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Flat knot 6.1178

Min(phi) over symmetries of the knot is: [-3,0,1,2,1,2,3,0,0,1]
Flat knots (up to 7 crossings) with same phi are :['6.1178']
Arrow polynomial of the knot is: 12K13 - 10K1*2 - 10K1K2 - 4K1 + 5K2 + 2K3 + 6
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.318', '6.1178', '6.1184']
Outer characteristic polynomial of the knot is: t^5+29t^3+10t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1178']
2-strand cable arrow polynomial of the knot is: 672K14K2 - 2848K14 + 672K1*3K2K3 - 1344K1*3K3 + 800K12K2*3 - 64K1*2K2*2K3*2 + 256K1*2K2*2K4 - 7280K12K2*2 + 192K1*2K2K32 - 1024K1*2K2K4 + 13056K1*2K2 - 1376K12K3*2 - 128K1*2K3K5 - 32K1*2K4*2 - 32K1*2K5*2 - 9924K1*2 + 544K1K23K3 + 64K1K2*2K3K4 - 2016K1K22K3 - 352K1K2*2K5 + 32K1K2K33 - 576K1K2K3K4 + 12056K1K2K3 - 64K1K2K4K5 + 2400K1K3K4 + 416K1K4K5 + 64K1K5K6 - 96K2*6 + 160K2*4K4 - 1592K24 - 32K2*3K6 - 768K22K3*2 - 136K2*2K4*2 + 2632K2*2K4 - 7496K22 - 32K2K32K4 + 928K2K3K5 + 128K2K4K6 + 8K32K6 - 3964K32 - 1266K4*2 - 336K5*2 - 48K6**2 + 7744
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1178']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.71390', 'vk6.71449', 'vk6.71916', 'vk6.71975', 'vk6.72444', 'vk6.72594', 'vk6.72711', 'vk6.72807', 'vk6.72870', 'vk6.73026', 'vk6.74229', 'vk6.74365', 'vk6.74429', 'vk6.74859', 'vk6.75045', 'vk6.76618', 'vk6.76911', 'vk6.77047', 'vk6.77415', 'vk6.77754', 'vk6.77805', 'vk6.79273', 'vk6.79403', 'vk6.79748', 'vk6.79821', 'vk6.79880', 'vk6.80849', 'vk6.80908', 'vk6.81390', 'vk6.85507', 'vk6.87199', 'vk6.89259']
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: [-3,0,1,2,1,2,3,0,0,1]

Flat knots (up to 7 crossings) with same phi are :['6.1178']

Arrow polynomial of the knot is: 12*K1**3 - 10*K1**2 - 10*K1*K2 - 4*K1 + 5*K2 + 2*K3 + 6

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.318', '6.1178', '6.1184']

Outer characteristic polynomial of the knot is: t^5+29t^3+10t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1178']

2-strand cable arrow polynomial of the knot is: 672*K1**4*K2 - 2848*K1**4 + 672*K1**3*K2*K3 - 1344*K1**3*K3 + 800*K1**2*K2**3 - 64*K1**2*K2**2*K3**2 + 256*K1**2*K2**2*K4 - 7280*K1**2*K2**2 + 192*K1**2*K2*K3**2 - 1024*K1**2*K2*K4 + 13056*K1**2*K2 - 1376*K1**2*K3**2 - 128*K1**2*K3*K5 - 32*K1**2*K4**2 - 32*K1**2*K5**2 - 9924*K1**2 + 544*K1*K2**3*K3 + 64*K1*K2**2*K3*K4 - 2016*K1*K2**2*K3 - 352*K1*K2**2*K5 + 32*K1*K2*K3**3 - 576*K1*K2*K3*K4 + 12056*K1*K2*K3 - 64*K1*K2*K4*K5 + 2400*K1*K3*K4 + 416*K1*K4*K5 + 64*K1*K5*K6 - 96*K2**6 + 160*K2**4*K4 - 1592*K2**4 - 32*K2**3*K6 - 768*K2**2*K3**2 - 136*K2**2*K4**2 + 2632*K2**2*K4 - 7496*K2**2 - 32*K2*K3**2*K4 + 928*K2*K3*K5 + 128*K2*K4*K6 + 8*K3**2*K6 - 3964*K3**2 - 1266*K4**2 - 336*K5**2 - 48*K6**2 + 7744

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1178']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.71390', 'vk6.71449', 'vk6.71916', 'vk6.71975', 'vk6.72444', 'vk6.72594', 'vk6.72711', 'vk6.72807', 'vk6.72870', 'vk6.73026', 'vk6.74229', 'vk6.74365', 'vk6.74429', 'vk6.74859', 'vk6.75045', 'vk6.76618', 'vk6.76911', 'vk6.77047', 'vk6.77415', 'vk6.77754', 'vk6.77805', 'vk6.79273', 'vk6.79403', 'vk6.79748', 'vk6.79821', 'vk6.79880', 'vk6.80849', 'vk6.80908', 'vk6.81390', 'vk6.85507', 'vk6.87199', 'vk6.89259']

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	O1O2O3O4U1U5U2O6O5U4U6U3
R3 orbit	{'O1O2O3O4U1U5U2O6O5U4U6U3'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4U2U5U1O6O5U3U6U4
Gauss code of K*	O1O2O3U4U2O5O4O6U1U3U6U5
Gauss code of -K*	O1O2O3U4U2O5O4O6U3U1U5U6
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 -3 0 2 1 0 0],[ 3 0 1 3 2 2 1],[ 0 -1 0 1 0 0 0],[-2 -3 -1 0 -1 -1 0],[-1 -2 0 1 0 -1 0],[ 0 -2 0 1 1 0 0],[ 0 -1 0 0 0 0 0]]
Primitive based matrix	[[ 0 2 1 0 -3],[-2 0 -1 0 -3],[-1 1 0 0 -2],[ 0 0 0 0 -1],[ 3 3 2 1 0]]
If based matrix primitive	False
Phi of primitive based matrix	[-2,-1,0,3,1,0,3,0,2,1]
Phi over symmetry	[-3,0,1,2,1,2,3,0,0,1]
Phi of -K	[-3,0,1,2,2,2,2,1,2,0]
Phi of K*	[-2,-1,0,3,0,2,2,1,2,2]
Phi of -K*	[-3,0,1,2,1,2,3,0,0,1]
Symmetry type of based matrix	c
u-polynomial	t^3-t^2-t
Normalized Jones-Krushkal polynomial	3z^2+24z+37
Enhanced Jones-Krushkal polynomial	3w^3z^2+24w^2z+37w
Inner characteristic polynomial	t^4+15t^2+1
Outer characteristic polynomial	t^5+29t^3+10t
Flat arrow polynomial	12K13 - 10K1*2 - 10K1K2 - 4K1 + 5K2 + 2K3 + 6
2-strand cable arrow polynomial	672K14K2 - 2848K14 + 672K1*3K2K3 - 1344K1*3K3 + 800K12K2*3 - 64K1*2K2*2K3*2 + 256K1*2K2*2K4 - 7280K12K2*2 + 192K1*2K2K32 - 1024K1*2K2K4 + 13056K1*2K2 - 1376K12K3*2 - 128K1*2K3K5 - 32K1*2K4*2 - 32K1*2K5*2 - 9924K1*2 + 544K1K23K3 + 64K1K2*2K3K4 - 2016K1K22K3 - 352K1K2*2K5 + 32K1K2K33 - 576K1K2K3K4 + 12056K1K2K3 - 64K1K2K4K5 + 2400K1K3K4 + 416K1K4K5 + 64K1K5K6 - 96K2*6 + 160K2*4K4 - 1592K24 - 32K2*3K6 - 768K22K3*2 - 136K2*2K4*2 + 2632K2*2K4 - 7496K22 - 32K2K32K4 + 928K2K3K5 + 128K2K4K6 + 8K32K6 - 3964K32 - 1266K4*2 - 336K5*2 - 48K6**2 + 7744
Genus of based matrix	1
Fillings of based matrix	[[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {5}, {3, 4}, {2}], [{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {5}, {1, 4}, {2}], [{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {2, 5}, {1, 3}], [{4, 6}, {2, 5}, {3}, {1}], [{4, 6}, {3, 5}, {1, 2}], [{4, 6}, {5}, {1, 3}, {2}]]
If K is slice	False

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