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Flat knot 6.27

Min(phi) over symmetries of the knot is: [-5,-2,1,1,2,3,1,3,5,2,4,2,3,1,3,0,0,1,1,2,0]
Flat knots (up to 7 crossings) with same phi are :['6.27']
Arrow polynomial of the knot is: 8K13 + 4K1*2K3 - 8K12 - 12K1K2 - 2K1K4 - K1 + 4K2 + 3*K3 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.27']
Outer characteristic polynomial of the knot is: t^7+128t^5+149t^3+8t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.27']
2-strand cable arrow polynomial of the knot is: -192K14K2*2 + 288K1*4K2 - 2176K14 + 128K1*3K2*3K3 + 928K13K2K3 - 480K1*3K3 - 448K12K2*4 + 768K1*2K2*3 - 512K1*2K2*2K3*2 - 8448K1*2K2*2 + 128K1*2K2K32 - 864K1*2K2K4 + 9656K1*2K2 - 1152K12K3*2 - 96K1*2K4*2 - 6424K1*2 + 3840K1K23K3 + 448K1K2*2K3K4 - 1696K1K22K3 + 32K1K2*2K4K5 + 64K1K22K5K6 - 1024K1K22K5 + 288K1K2K33 - 320K1K2K3K4 - 256K1K2K3K6 + 10704K1K2K3 - 160K1K2K4K5 + 1864K1K3K4 + 184K1K4K5 + 40K1K5K6 - 64K2*6 - 640K2*4K3*2 + 224K2*4K4 - 32K24K6*2 - 3536K2*4 + 768K2*3K3K5 + 128K2*3K4K6 + 32K2*3K6K8 - 96K2*3K6 - 128K22K3*4 + 32K2*2K3*2K4 + 160K22K3*2K6 - 3456K22K3*2 - 32K2*2K3K7 - 464K2*2K4*2 - 32K2*2K4K8 + 2912K2*2K4 - 320K22K5*2 - 160K2*2K6*2 - 8K2*2K8*2 - 3922K2*2 - 160K2K32K4 + 1936K2K3K5 + 440K2K4K6 + 32K2K5K7 + 32K2K6K8 - 96K34 - 32K3*2K4*2 + 128K3*2K6 - 3052K32 + 16K3K4K7 - 916K42 - 268K5*2 - 94K6*2 - 4K8**2 + 5558
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.27']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.19952', 'vk6.20051', 'vk6.21197', 'vk6.21335', 'vk6.26925', 'vk6.27112', 'vk6.28679', 'vk6.28803', 'vk6.38345', 'vk6.38501', 'vk6.40485', 'vk6.40704', 'vk6.45210', 'vk6.45397', 'vk6.47033', 'vk6.47147', 'vk6.56740', 'vk6.56851', 'vk6.57841', 'vk6.57992', 'vk6.61173', 'vk6.61374', 'vk6.62413', 'vk6.62539', 'vk6.66436', 'vk6.66564', 'vk6.67206', 'vk6.67357', 'vk6.69088', 'vk6.69212', 'vk6.69869', 'vk6.69955']
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,-2,1,1,2,3,1,3,5,2,4,2,3,1,3,0,0,1,1,2,0]

Flat knots (up to 7 crossings) with same phi are :['6.27']

Arrow polynomial of the knot is: 8*K1**3 + 4*K1**2*K3 - 8*K1**2 - 12*K1*K2 - 2*K1*K4 - K1 + 4*K2 + 3*K3 + 5

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.27']

Outer characteristic polynomial of the knot is: t^7+128t^5+149t^3+8t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.27']

2-strand cable arrow polynomial of the knot is: -192*K1**4*K2**2 + 288*K1**4*K2 - 2176*K1**4 + 128*K1**3*K2**3*K3 + 928*K1**3*K2*K3 - 480*K1**3*K3 - 448*K1**2*K2**4 + 768*K1**2*K2**3 - 512*K1**2*K2**2*K3**2 - 8448*K1**2*K2**2 + 128*K1**2*K2*K3**2 - 864*K1**2*K2*K4 + 9656*K1**2*K2 - 1152*K1**2*K3**2 - 96*K1**2*K4**2 - 6424*K1**2 + 3840*K1*K2**3*K3 + 448*K1*K2**2*K3*K4 - 1696*K1*K2**2*K3 + 32*K1*K2**2*K4*K5 + 64*K1*K2**2*K5*K6 - 1024*K1*K2**2*K5 + 288*K1*K2*K3**3 - 320*K1*K2*K3*K4 - 256*K1*K2*K3*K6 + 10704*K1*K2*K3 - 160*K1*K2*K4*K5 + 1864*K1*K3*K4 + 184*K1*K4*K5 + 40*K1*K5*K6 - 64*K2**6 - 640*K2**4*K3**2 + 224*K2**4*K4 - 32*K2**4*K6**2 - 3536*K2**4 + 768*K2**3*K3*K5 + 128*K2**3*K4*K6 + 32*K2**3*K6*K8 - 96*K2**3*K6 - 128*K2**2*K3**4 + 32*K2**2*K3**2*K4 + 160*K2**2*K3**2*K6 - 3456*K2**2*K3**2 - 32*K2**2*K3*K7 - 464*K2**2*K4**2 - 32*K2**2*K4*K8 + 2912*K2**2*K4 - 320*K2**2*K5**2 - 160*K2**2*K6**2 - 8*K2**2*K8**2 - 3922*K2**2 - 160*K2*K3**2*K4 + 1936*K2*K3*K5 + 440*K2*K4*K6 + 32*K2*K5*K7 + 32*K2*K6*K8 - 96*K3**4 - 32*K3**2*K4**2 + 128*K3**2*K6 - 3052*K3**2 + 16*K3*K4*K7 - 916*K4**2 - 268*K5**2 - 94*K6**2 - 4*K8**2 + 5558

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.27']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.19952', 'vk6.20051', 'vk6.21197', 'vk6.21335', 'vk6.26925', 'vk6.27112', 'vk6.28679', 'vk6.28803', 'vk6.38345', 'vk6.38501', 'vk6.40485', 'vk6.40704', 'vk6.45210', 'vk6.45397', 'vk6.47033', 'vk6.47147', 'vk6.56740', 'vk6.56851', 'vk6.57841', 'vk6.57992', 'vk6.61173', 'vk6.61374', 'vk6.62413', 'vk6.62539', 'vk6.66436', 'vk6.66564', 'vk6.67206', 'vk6.67357', 'vk6.69088', 'vk6.69212', 'vk6.69869', 'vk6.69955']

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	O1O2O3O4O5O6U1U3U6U4U5U2
R3 orbit	{'O1O2O3O4O5O6U1U3U6U4U5U2'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4O5O6U5U2U3U1U4U6
Gauss code of K*	O1O2O3O4O5O6U1U6U2U4U5U3
Gauss code of -K*	O1O2O3O4O5O6U4U2U3U5U1U6
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 1 -2 1 3 2],[ 5 0 5 1 3 4 2],[-1 -5 0 -3 0 2 1],[ 2 -1 3 0 2 3 1],[-1 -3 0 -2 0 1 0],[-3 -4 -2 -3 -1 0 0],[-2 -2 -1 -1 0 0 0]]
Primitive based matrix	[[ 0 3 2 1 1 -2 -5],[-3 0 0 -1 -2 -3 -4],[-2 0 0 0 -1 -1 -2],[-1 1 0 0 0 -2 -3],[-1 2 1 0 0 -3 -5],[ 2 3 1 2 3 0 -1],[ 5 4 2 3 5 1 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-3,-2,-1,-1,2,5,0,1,2,3,4,0,1,1,2,0,2,3,3,5,1]
Phi over symmetry	[-5,-2,1,1,2,3,1,3,5,2,4,2,3,1,3,0,0,1,1,2,0]
Phi of -K	[-5,-2,1,1,2,3,2,1,3,5,4,0,1,3,2,0,0,0,1,1,1]
Phi of K*	[-3,-2,-1,-1,2,5,1,0,1,2,4,0,1,3,5,0,0,1,1,3,2]
Phi of -K*	[-5,-2,1,1,2,3,1,3,5,2,4,2,3,1,3,0,0,1,1,2,0]
Symmetry type of based matrix	c
u-polynomial	t^5-t^3-2t
Normalized Jones-Krushkal polynomial	6z^2+27z+31
Enhanced Jones-Krushkal polynomial	6w^3z^2+27w^2z+31w
Inner characteristic polynomial	t^6+84t^4+33t^2+1
Outer characteristic polynomial	t^7+128t^5+149t^3+8t
Flat arrow polynomial	8K13 + 4K1*2K3 - 8K12 - 12K1K2 - 2K1K4 - K1 + 4K2 + 3*K3 + 5
2-strand cable arrow polynomial	-192K14K2*2 + 288K1*4K2 - 2176K14 + 128K1*3K2*3K3 + 928K13K2K3 - 480K1*3K3 - 448K12K2*4 + 768K1*2K2*3 - 512K1*2K2*2K3*2 - 8448K1*2K2*2 + 128K1*2K2K32 - 864K1*2K2K4 + 9656K1*2K2 - 1152K12K3*2 - 96K1*2K4*2 - 6424K1*2 + 3840K1K23K3 + 448K1K2*2K3K4 - 1696K1K22K3 + 32K1K2*2K4K5 + 64K1K22K5K6 - 1024K1K22K5 + 288K1K2K33 - 320K1K2K3K4 - 256K1K2K3K6 + 10704K1K2K3 - 160K1K2K4K5 + 1864K1K3K4 + 184K1K4K5 + 40K1K5K6 - 64K2*6 - 640K2*4K3*2 + 224K2*4K4 - 32K24K6*2 - 3536K2*4 + 768K2*3K3K5 + 128K2*3K4K6 + 32K2*3K6K8 - 96K2*3K6 - 128K22K3*4 + 32K2*2K3*2K4 + 160K22K3*2K6 - 3456K22K3*2 - 32K2*2K3K7 - 464K2*2K4*2 - 32K2*2K4K8 + 2912K2*2K4 - 320K22K5*2 - 160K2*2K6*2 - 8K2*2K8*2 - 3922K2*2 - 160K2K32K4 + 1936K2K3K5 + 440K2K4K6 + 32K2K5K7 + 32K2K6K8 - 96K34 - 32K3*2K4*2 + 128K3*2K6 - 3052K32 + 16K3K4K7 - 916K42 - 268K5*2 - 94K6*2 - 4K8**2 + 5558
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}], [{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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