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Flat knot 6.1163

Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,0,2,1,3,2,1,1,2,1,1,1,1,-1,0,1]
Flat knots (up to 7 crossings) with same phi are :['6.1163']
Arrow polynomial of the knot is: 8K13 + 4K1*2K2 - 12K12 - 10K1K2 - 2K1K3 - K1 - 2K2*2 + 5K2 + 3*K3 + K4 + 7
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.875', '6.1163']
Outer characteristic polynomial of the knot is: t^7+48t^5+60t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1163']
2-strand cable arrow polynomial of the knot is: -192K16 - 192K1*4K2*2 + 2272K1*4K2 - 4736K14 - 256K1*3K2*2K3 - 128K13K2K3K4 + 1216K13K2K3 + 288K1*3K3K4 - 1152K1*3K3 - 128K12K2*4 + 640K1*2K2*3 + 416K1*2K2*2K4 - 6784K12K2*2 + 448K1*2K2K32 + 128K1*2K2K42 - 928K1*2K2K4 + 11144K1*2K2 - 2048K12K3*2 - 128K1*2K3K5 - 448K1*2K4*2 - 7020K1*2 + 704K1K23K3 + 160K1K2*2K3K4 - 2240K1K22K3 + 32K1K2*2K4K5 - 288K1K22K5 + 64K1K2K33 + 32K1K2K3K42 - 864K1K2K3K4 + 10432K1K2K3 + 3200K1K3K4 + 528K1K4K5 + 8K1K5K6 - 64K2*6 - 64K2*4K3*2 - 32K2*4K4*2 + 160K2*4K4 - 1216K24 + 64K2*3K3K5 + 32K2*3K4K6 - 96K2*3K6 + 64K22K3*2K4 - 1056K22K3*2 - 32K2*2K3K7 + 32K2*2K4*3 - 344K2*2K4*2 + 2648K2*2K4 - 32K22K5*2 - 8K2*2K6*2 - 6522K2*2 - 96K2K32K4 - 96K2K3K4K5 + 952K2K3K5 - 32K2K42K6 + 280K2K4K6 + 48K2K5K7 + 8K2K6K8 - 64K3*4 - 48K3*2K4*2 + 104K3*2K6 - 3668K32 + 64K3K4K7 - 8K44 + 8K4*2K8 - 1554K42 - 296K5*2 - 110K6*2 - 24K7*2 - 2K8**2 + 6818
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1163']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4074', 'vk6.4105', 'vk6.5312', 'vk6.5343', 'vk6.7444', 'vk6.7473', 'vk6.8943', 'vk6.8974', 'vk6.10108', 'vk6.10279', 'vk6.10302', 'vk6.14545', 'vk6.15289', 'vk6.15416', 'vk6.15767', 'vk6.16184', 'vk6.29850', 'vk6.29881', 'vk6.33935', 'vk6.34011', 'vk6.34212', 'vk6.34393', 'vk6.48470', 'vk6.49173', 'vk6.50224', 'vk6.50255', 'vk6.51584', 'vk6.53970', 'vk6.54032', 'vk6.54171', 'vk6.54471', 'vk6.63297']
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,-1,0,1,1,2,0,2,1,3,2,1,1,2,1,1,1,1,-1,0,1]

Flat knots (up to 7 crossings) with same phi are :['6.1163']

Arrow polynomial of the knot is: 8*K1**3 + 4*K1**2*K2 - 12*K1**2 - 10*K1*K2 - 2*K1*K3 - K1 - 2*K2**2 + 5*K2 + 3*K3 + K4 + 7

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.875', '6.1163']

Outer characteristic polynomial of the knot is: t^7+48t^5+60t^3+4t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1163']

2-strand cable arrow polynomial of the knot is: -192*K1**6 - 192*K1**4*K2**2 + 2272*K1**4*K2 - 4736*K1**4 - 256*K1**3*K2**2*K3 - 128*K1**3*K2*K3*K4 + 1216*K1**3*K2*K3 + 288*K1**3*K3*K4 - 1152*K1**3*K3 - 128*K1**2*K2**4 + 640*K1**2*K2**3 + 416*K1**2*K2**2*K4 - 6784*K1**2*K2**2 + 448*K1**2*K2*K3**2 + 128*K1**2*K2*K4**2 - 928*K1**2*K2*K4 + 11144*K1**2*K2 - 2048*K1**2*K3**2 - 128*K1**2*K3*K5 - 448*K1**2*K4**2 - 7020*K1**2 + 704*K1*K2**3*K3 + 160*K1*K2**2*K3*K4 - 2240*K1*K2**2*K3 + 32*K1*K2**2*K4*K5 - 288*K1*K2**2*K5 + 64*K1*K2*K3**3 + 32*K1*K2*K3*K4**2 - 864*K1*K2*K3*K4 + 10432*K1*K2*K3 + 3200*K1*K3*K4 + 528*K1*K4*K5 + 8*K1*K5*K6 - 64*K2**6 - 64*K2**4*K3**2 - 32*K2**4*K4**2 + 160*K2**4*K4 - 1216*K2**4 + 64*K2**3*K3*K5 + 32*K2**3*K4*K6 - 96*K2**3*K6 + 64*K2**2*K3**2*K4 - 1056*K2**2*K3**2 - 32*K2**2*K3*K7 + 32*K2**2*K4**3 - 344*K2**2*K4**2 + 2648*K2**2*K4 - 32*K2**2*K5**2 - 8*K2**2*K6**2 - 6522*K2**2 - 96*K2*K3**2*K4 - 96*K2*K3*K4*K5 + 952*K2*K3*K5 - 32*K2*K4**2*K6 + 280*K2*K4*K6 + 48*K2*K5*K7 + 8*K2*K6*K8 - 64*K3**4 - 48*K3**2*K4**2 + 104*K3**2*K6 - 3668*K3**2 + 64*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 1554*K4**2 - 296*K5**2 - 110*K6**2 - 24*K7**2 - 2*K8**2 + 6818

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1163']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4074', 'vk6.4105', 'vk6.5312', 'vk6.5343', 'vk6.7444', 'vk6.7473', 'vk6.8943', 'vk6.8974', 'vk6.10108', 'vk6.10279', 'vk6.10302', 'vk6.14545', 'vk6.15289', 'vk6.15416', 'vk6.15767', 'vk6.16184', 'vk6.29850', 'vk6.29881', 'vk6.33935', 'vk6.34011', 'vk6.34212', 'vk6.34393', 'vk6.48470', 'vk6.49173', 'vk6.50224', 'vk6.50255', 'vk6.51584', 'vk6.53970', 'vk6.54032', 'vk6.54171', 'vk6.54471', 'vk6.63297']

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	O1O2O3O4U1U4U5O6O5U2U6U3
R3 orbit	{'O1O2O3O4U1U4U5O6O5U2U6U3'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4U2U5U3O6O5U6U1U4
Gauss code of K*	O1O2O3U4U3O5O4O6U1U5U6U2
Gauss code of -K*	O1O2O3U4U2O4O5O6U5U1U3U6
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 -1 2 1 1 0],[ 3 0 2 3 1 3 1],[ 1 -2 0 2 0 1 0],[-2 -3 -2 0 0 -1 -1],[-1 -1 0 0 0 -1 0],[-1 -3 -1 1 1 0 0],[ 0 -1 0 1 0 0 0]]
Primitive based matrix	[[ 0 2 1 1 0 -1 -3],[-2 0 0 -1 -1 -2 -3],[-1 0 0 -1 0 0 -1],[-1 1 1 0 0 -1 -3],[ 0 1 0 0 0 0 -1],[ 1 2 0 1 0 0 -2],[ 3 3 1 3 1 2 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-2,-1,-1,0,1,3,0,1,1,2,3,1,0,0,1,0,1,3,0,1,2]
Phi over symmetry	[-3,-1,0,1,1,2,0,2,1,3,2,1,1,2,1,1,1,1,-1,0,1]
Phi of -K	[-3,-1,0,1,1,2,0,2,1,3,2,1,1,2,1,1,1,1,-1,0,1]
Phi of K*	[-2,-1,-1,0,1,3,0,1,1,1,2,1,1,1,1,1,2,3,1,2,0]
Phi of -K*	[-3,-1,0,1,1,2,2,1,1,3,3,0,0,1,2,0,0,1,-1,0,1]
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^6+32t^4+31t^2+1
Outer characteristic polynomial	t^7+48t^5+60t^3+4t
Flat arrow polynomial	8K13 + 4K1*2K2 - 12K12 - 10K1K2 - 2K1K3 - K1 - 2K2*2 + 5K2 + 3*K3 + K4 + 7
2-strand cable arrow polynomial	-192K16 - 192K1*4K2*2 + 2272K1*4K2 - 4736K14 - 256K1*3K2*2K3 - 128K13K2K3K4 + 1216K13K2K3 + 288K1*3K3K4 - 1152K1*3K3 - 128K12K2*4 + 640K1*2K2*3 + 416K1*2K2*2K4 - 6784K12K2*2 + 448K1*2K2K32 + 128K1*2K2K42 - 928K1*2K2K4 + 11144K1*2K2 - 2048K12K3*2 - 128K1*2K3K5 - 448K1*2K4*2 - 7020K1*2 + 704K1K23K3 + 160K1K2*2K3K4 - 2240K1K22K3 + 32K1K2*2K4K5 - 288K1K22K5 + 64K1K2K33 + 32K1K2K3K42 - 864K1K2K3K4 + 10432K1K2K3 + 3200K1K3K4 + 528K1K4K5 + 8K1K5K6 - 64K2*6 - 64K2*4K3*2 - 32K2*4K4*2 + 160K2*4K4 - 1216K24 + 64K2*3K3K5 + 32K2*3K4K6 - 96K2*3K6 + 64K22K3*2K4 - 1056K22K3*2 - 32K2*2K3K7 + 32K2*2K4*3 - 344K2*2K4*2 + 2648K2*2K4 - 32K22K5*2 - 8K2*2K6*2 - 6522K2*2 - 96K2K32K4 - 96K2K3K4K5 + 952K2K3K5 - 32K2K42K6 + 280K2K4K6 + 48K2K5K7 + 8K2K6K8 - 64K3*4 - 48K3*2K4*2 + 104K3*2K6 - 3668K32 + 64K3K4K7 - 8K44 + 8K4*2K8 - 1554K42 - 296K5*2 - 110K6*2 - 24K7*2 - 2K8**2 + 6818
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}], [{6}, {5}, {4}, {2, 3}, {1}], [{6}, {5}, {4}, {3}, {1, 2}]]
If K is slice	False

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