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Flat knot 6.2

Min(phi) over symmetries of the knot is: [-5,-3,-1,2,3,4,1,2,3,5,4,1,2,4,3,1,3,2,1,1,1]
Flat knots (up to 7 crossings) with same phi are :['6.2']
Arrow polynomial of the knot is: -8K14 - 8K1*3K2 + 16K13 + 4K1*2K2 + 4K12K3 + 2K12 - 6K1K2 - 6K1 + K2 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.2']
Outer characteristic polynomial of the knot is: t^7+166t^5+210t^3+6t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.2']
2-strand cable arrow polynomial of the knot is: -384K14 - 512K1*2K2*6 + 1664K1*2K2*5 - 5376K1*2K2*4 - 128K1*2K2*3K4 + 4064K12K2*3 - 5312K1*2K2*2 - 256K1*2K2K4 + 3664K1*2K2 - 32K12K3*2 - 2236K1*2 + 2560K1K25K3 + 128K1K2*4K3K4 - 1408K1K24K3 - 384K1K2*4K5 + 4192K1K2*3K3 + 224K1K2*2K3K4 - 864K1K22K3 - 192K1K2*2K5 - 32K1K2K3K4 + 3080K1K2K3 + 240K1K3K4 + 8K1K4K5 - 128K2*8 - 512K2*6K3*2 - 128K2*6K4*2 + 512K2*6K4 - 3264K26 + 256K2*5K3K5 + 128K2*5K4K6 - 128K2*5K6 - 1152K24K3*2 - 480K2*4K4*2 + 2336K2*4K4 - 32K24K6*2 - 1592K2*4 + 224K2*3K3K5 + 224K2*3K4K6 - 64K2*3K6 - 1072K22K3*2 - 568K2*2K4*2 + 1600K2*2K4 - 32K22K6*2 + 88K2*2 + 184K2K3K5 + 120K2K4K6 - 624K3*2 - 326K4*2 - 12K5*2 - 8K6**2 + 1740
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.2']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.81817', 'vk6.81819', 'vk6.82034', 'vk6.82038', 'vk6.82526', 'vk6.82529', 'vk6.82768', 'vk6.82770', 'vk6.82964', 'vk6.82970', 'vk6.83045', 'vk6.83047', 'vk6.83510', 'vk6.83512', 'vk6.83888', 'vk6.83898', 'vk6.84511', 'vk6.84513', 'vk6.84877', 'vk6.84879', 'vk6.85791', 'vk6.85797', 'vk6.86048', 'vk6.86052', 'vk6.86368', 'vk6.86374', 'vk6.86827', 'vk6.86828', 'vk6.88798', 'vk6.88808', 'vk6.89860', 'vk6.89863']
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,-3,-1,2,3,4,1,2,3,5,4,1,2,4,3,1,3,2,1,1,1]

Flat knots (up to 7 crossings) with same phi are :['6.2']

Arrow polynomial of the knot is: -8*K1**4 - 8*K1**3*K2 + 16*K1**3 + 4*K1**2*K2 + 4*K1**2*K3 + 2*K1**2 - 6*K1*K2 - 6*K1 + K2 + 2

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.2']

Outer characteristic polynomial of the knot is: t^7+166t^5+210t^3+6t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.2']

2-strand cable arrow polynomial of the knot is: -384*K1**4 - 512*K1**2*K2**6 + 1664*K1**2*K2**5 - 5376*K1**2*K2**4 - 128*K1**2*K2**3*K4 + 4064*K1**2*K2**3 - 5312*K1**2*K2**2 - 256*K1**2*K2*K4 + 3664*K1**2*K2 - 32*K1**2*K3**2 - 2236*K1**2 + 2560*K1*K2**5*K3 + 128*K1*K2**4*K3*K4 - 1408*K1*K2**4*K3 - 384*K1*K2**4*K5 + 4192*K1*K2**3*K3 + 224*K1*K2**2*K3*K4 - 864*K1*K2**2*K3 - 192*K1*K2**2*K5 - 32*K1*K2*K3*K4 + 3080*K1*K2*K3 + 240*K1*K3*K4 + 8*K1*K4*K5 - 128*K2**8 - 512*K2**6*K3**2 - 128*K2**6*K4**2 + 512*K2**6*K4 - 3264*K2**6 + 256*K2**5*K3*K5 + 128*K2**5*K4*K6 - 128*K2**5*K6 - 1152*K2**4*K3**2 - 480*K2**4*K4**2 + 2336*K2**4*K4 - 32*K2**4*K6**2 - 1592*K2**4 + 224*K2**3*K3*K5 + 224*K2**3*K4*K6 - 64*K2**3*K6 - 1072*K2**2*K3**2 - 568*K2**2*K4**2 + 1600*K2**2*K4 - 32*K2**2*K6**2 + 88*K2**2 + 184*K2*K3*K5 + 120*K2*K4*K6 - 624*K3**2 - 326*K4**2 - 12*K5**2 - 8*K6**2 + 1740

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.2']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.81817', 'vk6.81819', 'vk6.82034', 'vk6.82038', 'vk6.82526', 'vk6.82529', 'vk6.82768', 'vk6.82770', 'vk6.82964', 'vk6.82970', 'vk6.83045', 'vk6.83047', 'vk6.83510', 'vk6.83512', 'vk6.83888', 'vk6.83898', 'vk6.84511', 'vk6.84513', 'vk6.84877', 'vk6.84879', 'vk6.85791', 'vk6.85797', 'vk6.86048', 'vk6.86052', 'vk6.86368', 'vk6.86374', 'vk6.86827', 'vk6.86828', 'vk6.88798', 'vk6.88808', 'vk6.89860', 'vk6.89863']

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	O1O2O3O4O5O6U1U2U3U5U6U4
R3 orbit	{'O1O2O3O4O5O6U1U2U3U5U6U4'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4O5O6U3U1U2U4U5U6
Gauss code of K*	O1O2O3O4O5O6U1U2U3U6U4U5
Gauss code of -K*	O1O2O3O4O5O6U2U3U1U4U5U6
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 -5 -3 -1 3 2 4],[ 5 0 1 2 5 3 4],[ 3 -1 0 1 4 2 3],[ 1 -2 -1 0 3 1 2],[-3 -5 -4 -3 0 -1 1],[-2 -3 -2 -1 1 0 1],[-4 -4 -3 -2 -1 -1 0]]
Primitive based matrix	[[ 0 4 3 2 -1 -3 -5],[-4 0 -1 -1 -2 -3 -4],[-3 1 0 -1 -3 -4 -5],[-2 1 1 0 -1 -2 -3],[ 1 2 3 1 0 -1 -2],[ 3 3 4 2 1 0 -1],[ 5 4 5 3 2 1 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-4,-3,-2,1,3,5,1,1,2,3,4,1,3,4,5,1,2,3,1,2,1]
Phi over symmetry	[-5,-3,-1,2,3,4,1,2,3,5,4,1,2,4,3,1,3,2,1,1,1]
Phi of -K	[-5,-3,-1,2,3,4,1,2,4,3,5,1,3,2,4,2,1,3,0,1,0]
Phi of K*	[-4,-3,-2,1,3,5,0,1,3,4,5,0,1,2,3,2,3,4,1,2,1]
Phi of -K*	[-5,-3,-1,2,3,4,1,2,3,5,4,1,2,4,3,1,3,2,1,1,1]
Symmetry type of based matrix	c
u-polynomial	t^5-t^4-t^2+t
Normalized Jones-Krushkal polynomial	z^2+6z+9
Enhanced Jones-Krushkal polynomial	-6w^4z^2+7w^3z^2-10w^3z+16w^2z+9w
Inner characteristic polynomial	t^6+102t^4+35t^2
Outer characteristic polynomial	t^7+166t^5+210t^3+6t
Flat arrow polynomial	-8K14 - 8K1*3K2 + 16K13 + 4K1*2K2 + 4K12K3 + 2K12 - 6K1K2 - 6K1 + K2 + 2
2-strand cable arrow polynomial	-384K14 - 512K1*2K2*6 + 1664K1*2K2*5 - 5376K1*2K2*4 - 128K1*2K2*3K4 + 4064K12K2*3 - 5312K1*2K2*2 - 256K1*2K2K4 + 3664K1*2K2 - 32K12K3*2 - 2236K1*2 + 2560K1K25K3 + 128K1K2*4K3K4 - 1408K1K24K3 - 384K1K2*4K5 + 4192K1K2*3K3 + 224K1K2*2K3K4 - 864K1K22K3 - 192K1K2*2K5 - 32K1K2K3K4 + 3080K1K2K3 + 240K1K3K4 + 8K1K4K5 - 128K2*8 - 512K2*6K3*2 - 128K2*6K4*2 + 512K2*6K4 - 3264K26 + 256K2*5K3K5 + 128K2*5K4K6 - 128K2*5K6 - 1152K24K3*2 - 480K2*4K4*2 + 2336K2*4K4 - 32K24K6*2 - 1592K2*4 + 224K2*3K3K5 + 224K2*3K4K6 - 64K2*3K6 - 1072K22K3*2 - 568K2*2K4*2 + 1600K2*2K4 - 32K22K6*2 + 88K2*2 + 184K2K3K5 + 120K2K4K6 - 624K3*2 - 326K4*2 - 12K5*2 - 8K6**2 + 1740
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}], [{4, 6}, {5}, {3}, {2}, {1}], [{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}], [{5, 6}, {4}, {3}, {2}, {1}], [{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}, {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}, {4}, {1, 3}, {2}]]
If K is slice	False

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