64787', 'vk6.64850', 'vk6.65594', 'vk6.65899', 'vk6.68085', 'vk6.68148', 'vk6.68665', 'vk6.68874']
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: [-4,0,0,1,1,2,1,3,1,3,4,1,0,0,0,0,0,1,0,-1,0]

Flat knots (up to 7 crossings) with same phi are :['6.277']

Arrow polynomial of the knot is: 8*K1**3 - 8*K1**2 - 8*K1*K2 - 2*K1*K3 - 2*K1 + 5*K2 + 2*K3 + K4 + 5

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.277', '6.387']

Outer characteristic polynomial of the knot is: t^7+71t^5+83t^3+9t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.277']

2-strand cable arrow polynomial of the knot is: -64*K1**6 - 192*K1**4*K2**2 + 1152*K1**4*K2 - 3072*K1**4 + 288*K1**3*K2*K3 + 32*K1**3*K3*K4 - 928*K1**3*K3 - 256*K1**2*K2**4 + 1248*K1**2*K2**3 - 6512*K1**2*K2**2 - 608*K1**2*K2*K4 + 10272*K1**2*K2 - 784*K1**2*K3**2 - 96*K1**2*K3*K5 - 128*K1**2*K4**2 - 7400*K1**2 + 864*K1*K2**3*K3 - 1216*K1*K2**2*K3 - 352*K1*K2**2*K5 + 64*K1*K2*K3**3 - 256*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 9032*K1*K2*K3 - 96*K1*K2*K4*K5 - 32*K1*K3**2*K5 + 1944*K1*K3*K4 + 520*K1*K4*K5 + 72*K1*K5*K6 - 192*K2**6 + 320*K2**4*K4 - 1584*K2**4 + 32*K2**3*K3*K5 - 64*K2**3*K6 - 928*K2**2*K3**2 - 32*K2**2*K3*K7 - 128*K2**2*K4**2 + 1848*K2**2*K4 - 32*K2**2*K5**2 - 8*K2**2*K6**2 - 5356*K2**2 - 32*K2*K3*K4*K5 + 1088*K2*K3*K5 + 120*K2*K4*K6 + 40*K2*K5*K7 + 8*K2*K6*K8 - 48*K3**4 + 56*K3**2*K6 - 3160*K3**2 + 16*K3*K4*K7 - 1118*K4**2 - 456*K5**2 - 60*K6**2 - 8*K7**2 - 2*K8**2 + 6214

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.277']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16582', 'vk6.16673', 'vk6.18139', 'vk6.18473', 'vk6.22981', 'vk6.23100', 'vk6.24594', 'vk6.25005', 'vk6.34974', 'vk6.35093', 'vk6.36729', 'vk6.37146', 'vk6.42543', 'vk6.42652', 'vk6.43997', 'vk6.44307', 'vk6.54813', 'vk6.54894', 'vk6.55941', 'vk6.56235', 'vk6.59241', 'vk6.59317', 'vk6.60475', 'vk6.60835', 'vk6.64787', 'vk6.64850', 'vk6.65594', 'vk6.65899', 'vk6.68085', 'vk6.68148', 'vk6.68665', 'vk6.68874']

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	O1O2O3O4O5U1U4O6U5U3U6U2
R3 orbit	{'O1O2O3O4O5U1U4O6U5U3U6U2'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4O5U4U6U3U1O6U2U5
Gauss code of K*	O1O2O3O4U5U4U2U6U1O5O6U3
Gauss code of -K*	O1O2O3O4U2O5O6U4U5U3U1U6
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 1 0 0 1 2],[ 4 0 4 3 1 2 2],[-1 -4 0 -1 -1 0 2],[ 0 -3 1 0 -1 1 2],[ 0 -1 1 1 0 1 1],[-1 -2 0 -1 -1 0 1],[-2 -2 -2 -2 -1 -1 0]]
Primitive based matrix	[[ 0 2 1 1 0 0 -4],[-2 0 -1 -2 -1 -2 -2],[-1 1 0 0 -1 -1 -2],[-1 2 0 0 -1 -1 -4],[ 0 1 1 1 0 1 -1],[ 0 2 1 1 -1 0 -3],[ 4 2 2 4 1 3 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-2,-1,-1,0,0,4,1,2,1,2,2,0,1,1,2,1,1,4,-1,1,3]
Phi over symmetry	[-4,0,0,1,1,2,1,3,1,3,4,1,0,0,0,0,0,1,0,-1,0]
Phi of -K	[-4,0,0,1,1,2,1,3,1,3,4,1,0,0,0,0,0,1,0,-1,0]
Phi of K*	[-2,-1,-1,0,0,4,-1,0,0,1,4,0,0,0,1,0,0,3,-1,1,3]
Phi of -K*	[-4,0,0,1,1,2,1,3,2,4,2,1,1,1,1,1,1,2,0,1,2]
Symmetry type of based matrix	c
u-polynomial	t^4-t^2-2t
Normalized Jones-Krushkal polynomial	z^2+20z+37
Enhanced Jones-Krushkal polynomial	w^3z^2-2w^3z+22w^2z+37w
Inner characteristic polynomial	t^6+49t^4+18t^2
Outer characteristic polynomial	t^7+71t^5+83t^3+9t
Flat arrow polynomial	8K13 - 8K1*2 - 8K1K2 - 2K1K3 - 2K1 + 5K2 + 2K3 + K4 + 5
2-strand cable arrow polynomial	-64K16 - 192K1*4K2*2 + 1152K1*4K2 - 3072K14 + 288K1*3K2K3 + 32K1*3K3K4 - 928K1*3K3 - 256K12K2*4 + 1248K1*2K2*3 - 6512K1*2K2*2 - 608K1*2K2K4 + 10272K1*2K2 - 784K12K3*2 - 96K1*2K3K5 - 128K1*2K4*2 - 7400K1*2 + 864K1K23K3 - 1216K1K2*2K3 - 352K1K2*2K5 + 64K1K2K33 - 256K1K2K3K4 - 32K1K2K3K6 + 9032K1K2K3 - 96K1K2K4K5 - 32K1K3*2K5 + 1944K1K3K4 + 520K1K4K5 + 72K1K5K6 - 192K2*6 + 320K2*4K4 - 1584K24 + 32K2*3K3K5 - 64K2*3K6 - 928K22K3*2 - 32K2*2K3K7 - 128K2*2K4*2 + 1848K2*2K4 - 32K22K5*2 - 8K2*2K6*2 - 5356K2*2 - 32K2K3K4K5 + 1088K2K3K5 + 120K2K4K6 + 40K2K5K7 + 8K2K6K8 - 48K3*4 + 56K3*2K6 - 3160K32 + 16K3K4K7 - 1118K42 - 456K5*2 - 60K6*2 - 8K7*2 - 2K8**2 + 6214
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}], [{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}, {1, 5}, {4}, {3}, {2}], [{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}, {1, 4}, {3}, {2}], [{6}, {5}, {2, 4}, {1, 3}], [{6}, {5}, {3, 4}, {1, 2}], [{6}, {5}, {4}, {2, 3}, {1}]]
If K is slice	False

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