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Flat knot 6.188

Min(phi) over symmetries of the knot is: [-4,-3,0,1,2,4,0,2,4,2,5,1,2,1,3,1,1,2,1,2,0]
Flat knots (up to 7 crossings) with same phi are :['6.188']
Arrow polynomial of the knot is: 4K12K2 - 6K12 - 2K1K2 - 4K1K3 + K1 + 3K2 + K3 + K4 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.188']
Outer characteristic polynomial of the knot is: t^7+121t^5+85t^3+7t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.188']
2-strand cable arrow polynomial of the knot is: 320K14K2 - 1216K14 + 64K1*3K2K3 - 672K1*3K3 + 192K12K2*3 - 64K1*2K2*2K3*2 + 320K1*2K2*2K4 - 3408K12K2*2 + 128K1*2K2K32 + 32K1*2K2K3K5 - 832K12K2K4 + 7232K1*2K2 - 1600K12K3*2 - 96K1*2K3K5 - 256K1*2K4*2 - 6960K1*2 + 448K1K23K3 + 192K1K2*2K3K4 - 1632K1K22K3 + 32K1K2*2K4K5 - 384K1K22K5 - 512K1K2K3K4 - 192K1K2K3K6 + 8760K1K2K3 + 3256K1K3K4 + 448K1K4K5 + 72K1K5K6 + 8K1K7K8 - 32K2*4K4*2 + 256K2*4K4 - 1368K24 + 32K2*3K4K6 - 128K2*3K6 - 1072K22K3*2 - 504K2*2K4*2 + 2776K2*2K4 - 32K22K5*2 - 16K2*2K6*2 - 5542K2*2 + 1184K2K3K5 + 400K2K4K6 + 24K2K5K7 + 16K2K6K8 - 160K3*4 + 152K3*2K6 - 3404K32 + 8K3K4K7 - 1666K42 - 364K5*2 - 130K6*2 - 16K7*2 - 10K8**2 + 5914
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.188']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.73329', 'vk6.73343', 'vk6.73491', 'vk6.73506', 'vk6.75248', 'vk6.75267', 'vk6.75497', 'vk6.75513', 'vk6.78219', 'vk6.78227', 'vk6.78458', 'vk6.78471', 'vk6.80042', 'vk6.80051', 'vk6.80191', 'vk6.80201', 'vk6.81930', 'vk6.81934', 'vk6.82197', 'vk6.82213', 'vk6.82658', 'vk6.82659', 'vk6.84718', 'vk6.84720', 'vk6.85018', 'vk6.85024', 'vk6.85751', 'vk6.86501', 'vk6.87326', 'vk6.87685', 'vk6.89628', 'vk6.90076']
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,-3,0,1,2,4,0,2,4,2,5,1,2,1,3,1,1,2,1,2,0]

Flat knots (up to 7 crossings) with same phi are :['6.188']

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

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.188']

Outer characteristic polynomial of the knot is: t^7+121t^5+85t^3+7t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.188']

2-strand cable arrow polynomial of the knot is: 320*K1**4*K2 - 1216*K1**4 + 64*K1**3*K2*K3 - 672*K1**3*K3 + 192*K1**2*K2**3 - 64*K1**2*K2**2*K3**2 + 320*K1**2*K2**2*K4 - 3408*K1**2*K2**2 + 128*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 832*K1**2*K2*K4 + 7232*K1**2*K2 - 1600*K1**2*K3**2 - 96*K1**2*K3*K5 - 256*K1**2*K4**2 - 6960*K1**2 + 448*K1*K2**3*K3 + 192*K1*K2**2*K3*K4 - 1632*K1*K2**2*K3 + 32*K1*K2**2*K4*K5 - 384*K1*K2**2*K5 - 512*K1*K2*K3*K4 - 192*K1*K2*K3*K6 + 8760*K1*K2*K3 + 3256*K1*K3*K4 + 448*K1*K4*K5 + 72*K1*K5*K6 + 8*K1*K7*K8 - 32*K2**4*K4**2 + 256*K2**4*K4 - 1368*K2**4 + 32*K2**3*K4*K6 - 128*K2**3*K6 - 1072*K2**2*K3**2 - 504*K2**2*K4**2 + 2776*K2**2*K4 - 32*K2**2*K5**2 - 16*K2**2*K6**2 - 5542*K2**2 + 1184*K2*K3*K5 + 400*K2*K4*K6 + 24*K2*K5*K7 + 16*K2*K6*K8 - 160*K3**4 + 152*K3**2*K6 - 3404*K3**2 + 8*K3*K4*K7 - 1666*K4**2 - 364*K5**2 - 130*K6**2 - 16*K7**2 - 10*K8**2 + 5914

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.188']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.73329', 'vk6.73343', 'vk6.73491', 'vk6.73506', 'vk6.75248', 'vk6.75267', 'vk6.75497', 'vk6.75513', 'vk6.78219', 'vk6.78227', 'vk6.78458', 'vk6.78471', 'vk6.80042', 'vk6.80051', 'vk6.80191', 'vk6.80201', 'vk6.81930', 'vk6.81934', 'vk6.82197', 'vk6.82213', 'vk6.82658', 'vk6.82659', 'vk6.84718', 'vk6.84720', 'vk6.85018', 'vk6.85024', 'vk6.85751', 'vk6.86501', 'vk6.87326', 'vk6.87685', 'vk6.89628', 'vk6.90076']

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	O1O2O3O4O5U2O6U1U4U6U3U5
R3 orbit	{'O1O2O3O4O5U2O6U1U4U6U3U5'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4O5U1U3U6U2U5O6U4
Gauss code of K*	O1O2O3O4O5U1U6U4U2U5O6U3
Gauss code of -K*	O1O2O3O4O5U3O6U1U4U2U6U5
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 -4 -3 1 0 4 2],[ 4 0 0 4 2 5 2],[ 3 0 0 2 1 3 1],[-1 -4 -2 0 -1 2 1],[ 0 -2 -1 1 0 2 1],[-4 -5 -3 -2 -2 0 0],[-2 -2 -1 -1 -1 0 0]]
Primitive based matrix	[[ 0 4 2 1 0 -3 -4],[-4 0 0 -2 -2 -3 -5],[-2 0 0 -1 -1 -1 -2],[-1 2 1 0 -1 -2 -4],[ 0 2 1 1 0 -1 -2],[ 3 3 1 2 1 0 0],[ 4 5 2 4 2 0 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-4,-2,-1,0,3,4,0,2,2,3,5,1,1,1,2,1,2,4,1,2,0]
Phi over symmetry	[-4,-3,0,1,2,4,0,2,4,2,5,1,2,1,3,1,1,2,1,2,0]
Phi of -K	[-4,-3,0,1,2,4,1,2,1,4,3,2,2,4,4,0,1,2,0,1,2]
Phi of K*	[-4,-2,-1,0,3,4,2,1,2,4,3,0,1,4,4,0,2,1,2,2,1]
Phi of -K*	[-4,-3,0,1,2,4,0,2,4,2,5,1,2,1,3,1,1,2,1,2,0]
Symmetry type of based matrix	c
u-polynomial	t^3-t^2-t
Normalized Jones-Krushkal polynomial	5z^2+26z+33
Enhanced Jones-Krushkal polynomial	5w^3z^2+26w^2z+33w
Inner characteristic polynomial	t^6+75t^4+12t^2
Outer characteristic polynomial	t^7+121t^5+85t^3+7t
Flat arrow polynomial	4K12K2 - 6K12 - 2K1K2 - 4K1K3 + K1 + 3K2 + K3 + K4 + 3
2-strand cable arrow polynomial	320K14K2 - 1216K14 + 64K1*3K2K3 - 672K1*3K3 + 192K12K2*3 - 64K1*2K2*2K3*2 + 320K1*2K2*2K4 - 3408K12K2*2 + 128K1*2K2K32 + 32K1*2K2K3K5 - 832K12K2K4 + 7232K1*2K2 - 1600K12K3*2 - 96K1*2K3K5 - 256K1*2K4*2 - 6960K1*2 + 448K1K23K3 + 192K1K2*2K3K4 - 1632K1K22K3 + 32K1K2*2K4K5 - 384K1K22K5 - 512K1K2K3K4 - 192K1K2K3K6 + 8760K1K2K3 + 3256K1K3K4 + 448K1K4K5 + 72K1K5K6 + 8K1K7K8 - 32K2*4K4*2 + 256K2*4K4 - 1368K24 + 32K2*3K4K6 - 128K2*3K6 - 1072K22K3*2 - 504K2*2K4*2 + 2776K2*2K4 - 32K22K5*2 - 16K2*2K6*2 - 5542K2*2 + 1184K2K3K5 + 400K2K4K6 + 24K2K5K7 + 16K2K6K8 - 160K3*4 + 152K3*2K6 - 3404K32 + 8K3K4K7 - 1666K42 - 364K5*2 - 130K6*2 - 16K7*2 - 10K8**2 + 5914
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}, {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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