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Flat knot 6.537

Min(phi) over symmetries of the knot is: [-3,-2,-1,2,2,2,0,2,2,2,4,1,1,1,2,1,2,2,0,0,-1]
Flat knots (up to 7 crossings) with same phi are :['6.537']
Arrow polynomial of the knot is: 4K13 - 6K1*2 - 6K1K2 - 2K1K3 - 2K2*2 + 4K2 + 2K3 + 2K4 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.537']
Outer characteristic polynomial of the knot is: t^7+71t^5+72t^3+5t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.537']
2-strand cable arrow polynomial of the knot is: -864K14 + 128K1*3K2K3 - 704K1*3K3 + 160K12K2*3 - 64K1*2K2*2K3*2 - 2240K1*2K2*2 + 96K1*2K2K32 + 32K1*2K2K3K5 - 288K12K2K4 + 6840K1*2K2 - 896K12K3*2 - 224K1*2K3K5 - 48K1*2K4*2 - 6784K1*2 + 256K1K23K3 - 832K1K2*2K3 - 32K1K2*2K5 + 64K1K2K33 - 448K1K2K3K4 + 6248K1K2K3 + 64K1K3*3K4 - 96K1K3*2K5 - 32K1K3K4K6 + 2136K1K3K4 + 440K1K4K5 + 48K1K5K6 - 32K2*6 + 64K2*4K4 - 536K24 + 32K2*3K3K5 - 672K2*2K3*2 - 56K2*2K4*2 + 1032K2*2K4 - 32K22K5*2 - 8K2*2K6*2 - 4572K2*2 - 64K2K32K4 - 32K2K3K4K5 + 1040K2K3K5 + 72K2K4K6 + 40K2K5K7 + 16K2K6K8 - 96K34 - 64K3*2K4*2 + 152K3*2K6 - 2852K32 + 48K3K4K7 - 8K44 + 16K4*2K8 - 1028K42 - 444K5*2 - 76K6*2 - 16K7*2 - 12K8**2 + 5126
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.537']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.71385', 'vk6.71446', 'vk6.71911', 'vk6.71972', 'vk6.72443', 'vk6.72590', 'vk6.72707', 'vk6.72800', 'vk6.72865', 'vk6.73022', 'vk6.74244', 'vk6.74359', 'vk6.74444', 'vk6.74873', 'vk6.75057', 'vk6.76629', 'vk6.76922', 'vk6.77050', 'vk6.77420', 'vk6.77743', 'vk6.77796', 'vk6.79290', 'vk6.79405', 'vk6.79764', 'vk6.79823', 'vk6.79900', 'vk6.80855', 'vk6.80922', 'vk6.81376', 'vk6.85514', 'vk6.87208', 'vk6.89272']
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,-1,2,2,2,0,2,2,2,4,1,1,1,2,1,2,2,0,0,-1]

Flat knots (up to 7 crossings) with same phi are :['6.537']

Arrow polynomial of the knot is: 4*K1**3 - 6*K1**2 - 6*K1*K2 - 2*K1*K3 - 2*K2**2 + 4*K2 + 2*K3 + 2*K4 + 5

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.537']

Outer characteristic polynomial of the knot is: t^7+71t^5+72t^3+5t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.537']

2-strand cable arrow polynomial of the knot is: -864*K1**4 + 128*K1**3*K2*K3 - 704*K1**3*K3 + 160*K1**2*K2**3 - 64*K1**2*K2**2*K3**2 - 2240*K1**2*K2**2 + 96*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 288*K1**2*K2*K4 + 6840*K1**2*K2 - 896*K1**2*K3**2 - 224*K1**2*K3*K5 - 48*K1**2*K4**2 - 6784*K1**2 + 256*K1*K2**3*K3 - 832*K1*K2**2*K3 - 32*K1*K2**2*K5 + 64*K1*K2*K3**3 - 448*K1*K2*K3*K4 + 6248*K1*K2*K3 + 64*K1*K3**3*K4 - 96*K1*K3**2*K5 - 32*K1*K3*K4*K6 + 2136*K1*K3*K4 + 440*K1*K4*K5 + 48*K1*K5*K6 - 32*K2**6 + 64*K2**4*K4 - 536*K2**4 + 32*K2**3*K3*K5 - 672*K2**2*K3**2 - 56*K2**2*K4**2 + 1032*K2**2*K4 - 32*K2**2*K5**2 - 8*K2**2*K6**2 - 4572*K2**2 - 64*K2*K3**2*K4 - 32*K2*K3*K4*K5 + 1040*K2*K3*K5 + 72*K2*K4*K6 + 40*K2*K5*K7 + 16*K2*K6*K8 - 96*K3**4 - 64*K3**2*K4**2 + 152*K3**2*K6 - 2852*K3**2 + 48*K3*K4*K7 - 8*K4**4 + 16*K4**2*K8 - 1028*K4**2 - 444*K5**2 - 76*K6**2 - 16*K7**2 - 12*K8**2 + 5126

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.537']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.71385', 'vk6.71446', 'vk6.71911', 'vk6.71972', 'vk6.72443', 'vk6.72590', 'vk6.72707', 'vk6.72800', 'vk6.72865', 'vk6.73022', 'vk6.74244', 'vk6.74359', 'vk6.74444', 'vk6.74873', 'vk6.75057', 'vk6.76629', 'vk6.76922', 'vk6.77050', 'vk6.77420', 'vk6.77743', 'vk6.77796', 'vk6.79290', 'vk6.79405', 'vk6.79764', 'vk6.79823', 'vk6.79900', 'vk6.80855', 'vk6.80922', 'vk6.81376', 'vk6.85514', 'vk6.87208', 'vk6.89272']

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	O1O2O3O4O5U3U1U5O6U2U4U6
R3 orbit	{'O1O2O3O4O5U3U1U5O6U2U4U6'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4O5U6U2U4O6U1U5U3
Gauss code of K*	O1O2O3U4O5O6O4U2U5U1U6U3
Gauss code of -K*	O1O2O3U1O4O5O6U4U2U6U3U5
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 -1 -2 2 2 2],[ 3 0 2 0 4 2 2],[ 1 -2 0 -1 2 1 2],[ 2 0 1 0 2 1 1],[-2 -4 -2 -2 0 0 1],[-2 -2 -1 -1 0 0 0],[-2 -2 -2 -1 -1 0 0]]
Primitive based matrix	[[ 0 2 2 2 -1 -2 -3],[-2 0 1 0 -2 -2 -4],[-2 -1 0 0 -2 -1 -2],[-2 0 0 0 -1 -1 -2],[ 1 2 2 1 0 -1 -2],[ 2 2 1 1 1 0 0],[ 3 4 2 2 2 0 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-2,-2,-2,1,2,3,-1,0,2,2,4,0,2,1,2,1,1,2,1,2,0]
Phi over symmetry	[-3,-2,-1,2,2,2,0,2,2,2,4,1,1,1,2,1,2,2,0,0,-1]
Phi of -K	[-3,-2,-1,2,2,2,1,0,1,3,3,0,2,3,3,1,1,2,-1,0,0]
Phi of K*	[-2,-2,-2,1,2,3,-1,0,1,3,3,0,1,2,1,2,3,3,0,0,1]
Phi of -K*	[-3,-2,-1,2,2,2,0,2,2,2,4,1,1,1,2,1,2,2,0,0,-1]
Symmetry type of based matrix	c
u-polynomial	t^3-2t^2+t
Normalized Jones-Krushkal polynomial	2z^2+21z+35
Enhanced Jones-Krushkal polynomial	2w^3z^2+21w^2z+35w
Inner characteristic polynomial	t^6+45t^4+16t^2
Outer characteristic polynomial	t^7+71t^5+72t^3+5t
Flat arrow polynomial	4K13 - 6K1*2 - 6K1K2 - 2K1K3 - 2K2*2 + 4K2 + 2K3 + 2K4 + 5
2-strand cable arrow polynomial	-864K14 + 128K1*3K2K3 - 704K1*3K3 + 160K12K2*3 - 64K1*2K2*2K3*2 - 2240K1*2K2*2 + 96K1*2K2K32 + 32K1*2K2K3K5 - 288K12K2K4 + 6840K1*2K2 - 896K12K3*2 - 224K1*2K3K5 - 48K1*2K4*2 - 6784K1*2 + 256K1K23K3 - 832K1K2*2K3 - 32K1K2*2K5 + 64K1K2K33 - 448K1K2K3K4 + 6248K1K2K3 + 64K1K3*3K4 - 96K1K3*2K5 - 32K1K3K4K6 + 2136K1K3K4 + 440K1K4K5 + 48K1K5K6 - 32K2*6 + 64K2*4K4 - 536K24 + 32K2*3K3K5 - 672K2*2K3*2 - 56K2*2K4*2 + 1032K2*2K4 - 32K22K5*2 - 8K2*2K6*2 - 4572K2*2 - 64K2K32K4 - 32K2K3K4K5 + 1040K2K3K5 + 72K2K4K6 + 40K2K5K7 + 16K2K6K8 - 96K34 - 64K3*2K4*2 + 152K3*2K6 - 2852K32 + 48K3K4K7 - 8K44 + 16K4*2K8 - 1028K42 - 444K5*2 - 76K6*2 - 16K7*2 - 12K8**2 + 5126
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}, {2, 5}, {1, 4}, {3}], [{6}, {2, 5}, {3, 4}, {1}], [{6}, {2, 5}, {4}, {1, 3}], [{6}, {2, 5}, {4}, {3}, {1}], [{6}, {3, 5}, {1, 4}, {2}], [{6}, {3, 5}, {2, 4}, {1}], [{6}, {3, 5}, {4}, {1, 2}], [{6}, {3, 5}, {4}, {2}, {1}], [{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}, {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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