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Flat knot 6.333

Min(phi) over symmetries of the knot is: [-3,-2,0,1,1,3,-1,1,1,3,4,1,1,2,2,1,1,2,0,0,2]
Flat knots (up to 7 crossings) with same phi are :['6.333']
Arrow polynomial of the knot is: 8K13 + 4K1*2K2 - 8K12 - 8K1K2 - 2K1K3 - 2K1 - 2K22 + 3K2 + 2*K3 + K4 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.333']
Outer characteristic polynomial of the knot is: t^7+66t^5+61t^3+8t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.333']
2-strand cable arrow polynomial of the knot is: -64K16 + 928K1*4K2 - 3904K14 + 800K1*3K2K3 + 32K1*3K3K4 - 1216K1*3K3 - 128K12K2*4 + 512K1*2K2*3 + 160K1*2K2*2K4 - 5840K12K2*2 + 128K1*2K2K32 + 64K1*2K2K42 - 1024K1*2K2K4 + 10360K1*2K2 - 1664K12K3*2 - 32K1*2K3K5 - 464K1*2K4*2 - 32K1*2K4K6 - 6788K1*2 + 800K1K23K3 + 128K1K2*2K3K4 - 1056K1K22K3 + 64K1K2*2K4K5 - 736K1K22K5 + 64K1K2K33 + 32K1K2K3K42 - 576K1K2K3K4 - 64K1K2K3K6 + 9336K1K2K3 - 128K1K2K4K5 - 64K1K2K4K7 + 2936K1K3K4 + 840K1K4K5 + 56K1K5K6 + 8K1K6K7 - 64K26 - 64K2*4K3*2 - 32K2*4K4*2 + 128K2*4K4 - 1216K24 + 64K2*3K3K5 + 32K2*3K4K6 - 32K2*3K6 + 64K22K3*2K4 - 976K22K3*2 - 32K2*2K3K7 + 32K2*2K4*3 - 352K2*2K4*2 - 32K2*2K4K8 + 2016K2*2K4 - 96K22K5*2 - 8K2*2K6*2 - 5312K2*2 - 64K2K32K4 + 1112K2K3K5 + 280K2K4K6 + 40K2K5K7 + 8K2K6K8 - 96K34 - 80K3*2K4*2 + 48K3*2K6 - 3036K32 + 56K3K4K7 - 8K44 + 8K4*2K8 - 1378K42 - 400K5*2 - 40K6*2 - 8K7*2 - 2K8**2 + 5970
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.333']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11528', 'vk6.11861', 'vk6.12874', 'vk6.13183', 'vk6.20357', 'vk6.21698', 'vk6.27657', 'vk6.29201', 'vk6.31299', 'vk6.31696', 'vk6.32453', 'vk6.32870', 'vk6.39091', 'vk6.41345', 'vk6.45843', 'vk6.47508', 'vk6.52299', 'vk6.52565', 'vk6.53139', 'vk6.53445', 'vk6.57228', 'vk6.58453', 'vk6.61838', 'vk6.62973', 'vk6.63804', 'vk6.63938', 'vk6.64246', 'vk6.64444', 'vk6.66837', 'vk6.67705', 'vk6.69473', 'vk6.70195']
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,-2,0,1,1,3,-1,1,1,3,4,1,1,2,2,1,1,2,0,0,2]

Flat knots (up to 7 crossings) with same phi are :['6.333']

Arrow polynomial of the knot is: 8*K1**3 + 4*K1**2*K2 - 8*K1**2 - 8*K1*K2 - 2*K1*K3 - 2*K1 - 2*K2**2 + 3*K2 + 2*K3 + K4 + 5

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.333']

Outer characteristic polynomial of the knot is: t^7+66t^5+61t^3+8t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.333']

2-strand cable arrow polynomial of the knot is: -64*K1**6 + 928*K1**4*K2 - 3904*K1**4 + 800*K1**3*K2*K3 + 32*K1**3*K3*K4 - 1216*K1**3*K3 - 128*K1**2*K2**4 + 512*K1**2*K2**3 + 160*K1**2*K2**2*K4 - 5840*K1**2*K2**2 + 128*K1**2*K2*K3**2 + 64*K1**2*K2*K4**2 - 1024*K1**2*K2*K4 + 10360*K1**2*K2 - 1664*K1**2*K3**2 - 32*K1**2*K3*K5 - 464*K1**2*K4**2 - 32*K1**2*K4*K6 - 6788*K1**2 + 800*K1*K2**3*K3 + 128*K1*K2**2*K3*K4 - 1056*K1*K2**2*K3 + 64*K1*K2**2*K4*K5 - 736*K1*K2**2*K5 + 64*K1*K2*K3**3 + 32*K1*K2*K3*K4**2 - 576*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 9336*K1*K2*K3 - 128*K1*K2*K4*K5 - 64*K1*K2*K4*K7 + 2936*K1*K3*K4 + 840*K1*K4*K5 + 56*K1*K5*K6 + 8*K1*K6*K7 - 64*K2**6 - 64*K2**4*K3**2 - 32*K2**4*K4**2 + 128*K2**4*K4 - 1216*K2**4 + 64*K2**3*K3*K5 + 32*K2**3*K4*K6 - 32*K2**3*K6 + 64*K2**2*K3**2*K4 - 976*K2**2*K3**2 - 32*K2**2*K3*K7 + 32*K2**2*K4**3 - 352*K2**2*K4**2 - 32*K2**2*K4*K8 + 2016*K2**2*K4 - 96*K2**2*K5**2 - 8*K2**2*K6**2 - 5312*K2**2 - 64*K2*K3**2*K4 + 1112*K2*K3*K5 + 280*K2*K4*K6 + 40*K2*K5*K7 + 8*K2*K6*K8 - 96*K3**4 - 80*K3**2*K4**2 + 48*K3**2*K6 - 3036*K3**2 + 56*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 1378*K4**2 - 400*K5**2 - 40*K6**2 - 8*K7**2 - 2*K8**2 + 5970

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.333']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11528', 'vk6.11861', 'vk6.12874', 'vk6.13183', 'vk6.20357', 'vk6.21698', 'vk6.27657', 'vk6.29201', 'vk6.31299', 'vk6.31696', 'vk6.32453', 'vk6.32870', 'vk6.39091', 'vk6.41345', 'vk6.45843', 'vk6.47508', 'vk6.52299', 'vk6.52565', 'vk6.53139', 'vk6.53445', 'vk6.57228', 'vk6.58453', 'vk6.61838', 'vk6.62973', 'vk6.63804', 'vk6.63938', 'vk6.64246', 'vk6.64444', 'vk6.66837', 'vk6.67705', 'vk6.69473', 'vk6.70195']

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	O1O2O3O4O5U2U5O6U4U1U3U6
R3 orbit	{'O1O2O3O4O5U2U5O6U4U1U3U6'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4O5U6U3U5U2O6U1U4
Gauss code of K*	O1O2O3O4U2U5U3U1U6O5O6U4
Gauss code of -K*	O1O2O3O4U1O5O6U5U4U2U6U3
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 -3 1 0 1 3],[ 2 0 -2 2 1 1 3],[ 3 2 0 3 2 1 2],[-1 -2 -3 0 0 0 2],[ 0 -1 -2 0 0 0 1],[-1 -1 -1 0 0 0 0],[-3 -3 -2 -2 -1 0 0]]
Primitive based matrix	[[ 0 3 1 1 0 -2 -3],[-3 0 0 -2 -1 -3 -2],[-1 0 0 0 0 -1 -1],[-1 2 0 0 0 -2 -3],[ 0 1 0 0 0 -1 -2],[ 2 3 1 2 1 0 -2],[ 3 2 1 3 2 2 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-3,-1,-1,0,2,3,0,2,1,3,2,0,0,1,1,0,2,3,1,2,2]
Phi over symmetry	[-3,-2,0,1,1,3,-1,1,1,3,4,1,1,2,2,1,1,2,0,0,2]
Phi of -K	[-3,-2,0,1,1,3,-1,1,1,3,4,1,1,2,2,1,1,2,0,0,2]
Phi of K*	[-3,-1,-1,0,2,3,0,2,2,2,4,0,1,1,1,1,2,3,1,1,-1]
Phi of -K*	[-3,-2,0,1,1,3,2,2,1,3,2,1,1,2,3,0,0,1,0,0,2]
Symmetry type of based matrix	c
u-polynomial	t^2-2t
Normalized Jones-Krushkal polynomial	4z^2+25z+35
Enhanced Jones-Krushkal polynomial	4w^3z^2+25w^2z+35w
Inner characteristic polynomial	t^6+42t^4+20t^2+1
Outer characteristic polynomial	t^7+66t^5+61t^3+8t
Flat arrow polynomial	8K13 + 4K1*2K2 - 8K12 - 8K1K2 - 2K1K3 - 2K1 - 2K22 + 3K2 + 2*K3 + K4 + 5
2-strand cable arrow polynomial	-64K16 + 928K1*4K2 - 3904K14 + 800K1*3K2K3 + 32K1*3K3K4 - 1216K1*3K3 - 128K12K2*4 + 512K1*2K2*3 + 160K1*2K2*2K4 - 5840K12K2*2 + 128K1*2K2K32 + 64K1*2K2K42 - 1024K1*2K2K4 + 10360K1*2K2 - 1664K12K3*2 - 32K1*2K3K5 - 464K1*2K4*2 - 32K1*2K4K6 - 6788K1*2 + 800K1K23K3 + 128K1K2*2K3K4 - 1056K1K22K3 + 64K1K2*2K4K5 - 736K1K22K5 + 64K1K2K33 + 32K1K2K3K42 - 576K1K2K3K4 - 64K1K2K3K6 + 9336K1K2K3 - 128K1K2K4K5 - 64K1K2K4K7 + 2936K1K3K4 + 840K1K4K5 + 56K1K5K6 + 8K1K6K7 - 64K26 - 64K2*4K3*2 - 32K2*4K4*2 + 128K2*4K4 - 1216K24 + 64K2*3K3K5 + 32K2*3K4K6 - 32K2*3K6 + 64K22K3*2K4 - 976K22K3*2 - 32K2*2K3K7 + 32K2*2K4*3 - 352K2*2K4*2 - 32K2*2K4K8 + 2016K2*2K4 - 96K22K5*2 - 8K2*2K6*2 - 5312K2*2 - 64K2K32K4 + 1112K2K3K5 + 280K2K4K6 + 40K2K5K7 + 8K2K6K8 - 96K34 - 80K3*2K4*2 + 48K3*2K6 - 3036K32 + 56K3K4K7 - 8K44 + 8K4*2K8 - 1378K42 - 400K5*2 - 40K6*2 - 8K7*2 - 2K8**2 + 5970
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}, {5}, {1, 4}, {2, 3}], [{6}, {5}, {2, 4}, {1, 3}], [{6}, {5}, {2, 4}, {3}, {1}], [{6}, {5}, {3, 4}, {1, 2}]]
If K is slice	False

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