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Flat knot 6.91

Min(phi) over symmetries of the knot is: [-4,-2,0,1,2,3,0,3,2,4,3,2,1,2,2,0,1,2,0,1,2]
Flat knots (up to 7 crossings) with same phi are :['6.91']
Arrow polynomial of the knot is: 4K13 + 4K1*2K2 - 6K12 - 6K1K2 - 2K1K3 + 2K2 + 2*K3 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.91', '6.202', '6.210']
Outer characteristic polynomial of the knot is: t^7+91t^5+62t^3+3t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.91']
2-strand cable arrow polynomial of the knot is: -64K14 + 128K1*3K2K3 - 192K1*3K3 - 128K12K2*4 + 576K1*2K2*3 - 128K1*2K2*2K3*2 - 2592K1*2K2*2 + 64K1*2K2K32 - 256K1*2K2K4 + 2488K1*2K2 - 272K12K3*2 - 32K1*2K3K5 - 1992K1*2 + 1504K1K23K3 + 224K1K2*2K3K4 - 928K1K22K3 + 32K1K2*2K4K5 - 128K1K22K5 + 64K1K2K33 - 288K1K2K3K4 - 64K1K2K3K6 + 3176K1K2K3 + 440K1K3K4 + 40K1K4K5 - 32K26 - 256K2*4K3*2 - 32K2*4K4*2 + 192K2*4K4 - 1464K24 + 128K2*3K3K5 + 32K2*3K4K6 - 32K2*3K6 + 64K22K3*2K4 - 1088K22K3*2 - 32K2*2K3K7 - 232K2*2K4*2 + 1120K2*2K4 - 32K22K5*2 - 8K2*2K6*2 - 872K2*2 + 464K2K3K5 + 72K2K4K6 + 8K2K5K7 - 16K34 + 8K3*2K6 - 880K32 - 228K4*2 - 40K5*2 - 8K6**2 + 1490
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.91']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.70462', 'vk6.70477', 'vk6.70526', 'vk6.70601', 'vk6.70632', 'vk6.70657', 'vk6.70755', 'vk6.70841', 'vk6.70913', 'vk6.70941', 'vk6.71003', 'vk6.71106', 'vk6.71152', 'vk6.71167', 'vk6.71238', 'vk6.71297', 'vk6.71315', 'vk6.71330', 'vk6.73547', 'vk6.74219', 'vk6.74358', 'vk6.75007', 'vk6.75304', 'vk6.76416', 'vk6.76578', 'vk6.76649', 'vk6.76976', 'vk6.78284', 'vk6.79263', 'vk6.79398', 'vk6.79941', 'vk6.80752', 'vk6.81493', 'vk6.83989', 'vk6.86355', 'vk6.86863', 'vk6.87278', 'vk6.88064', 'vk6.88239', 'vk6.88247']
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,-2,0,1,2,3,0,3,2,4,3,2,1,2,2,0,1,2,0,1,2]

Flat knots (up to 7 crossings) with same phi are :['6.91']

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

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.91', '6.202', '6.210']

Outer characteristic polynomial of the knot is: t^7+91t^5+62t^3+3t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.91']

2-strand cable arrow polynomial of the knot is: -64*K1**4 + 128*K1**3*K2*K3 - 192*K1**3*K3 - 128*K1**2*K2**4 + 576*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 - 2592*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 256*K1**2*K2*K4 + 2488*K1**2*K2 - 272*K1**2*K3**2 - 32*K1**2*K3*K5 - 1992*K1**2 + 1504*K1*K2**3*K3 + 224*K1*K2**2*K3*K4 - 928*K1*K2**2*K3 + 32*K1*K2**2*K4*K5 - 128*K1*K2**2*K5 + 64*K1*K2*K3**3 - 288*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 3176*K1*K2*K3 + 440*K1*K3*K4 + 40*K1*K4*K5 - 32*K2**6 - 256*K2**4*K3**2 - 32*K2**4*K4**2 + 192*K2**4*K4 - 1464*K2**4 + 128*K2**3*K3*K5 + 32*K2**3*K4*K6 - 32*K2**3*K6 + 64*K2**2*K3**2*K4 - 1088*K2**2*K3**2 - 32*K2**2*K3*K7 - 232*K2**2*K4**2 + 1120*K2**2*K4 - 32*K2**2*K5**2 - 8*K2**2*K6**2 - 872*K2**2 + 464*K2*K3*K5 + 72*K2*K4*K6 + 8*K2*K5*K7 - 16*K3**4 + 8*K3**2*K6 - 880*K3**2 - 228*K4**2 - 40*K5**2 - 8*K6**2 + 1490

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.91']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.70462', 'vk6.70477', 'vk6.70526', 'vk6.70601', 'vk6.70632', 'vk6.70657', 'vk6.70755', 'vk6.70841', 'vk6.70913', 'vk6.70941', 'vk6.71003', 'vk6.71106', 'vk6.71152', 'vk6.71167', 'vk6.71238', 'vk6.71297', 'vk6.71315', 'vk6.71330', 'vk6.73547', 'vk6.74219', 'vk6.74358', 'vk6.75007', 'vk6.75304', 'vk6.76416', 'vk6.76578', 'vk6.76649', 'vk6.76976', 'vk6.78284', 'vk6.79263', 'vk6.79398', 'vk6.79941', 'vk6.80752', 'vk6.81493', 'vk6.83989', 'vk6.86355', 'vk6.86863', 'vk6.87278', 'vk6.88064', 'vk6.88239', 'vk6.88247']

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	O1O2O3O4O5O6U2U5U6U1U3U4
R3 orbit	{'O1O2O3O4O5U1O6U5U2U6U3U4', 'O1O2O3O4O5O6U2U5U6U1U3U4'}
R3 orbit length	2
Gauss code of -K	O1O2O3O4O5O6U3U4U6U1U2U5
Gauss code of K*	O1O2O3O4O5O6U4U1U5U6U2U3
Gauss code of -K*	O1O2O3O4O5O6U4U5U1U2U6U3
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 -4 1 3 0 2],[ 2 0 -2 2 3 0 2],[ 4 2 0 3 4 1 2],[-1 -2 -3 0 1 -1 1],[-3 -3 -4 -1 0 -1 1],[ 0 0 -1 1 1 0 1],[-2 -2 -2 -1 -1 -1 0]]
Primitive based matrix	[[ 0 3 2 1 0 -2 -4],[-3 0 1 -1 -1 -3 -4],[-2 -1 0 -1 -1 -2 -2],[-1 1 1 0 -1 -2 -3],[ 0 1 1 1 0 0 -1],[ 2 3 2 2 0 0 -2],[ 4 4 2 3 1 2 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-3,-2,-1,0,2,4,-1,1,1,3,4,1,1,2,2,1,2,3,0,1,2]
Phi over symmetry	[-4,-2,0,1,2,3,0,3,2,4,3,2,1,2,2,0,1,2,0,1,2]
Phi of -K	[-4,-2,0,1,2,3,0,3,2,4,3,2,1,2,2,0,1,2,0,1,2]
Phi of K*	[-3,-2,-1,0,2,4,2,1,2,2,3,0,1,2,4,0,1,2,2,3,0]
Phi of -K*	[-4,-2,0,1,2,3,2,1,3,2,4,0,2,2,3,1,1,1,1,1,-1]
Symmetry type of based matrix	c
u-polynomial	t^4-t^3-t
Normalized Jones-Krushkal polynomial	5z^2+18z+17
Enhanced Jones-Krushkal polynomial	5w^3z^2+18w^2z+17w
Inner characteristic polynomial	t^6+57t^4+12t^2
Outer characteristic polynomial	t^7+91t^5+62t^3+3t
Flat arrow polynomial	4K13 + 4K1*2K2 - 6K12 - 6K1K2 - 2K1K3 + 2K2 + 2*K3 + 3
2-strand cable arrow polynomial	-64K14 + 128K1*3K2K3 - 192K1*3K3 - 128K12K2*4 + 576K1*2K2*3 - 128K1*2K2*2K3*2 - 2592K1*2K2*2 + 64K1*2K2K32 - 256K1*2K2K4 + 2488K1*2K2 - 272K12K3*2 - 32K1*2K3K5 - 1992K1*2 + 1504K1K23K3 + 224K1K2*2K3K4 - 928K1K22K3 + 32K1K2*2K4K5 - 128K1K22K5 + 64K1K2K33 - 288K1K2K3K4 - 64K1K2K3K6 + 3176K1K2K3 + 440K1K3K4 + 40K1K4K5 - 32K26 - 256K2*4K3*2 - 32K2*4K4*2 + 192K2*4K4 - 1464K24 + 128K2*3K3K5 + 32K2*3K4K6 - 32K2*3K6 + 64K22K3*2K4 - 1088K22K3*2 - 32K2*2K3K7 - 232K2*2K4*2 + 1120K2*2K4 - 32K22K5*2 - 8K2*2K6*2 - 872K2*2 + 464K2K3K5 + 72K2K4K6 + 8K2K5K7 - 16K34 + 8K3*2K6 - 880K32 - 228K4*2 - 40K5*2 - 8K6**2 + 1490
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}], [{2, 6}, {5}, {4}, {3}, {1}], [{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}, {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}, {4, 5}, {3}, {2}, {1}], [{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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