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Flat knot 6.350

Min(phi) over symmetries of the knot is: [-4,0,2,2,2,2,3,1,2,0]
Flat knots (up to 7 crossings) with same phi are :['6.350']
Arrow polynomial of the knot is: -2K12 - 4K1K2 - 2K1K3 + 2K1 + 2K2 + 2K3 + K4 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.180', '6.263', '6.295', '6.317', '6.350', '6.473', '6.504']
Outer characteristic polynomial of the knot is: t^5+46t^3+25t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.350']
2-strand cable arrow polynomial of the knot is: 1152K14K2 - 1936K14 + 576K1*3K2K3 - 736K1*3K3 + 128K12K2*2K4 - 2928K12K2*2 - 192K1*2K2K4 + 5128K1*2K2 - 784K12K3*2 - 96K1*2K3K5 - 32K1*2K4*2 - 4100K1*2 + 64K1K23K3 - 896K1K2*2K3 - 192K1K2*2K5 - 640K1K2K3K4 + 5584K1K2K3 - 32K1K32K5 + 1712K1K3K4 + 512K1K4K5 + 32K1K5K6 - 40K2*4 - 32K2*3K6 - 288K22K3*2 - 48K2*2K4*2 + 1160K2*2K4 - 16K22K5*2 - 8K2*2K6*2 - 4208K2*2 - 96K2K32K4 - 32K2K3K4K5 + 1144K2K3K5 + 272K2K4K6 + 24K2K5K7 + 8K2K6K8 - 64K34 + 152K3*2K6 - 2608K32 + 8K3K4K7 - 1112K42 - 544K5*2 - 160K6*2 - 4K7*2 - 2K8**2 + 4168
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.350']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.14116', 'vk6.14117', 'vk6.14337', 'vk6.14338', 'vk6.15567', 'vk6.15568', 'vk6.16039', 'vk6.16040', 'vk6.16436', 'vk6.16445', 'vk6.16449', 'vk6.22840', 'vk6.22844', 'vk6.34063', 'vk6.34127', 'vk6.34465', 'vk6.34500', 'vk6.34790', 'vk6.34809', 'vk6.34813', 'vk6.42403', 'vk6.42407', 'vk6.54081', 'vk6.54082', 'vk6.54311', 'vk6.54312', 'vk6.54669', 'vk6.54688', 'vk6.54692', 'vk6.64535', 'vk6.64536', 'vk6.64737']
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: [-4,0,2,2,2,2,3,1,2,0]

Flat knots (up to 7 crossings) with same phi are :['6.350']

Arrow polynomial of the knot is: -2*K1**2 - 4*K1*K2 - 2*K1*K3 + 2*K1 + 2*K2 + 2*K3 + K4 + 2

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.180', '6.263', '6.295', '6.317', '6.350', '6.473', '6.504']

Outer characteristic polynomial of the knot is: t^5+46t^3+25t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.350']

2-strand cable arrow polynomial of the knot is: 1152*K1**4*K2 - 1936*K1**4 + 576*K1**3*K2*K3 - 736*K1**3*K3 + 128*K1**2*K2**2*K4 - 2928*K1**2*K2**2 - 192*K1**2*K2*K4 + 5128*K1**2*K2 - 784*K1**2*K3**2 - 96*K1**2*K3*K5 - 32*K1**2*K4**2 - 4100*K1**2 + 64*K1*K2**3*K3 - 896*K1*K2**2*K3 - 192*K1*K2**2*K5 - 640*K1*K2*K3*K4 + 5584*K1*K2*K3 - 32*K1*K3**2*K5 + 1712*K1*K3*K4 + 512*K1*K4*K5 + 32*K1*K5*K6 - 40*K2**4 - 32*K2**3*K6 - 288*K2**2*K3**2 - 48*K2**2*K4**2 + 1160*K2**2*K4 - 16*K2**2*K5**2 - 8*K2**2*K6**2 - 4208*K2**2 - 96*K2*K3**2*K4 - 32*K2*K3*K4*K5 + 1144*K2*K3*K5 + 272*K2*K4*K6 + 24*K2*K5*K7 + 8*K2*K6*K8 - 64*K3**4 + 152*K3**2*K6 - 2608*K3**2 + 8*K3*K4*K7 - 1112*K4**2 - 544*K5**2 - 160*K6**2 - 4*K7**2 - 2*K8**2 + 4168

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.350']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.14116', 'vk6.14117', 'vk6.14337', 'vk6.14338', 'vk6.15567', 'vk6.15568', 'vk6.16039', 'vk6.16040', 'vk6.16436', 'vk6.16445', 'vk6.16449', 'vk6.22840', 'vk6.22844', 'vk6.34063', 'vk6.34127', 'vk6.34465', 'vk6.34500', 'vk6.34790', 'vk6.34809', 'vk6.34813', 'vk6.42403', 'vk6.42407', 'vk6.54081', 'vk6.54082', 'vk6.54311', 'vk6.54312', 'vk6.54669', 'vk6.54688', 'vk6.54692', 'vk6.64535', 'vk6.64536', 'vk6.64737']

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	O1O2O3O4O5U3U2O6U4U6U1U5
R3 orbit	{'O1O2O3O4O5U3U2O6U4U6U1U5'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4O5U1U5U6U2O6U4U3
Gauss code of K*	O1O2O3O4U3U5U6U1U4O6O5U2
Gauss code of -K*	O1O2O3O4U3O5O6U1U4U6U5U2
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 -1 -2 -2 0 4 1],[ 1 0 -1 -1 1 4 1],[ 2 1 0 0 2 3 1],[ 2 1 0 0 1 2 1],[ 0 -1 -2 -1 0 2 1],[-4 -4 -3 -2 -2 0 0],[-1 -1 -1 -1 -1 0 0]]
Primitive based matrix	[[ 0 4 0 -2 -2],[-4 0 -2 -2 -3],[ 0 2 0 -1 -2],[ 2 2 1 0 0],[ 2 3 2 0 0]]
If based matrix primitive	False
Phi of primitive based matrix	[-4,0,2,2,2,2,3,1,2,0]
Phi over symmetry	[-4,0,2,2,2,2,3,1,2,0]
Phi of -K	[-2,-2,0,4,0,0,3,1,4,2]
Phi of K*	[-4,0,2,2,2,3,4,0,1,0]
Phi of -K*	[-2,-2,0,4,0,1,2,2,3,2]
Symmetry type of based matrix	c
u-polynomial	-t^4+2t^2
Normalized Jones-Krushkal polynomial	6z^2+27z+31
Enhanced Jones-Krushkal polynomial	6w^3z^2+27w^2z+31w
Inner characteristic polynomial	t^4+22t^2+1
Outer characteristic polynomial	t^5+46t^3+25t
Flat arrow polynomial	-2K12 - 4K1K2 - 2K1K3 + 2K1 + 2K2 + 2K3 + K4 + 2
2-strand cable arrow polynomial	1152K14K2 - 1936K14 + 576K1*3K2K3 - 736K1*3K3 + 128K12K2*2K4 - 2928K12K2*2 - 192K1*2K2K4 + 5128K1*2K2 - 784K12K3*2 - 96K1*2K3K5 - 32K1*2K4*2 - 4100K1*2 + 64K1K23K3 - 896K1K2*2K3 - 192K1K2*2K5 - 640K1K2K3K4 + 5584K1K2K3 - 32K1K32K5 + 1712K1K3K4 + 512K1K4K5 + 32K1K5K6 - 40K2*4 - 32K2*3K6 - 288K22K3*2 - 48K2*2K4*2 + 1160K2*2K4 - 16K22K5*2 - 8K2*2K6*2 - 4208K2*2 - 96K2K32K4 - 32K2K3K4K5 + 1144K2K3K5 + 272K2K4K6 + 24K2K5K7 + 8K2K6K8 - 64K34 + 152K3*2K6 - 2608K32 + 8K3K4K7 - 1112K42 - 544K5*2 - 160K6*2 - 4K7*2 - 2K8**2 + 4168
Genus of based matrix	1
Fillings of based matrix	[[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {3, 5}, {2, 4}], [{1, 6}, {4, 5}, {2, 3}], [{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {4, 5}, {1, 3}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {2, 4}, {1, 3}]]
If K is slice	False

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