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Flat knot 6.14

Min(phi) over symmetries of the knot is: [-5,-2,-2,2,3,4,1,2,3,5,4,0,1,3,2,2,4,3,1,1,1]
Flat knots (up to 7 crossings) with same phi are :['6.14']
Arrow polynomial of the knot is: 4K1K2*2 - 4K1K2 - 2K1K3 - 2K1K4 + K2 + 2K3 + K4 + 1
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.14']
Outer characteristic polynomial of the knot is: t^7+163t^5+236t^3+8t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.14']
2-strand cable arrow polynomial of the knot is: 96K13K2K3 - 96K1*3K3 - 256K12K2*2K3*2 - 864K1*2K2*2 + 384K1*2K2K32 + 32K1*2K2K3K5 - 256K12K2K4 + 1888K1*2K2 - 1184K12K3*2 - 96K1*2K3K5 - 64K1*2K4*2 - 2716K1*2 + 224K1K23K3 + 1152K1K2*2K3K4 - 1216K1K22K3 + 192K1K2*2K4K5 - 128K1K22K5 - 1344K1K2K3K4 - 128K1K2K3K6 + 4376K1K2K3 - 128K1K2K4K5 + 2232K1K3K4 + 480K1K4K5 + 40K1K5K6 + 8K1K6K7 + 8K1K7K8 - 64K24 + 64K2*3K3K5 - 1648K2*2K3*2 + 32K2*2K3K4K7 - 32K22K3K7 - 32K2*2K4*4 + 64K2*2K4*3 + 32K2*2K4*2K8 - 944K22K4*2 - 32K2*2K4K8 + 1736K2*2K4 - 128K22K5*2 - 8K2*2K6*2 - 32K2*2K7*2 - 8K2*2K8*2 - 2608K2*2 - 96K2K32K4 - 32K2K3K4K5 + 1592K2K3K5 - 32K2K42K6 + 376K2K4K6 + 72K2K5K7 + 24K2K6K8 + 8K2K7K9 - 2032K32 + 16K3K4K7 - 16K44 + 24K4*2K8 - 1170K42 - 388K5*2 - 48K6*2 - 32K7*2 - 14K8**2 + 2718
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.14']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.81845', 'vk6.81891', 'vk6.82069', 'vk6.82079', 'vk6.82564', 'vk6.82607', 'vk6.82778', 'vk6.82783', 'vk6.82830', 'vk6.82844', 'vk6.82947', 'vk6.83059', 'vk6.83065', 'vk6.83275', 'vk6.83320', 'vk6.83358', 'vk6.83522', 'vk6.84545', 'vk6.84642', 'vk6.84919', 'vk6.84957', 'vk6.85829', 'vk6.86106', 'vk6.86120', 'vk6.86156', 'vk6.86835', 'vk6.88458', 'vk6.88896', 'vk6.89028', 'vk6.89701', 'vk6.89922', 'vk6.90020']
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,-2,2,3,4,1,2,3,5,4,0,1,3,2,2,4,3,1,1,1]

Flat knots (up to 7 crossings) with same phi are :['6.14']

Arrow polynomial of the knot is: 4*K1*K2**2 - 4*K1*K2 - 2*K1*K3 - 2*K1*K4 + K2 + 2*K3 + K4 + 1

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.14']

Outer characteristic polynomial of the knot is: t^7+163t^5+236t^3+8t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.14']

2-strand cable arrow polynomial of the knot is: 96*K1**3*K2*K3 - 96*K1**3*K3 - 256*K1**2*K2**2*K3**2 - 864*K1**2*K2**2 + 384*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 256*K1**2*K2*K4 + 1888*K1**2*K2 - 1184*K1**2*K3**2 - 96*K1**2*K3*K5 - 64*K1**2*K4**2 - 2716*K1**2 + 224*K1*K2**3*K3 + 1152*K1*K2**2*K3*K4 - 1216*K1*K2**2*K3 + 192*K1*K2**2*K4*K5 - 128*K1*K2**2*K5 - 1344*K1*K2*K3*K4 - 128*K1*K2*K3*K6 + 4376*K1*K2*K3 - 128*K1*K2*K4*K5 + 2232*K1*K3*K4 + 480*K1*K4*K5 + 40*K1*K5*K6 + 8*K1*K6*K7 + 8*K1*K7*K8 - 64*K2**4 + 64*K2**3*K3*K5 - 1648*K2**2*K3**2 + 32*K2**2*K3*K4*K7 - 32*K2**2*K3*K7 - 32*K2**2*K4**4 + 64*K2**2*K4**3 + 32*K2**2*K4**2*K8 - 944*K2**2*K4**2 - 32*K2**2*K4*K8 + 1736*K2**2*K4 - 128*K2**2*K5**2 - 8*K2**2*K6**2 - 32*K2**2*K7**2 - 8*K2**2*K8**2 - 2608*K2**2 - 96*K2*K3**2*K4 - 32*K2*K3*K4*K5 + 1592*K2*K3*K5 - 32*K2*K4**2*K6 + 376*K2*K4*K6 + 72*K2*K5*K7 + 24*K2*K6*K8 + 8*K2*K7*K9 - 2032*K3**2 + 16*K3*K4*K7 - 16*K4**4 + 24*K4**2*K8 - 1170*K4**2 - 388*K5**2 - 48*K6**2 - 32*K7**2 - 14*K8**2 + 2718

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.14']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.81845', 'vk6.81891', 'vk6.82069', 'vk6.82079', 'vk6.82564', 'vk6.82607', 'vk6.82778', 'vk6.82783', 'vk6.82830', 'vk6.82844', 'vk6.82947', 'vk6.83059', 'vk6.83065', 'vk6.83275', 'vk6.83320', 'vk6.83358', 'vk6.83522', 'vk6.84545', 'vk6.84642', 'vk6.84919', 'vk6.84957', 'vk6.85829', 'vk6.86106', 'vk6.86120', 'vk6.86156', 'vk6.86835', 'vk6.88458', 'vk6.88896', 'vk6.89028', 'vk6.89701', 'vk6.89922', 'vk6.90020']

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	O1O2O3O4O5O6U1U3U2U5U6U4
R3 orbit	{'O1O2O3O4O5O6U1U3U2U5U6U4'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4O5O6U3U1U2U5U4U6
Gauss code of K*	O1O2O3O4O5O6U1U3U2U6U4U5
Gauss code of -K*	O1O2O3O4O5O6U2U3U1U5U4U6
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 -2 -2 3 2 4],[ 5 0 2 1 5 3 4],[ 2 -2 0 0 4 2 3],[ 2 -1 0 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 -2 -2 -5],[-4 0 -1 -1 -2 -3 -4],[-3 1 0 -1 -3 -4 -5],[-2 1 1 0 -1 -2 -3],[ 2 2 3 1 0 0 -1],[ 2 3 4 2 0 0 -2],[ 5 4 5 3 1 2 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-4,-3,-2,2,2,5,1,1,2,3,4,1,3,4,5,1,2,3,0,1,2]
Phi over symmetry	[-5,-2,-2,2,3,4,1,2,3,5,4,0,1,3,2,2,4,3,1,1,1]
Phi of -K	[-5,-2,-2,2,3,4,1,2,4,3,5,0,2,1,3,3,2,4,0,1,0]
Phi of K*	[-4,-3,-2,2,2,5,0,1,3,4,5,0,1,2,3,2,3,4,0,1,2]
Phi of -K*	[-5,-2,-2,2,3,4,1,2,3,5,4,0,1,3,2,2,4,3,1,1,1]
Symmetry type of based matrix	c
u-polynomial	t^5-t^4-t^3+t^2
Normalized Jones-Krushkal polynomial	4z^2+17z+19
Enhanced Jones-Krushkal polynomial	-2w^4z^2+6w^3z^2-6w^3z+23w^2z+19w
Inner characteristic polynomial	t^6+101t^4+45t^2+1
Outer characteristic polynomial	t^7+163t^5+236t^3+8t
Flat arrow polynomial	4K1K2*2 - 4K1K2 - 2K1K3 - 2K1K4 + K2 + 2K3 + K4 + 1
2-strand cable arrow polynomial	96K13K2K3 - 96K1*3K3 - 256K12K2*2K3*2 - 864K1*2K2*2 + 384K1*2K2K32 + 32K1*2K2K3K5 - 256K12K2K4 + 1888K1*2K2 - 1184K12K3*2 - 96K1*2K3K5 - 64K1*2K4*2 - 2716K1*2 + 224K1K23K3 + 1152K1K2*2K3K4 - 1216K1K22K3 + 192K1K2*2K4K5 - 128K1K22K5 - 1344K1K2K3K4 - 128K1K2K3K6 + 4376K1K2K3 - 128K1K2K4K5 + 2232K1K3K4 + 480K1K4K5 + 40K1K5K6 + 8K1K6K7 + 8K1K7K8 - 64K24 + 64K2*3K3K5 - 1648K2*2K3*2 + 32K2*2K3K4K7 - 32K22K3K7 - 32K2*2K4*4 + 64K2*2K4*3 + 32K2*2K4*2K8 - 944K22K4*2 - 32K2*2K4K8 + 1736K2*2K4 - 128K22K5*2 - 8K2*2K6*2 - 32K2*2K7*2 - 8K2*2K8*2 - 2608K2*2 - 96K2K32K4 - 32K2K3K4K5 + 1592K2K3K5 - 32K2K42K6 + 376K2K4K6 + 72K2K5K7 + 24K2K6K8 + 8K2K7K9 - 2032K32 + 16K3K4K7 - 16K44 + 24K4*2K8 - 1170K42 - 388K5*2 - 48K6*2 - 32K7*2 - 14K8**2 + 2718
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}], [{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}, {5}, {1, 4}, {2, 3}], [{6}, {5}, {1, 4}, {3}, {2}], [{6}, {5}, {2, 4}, {1, 3}], [{6}, {5}, {2, 4}, {3}, {1}], [{6}, {5}, {3, 4}, {1, 2}], [{6}, {5}, {4}, {3}, {1, 2}]]
If K is slice	False

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