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Flat knot 6.26

Min(phi) over symmetries of the knot is: [-5,-2,0,1,2,4,1,4,3,2,5,2,2,1,3,1,1,3,0,1,0]
Flat knots (up to 7 crossings) with same phi are :['6.26']
Arrow polynomial of the knot is: -2K12 - 2K1K3 + K1 - 2K2K3 + 2K2 + K4 + K5 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.26']
Outer characteristic polynomial of the knot is: t^7+135t^5+109t^3
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.26']
2-strand cable arrow polynomial of the knot is: -528K14 + 64K1*3K2K3 + 32K1*3K3K4 - 368K1*2K2*2 + 832K1*2K2 - 864K12K3*2 - 64K1*2K4*2 - 32K1*2K6*2 - 1332K1*2 + 96K1K2K3*3 + 2104K1K2K3 + 64K1K3*3K4 + 1248K1K3K4 + 48K1K4K5 + 64K1K5K6 + 48K1K6K7 - 2K102 + 8K10K4K6 - 24K24 - 224K2*2K3*2 + 56K2*2K4 - 8K22K6*2 - 1058K2*2 + 176K2K3K5 + 64K2K4K6 + 8K2K6K8 - 400K34 - 128K3*2K4*2 - 32K3*2K6*2 + 328K3*2K6 - 1116K32 + 80K3K4K7 + 16K3K6K9 - 8K4*2K6*2 - 484K4*2 - 80K5*2 - 148K6*2 - 32K7*2 - 2K8**2 + 1596
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.26']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.73290', 'vk6.73298', 'vk6.73431', 'vk6.73439', 'vk6.74089', 'vk6.74105', 'vk6.74658', 'vk6.74674', 'vk6.75431', 'vk6.75439', 'vk6.76123', 'vk6.76139', 'vk6.78163', 'vk6.78171', 'vk6.78393', 'vk6.78401', 'vk6.79091', 'vk6.79107', 'vk6.79984', 'vk6.79992', 'vk6.80135', 'vk6.80143', 'vk6.80595', 'vk6.80611', 'vk6.83795', 'vk6.83797', 'vk6.85109', 'vk6.85113', 'vk6.86595', 'vk6.86599', 'vk6.87378', 'vk6.87386']
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: [-5,-2,0,1,2,4,1,4,3,2,5,2,2,1,3,1,1,3,0,1,0]

Flat knots (up to 7 crossings) with same phi are :['6.26']

Arrow polynomial of the knot is: -2*K1**2 - 2*K1*K3 + K1 - 2*K2*K3 + 2*K2 + K4 + K5 + 2

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.26']

Outer characteristic polynomial of the knot is: t^7+135t^5+109t^3

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.26']

2-strand cable arrow polynomial of the knot is: -528*K1**4 + 64*K1**3*K2*K3 + 32*K1**3*K3*K4 - 368*K1**2*K2**2 + 832*K1**2*K2 - 864*K1**2*K3**2 - 64*K1**2*K4**2 - 32*K1**2*K6**2 - 1332*K1**2 + 96*K1*K2*K3**3 + 2104*K1*K2*K3 + 64*K1*K3**3*K4 + 1248*K1*K3*K4 + 48*K1*K4*K5 + 64*K1*K5*K6 + 48*K1*K6*K7 - 2*K10**2 + 8*K10*K4*K6 - 24*K2**4 - 224*K2**2*K3**2 + 56*K2**2*K4 - 8*K2**2*K6**2 - 1058*K2**2 + 176*K2*K3*K5 + 64*K2*K4*K6 + 8*K2*K6*K8 - 400*K3**4 - 128*K3**2*K4**2 - 32*K3**2*K6**2 + 328*K3**2*K6 - 1116*K3**2 + 80*K3*K4*K7 + 16*K3*K6*K9 - 8*K4**2*K6**2 - 484*K4**2 - 80*K5**2 - 148*K6**2 - 32*K7**2 - 2*K8**2 + 1596

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.26']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.73290', 'vk6.73298', 'vk6.73431', 'vk6.73439', 'vk6.74089', 'vk6.74105', 'vk6.74658', 'vk6.74674', 'vk6.75431', 'vk6.75439', 'vk6.76123', 'vk6.76139', 'vk6.78163', 'vk6.78171', 'vk6.78393', 'vk6.78401', 'vk6.79091', 'vk6.79107', 'vk6.79984', 'vk6.79992', 'vk6.80135', 'vk6.80143', 'vk6.80595', 'vk6.80611', 'vk6.83795', 'vk6.83797', 'vk6.85109', 'vk6.85113', 'vk6.86595', 'vk6.86599', 'vk6.87378', 'vk6.87386']

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	O1O2O3O4O5O6U1U3U6U4U2U5
R3 orbit	{'O1O2O3O4O5O6U1U3U6U4U2U5'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4O5O6U2U5U3U1U4U6
Gauss code of K*	O1O2O3O4O5O6U1U5U2U4U6U3
Gauss code of -K*	O1O2O3O4O5O6U4U1U3U5U2U6
Diagrammatic symmetry type	c
Flat genus of the diagram	2
If K is checkerboard colorable	False
If K is almost classical	False
Based matrix from Gauss code	[[ 0 -5 0 -2 1 4 2],[ 5 0 4 1 3 5 2],[ 0 -4 0 -2 1 3 1],[ 2 -1 2 0 2 3 1],[-1 -3 -1 -2 0 1 0],[-4 -5 -3 -3 -1 0 0],[-2 -2 -1 -1 0 0 0]]
Primitive based matrix	[[ 0 4 2 1 0 -2 -5],[-4 0 0 -1 -3 -3 -5],[-2 0 0 0 -1 -1 -2],[-1 1 0 0 -1 -2 -3],[ 0 3 1 1 0 -2 -4],[ 2 3 1 2 2 0 -1],[ 5 5 2 3 4 1 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-4,-2,-1,0,2,5,0,1,3,3,5,0,1,1,2,1,2,3,2,4,1]
Phi over symmetry	[-5,-2,0,1,2,4,1,4,3,2,5,2,2,1,3,1,1,3,0,1,0]
Phi of -K	[-5,-2,0,1,2,4,2,1,3,5,4,0,1,3,3,0,1,1,1,2,2]
Phi of K*	[-4,-2,-1,0,2,5,2,2,1,3,4,1,1,3,5,0,1,3,0,1,2]
Phi of -K*	[-5,-2,0,1,2,4,1,4,3,2,5,2,2,1,3,1,1,3,0,1,0]
Symmetry type of based matrix	c
u-polynomial	t^5-t^4-t
Normalized Jones-Krushkal polynomial	13z+27
Enhanced Jones-Krushkal polynomial	13w^2z+27w
Inner characteristic polynomial	t^6+85t^4+19t^2
Outer characteristic polynomial	t^7+135t^5+109t^3
Flat arrow polynomial	-2K12 - 2K1K3 + K1 - 2K2K3 + 2K2 + K4 + K5 + 2
2-strand cable arrow polynomial	-528K14 + 64K1*3K2K3 + 32K1*3K3K4 - 368K1*2K2*2 + 832K1*2K2 - 864K12K3*2 - 64K1*2K4*2 - 32K1*2K6*2 - 1332K1*2 + 96K1K2K3*3 + 2104K1K2K3 + 64K1K3*3K4 + 1248K1K3K4 + 48K1K4K5 + 64K1K5K6 + 48K1K6K7 - 2K102 + 8K10K4K6 - 24K24 - 224K2*2K3*2 + 56K2*2K4 - 8K22K6*2 - 1058K2*2 + 176K2K3K5 + 64K2K4K6 + 8K2K6K8 - 400K34 - 128K3*2K4*2 - 32K3*2K6*2 + 328K3*2K6 - 1116K32 + 80K3K4K7 + 16K3K6K9 - 8K4*2K6*2 - 484K4*2 - 80K5*2 - 148K6*2 - 32K7*2 - 2K8**2 + 1596
Genus of based matrix	1
Fillings of based matrix	[[{2, 6}, {3, 5}, {1, 4}], [{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {5}, {2, 3}, {1}]]
If K is slice	False

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