10 136: Difference between revisions

Revision as of 10:39, 30 August 2005

10_135

10_137

(KnotPlot image)

See the full Rolfsen Knot Table.

Visit 10 136's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 10 136 at Knotilus!

[edit 10 136 Quick Notes]

[edit 10 136 Further Notes and Views]

Knot presentations

Planar diagram presentation	X₁₄₂₅ X_5,10,6,11 X₃₉₄₈ X_9,3,10,2 X_14,8,15,7 X_18,12,19,11 X_20,15,1,16 X_16,19,17,20 X_12,18,13,17 X_6,14,7,13
Gauss code	-1, 4, -3, 1, -2, -10, 5, 3, -4, 2, 6, -9, 10, -5, 7, -8, 9, -6, 8, -7
Dowker-Thistlethwaite code	4 8 10 -14 2 -18 -6 -20 -12 -16
Conway Notation	[22,22,2-]

Minimum Braid Representative

A Morse Link Presentation

An Arc Presentation

Length is 10, width is 5,

Braid index is 4

[{11, 2}, {1, 7}, {9, 5}, {7, 11}, {8, 10}, {2, 9}, {6, 4}, {5, 8}, {3, 6}, {4, 1}, {10, 3}]

[edit Notes on presentations of 10 136] The knot 10_136 is the only knot in the Rolfsen Knot Table whose braid index is smaller than the width of its minimum braid.

The next such knot is K11n8.

Computer Talk

The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:= AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of May 31, 2006, 14:15:20.091.

Read more at http://katlas.org/wiki/KnotTheory.

In[3]:= K = Knot["10 136"];

In[4]:= PD[K]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]= X₁₄₂₅ X_5,10,6,11 X₃₉₄₈ X_9,3,10,2 X_14,8,15,7 X_18,12,19,11 X_20,15,1,16 X_16,19,17,20 X_12,18,13,17 X_6,14,7,13

In[5]:= GaussCode[K]

Out[5]= -1, 4, -3, 1, -2, -10, 5, 3, -4, 2, 6, -9, 10, -5, 7, -8, 9, -6, 8, -7

In[6]:= DTCode[K]

Out[6]= 4 8 10 -14 2 -18 -6 -20 -12 -16

(The path below may be different on your system)

In[7]:= AppendTo[$Path, "C:/bin/LinKnot/"];

In[8]:= ConwayNotation[K]

Out[8]= [22,22,2-]

In[9]:= br = BR[K]

KnotTheory::credits: The minimum braids representing the knots with up to 10 crossings were provided by Thomas Gittings. See arXiv:math.GT/0401051.

Out[9]= ${\textrm {BR}}(5,\{1,-2,1,-2,-3,2,2,4,-3,4\})$

In[10]:= {First[br], Crossings[br], BraidIndex[K]}

KnotTheory::credits: The braid index data known to KnotTheory` is taken from Charles Livingston's http://www.indiana.edu/~knotinfo/.

KnotTheory::loading: Loading precomputed data in IndianaData`.

Out[10]= { 5, 10, 4 }

In[11]:= Show[BraidPlot[br]]

Out[11]= -Graphics-

In[12]:= Show[DrawMorseLink[K]]

KnotTheory::credits: "MorseLink was added to KnotTheory` by Siddarth Sankaran at the University of Toronto in the summer of 2005."

KnotTheory::credits: "DrawMorseLink was written by Siddarth Sankaran at the University of Toronto in the summer of 2005."

Out[12]= -Graphics-

In[13]:= ap = ArcPresentation[K]

Out[13]= ArcPresentation[{11, 2}, {1, 7}, {9, 5}, {7, 11}, {8, 10}, {2, 9}, {6, 4}, {5, 8}, {3, 6}, {4, 1}, {10, 3}]

In[14]:= Draw[ap]

Out[14]= -Graphics-

Three dimensional invariants

Symmetry type	Reversible
Unknotting number	1
3-genus	2
Bridge index	3
Super bridge index	Missing
Nakanishi index	1
Maximal Thurston-Bennequin number	[-3][-6]
Hyperbolic Volume	7.74627
A-Polynomial	See Data:10 136/A-polynomial

[edit Notes for 10 136's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus	$1$
Topological 4 genus	$1$
Concordance genus	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2}
Rasmussen s-Invariant	0

[edit Notes for 10 136's four dimensional invariants]

Polynomial invariants

Alexander polynomial	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -t^2+4 t-5+4 t^{-1} - t^{-2} }
Conway polynomial	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1-z^4}
2nd Alexander ideal (db, data sources)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{1\}}
Determinant and Signature	{ 15, 2 }
Jones polynomial	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^4+2 q^3-2 q^2+3 q-2+2 q^{-1} -2 q^{-2} + q^{-3} }
HOMFLY-PT polynomial (db, data sources)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -z^4+a^2 z^2+2 z^2 a^{-2} -3 z^2+a^2+3 a^{-2} - a^{-4} -2}
Kauffman polynomial (db, data sources)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^8 a^{-2} +z^8+2 a z^7+3 z^7 a^{-1} +z^7 a^{-3} +a^2 z^6-4 z^6 a^{-2} -3 z^6-9 a z^5-14 z^5 a^{-1} -5 z^5 a^{-3} -4 a^2 z^4+2 z^4 a^{-2} -2 z^4+9 a z^3+16 z^3 a^{-1} +7 z^3 a^{-3} +3 a^2 z^2+4 z^2 a^{-2} +z^2 a^{-4} +6 z^2-2 a z-4 z a^{-1} -2 z a^{-3} -a^2-3 a^{-2} - a^{-4} -2}
The A2 invariant	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{10}-q^2+ q^{-4} +2 q^{-6} + q^{-8} + q^{-10} - q^{-12} - q^{-14} }
The G2 invariant	$q^{46}-q^{44}+2q^{42}-2q^{40}+q^{38}-2q^{34}+6q^{32}-4q^{30}+3q^{28}-2q^{24}+3q^{22}-2q^{20}-q^{18}+3q^{16}-3q^{14}+3q^{10}-6q^{8}+6q^{6}-7q^{4}+3-5q^{-2}+4q^{-4}-3q^{-6}+3q^{-8}+q^{-12}-q^{-14}+2q^{-18}+q^{-20}+q^{-24}+3q^{-26}+4q^{-30}-6q^{-32}+6q^{-34}-2q^{-36}+4q^{-40}-7q^{-42}+6q^{-44}-2q^{-48}+q^{-50}-2q^{-52}-2q^{-54}+3q^{-56}-3q^{-58}+q^{-60}+q^{-62}-3q^{-64}+3q^{-66}-3q^{-68}+q^{-70}+q^{-72}-2q^{-74}+q^{-76}$

Further Quantum Invariants

Further quantum knot invariants for 10_136.

A1 Invariants.

Weight	Invariant
1	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^7-q^5+ q^{-1} + q^{-3} + q^{-7} - q^{-9} }
2	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{22}-q^{20}-2 q^{18}+2 q^{16}+q^{14}-q^{12}+q^8+q^6-2 q^4+2- q^{-2} +2 q^{-6} - q^{-8} +2 q^{-12} + q^{-14} - q^{-16} - q^{-18} + q^{-20} -2 q^{-24} + q^{-28} }
3	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{45}-q^{43}-2 q^{41}+3 q^{37}+3 q^{35}-3 q^{33}-3 q^{31}+2 q^{27}+2 q^{25}-2 q^{21}-4 q^{19}+q^{17}+6 q^{15}+2 q^{13}-6 q^{11}-4 q^9+5 q^7+6 q^5-3 q^3-5 q+2 q^{-1} +6 q^{-3} - q^{-5} -4 q^{-7} + q^{-9} +3 q^{-11} - q^{-13} -2 q^{-15} + q^{-17} + q^{-19} +2 q^{-21} -4 q^{-25} - q^{-27} +7 q^{-29} +5 q^{-31} -7 q^{-33} -10 q^{-35} +5 q^{-37} +8 q^{-39} -2 q^{-41} -7 q^{-43} - q^{-45} +5 q^{-47} +2 q^{-49} - q^{-51} - q^{-53} }

A2 Invariants.

Weight

Invariant

1,0

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{10}-q^2+ q^{-4} +2 q^{-6} + q^{-8} + q^{-10} - q^{-12} - q^{-14} }

2,0

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{28}-q^{24}-q^{22}+q^{18}+2 q^8+q^6+q^2+1-2 q^{-4} - q^{-6} - q^{-8} - q^{-10} +2 q^{-14} +4 q^{-16} +3 q^{-18} +4 q^{-20} - q^{-22} - q^{-24} -2 q^{-26} -2 q^{-28} -2 q^{-30} -2 q^{-32} + q^{-34} + q^{-36} + q^{-38} }

A3 Invariants.

Weight	Invariant
0,1,0	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{20}-q^{18}+q^{14}+q^{10}+q^6-q^4-q^2- q^{-2} + q^{-6} +3 q^{-8} + q^{-10} +3 q^{-12} + q^{-14} - q^{-18} - q^{-20} - q^{-24} }
1,0,0	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{13}+q^9-q^3-q- q^{-1} + q^{-5} +2 q^{-7} +3 q^{-9} + q^{-11} + q^{-13} - q^{-15} - q^{-17} - q^{-19} }

B2 Invariants.

Weight	Invariant
0,1	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{20}-q^{18}+2 q^{16}-q^{14}+2 q^{12}-q^{10}-q^6-q^4+q^2-4+3 q^{-2} -2 q^{-4} +3 q^{-6} - q^{-8} +3 q^{-10} + q^{-12} + q^{-14} +2 q^{-16} - q^{-18} + q^{-20} -2 q^{-22} + q^{-24} -2 q^{-26} }
1,0	$q^{34}-q^{30}-q^{28}+q^{26}+q^{24}-q^{22}-q^{20}+2q^{18}+2q^{16}-q^{14}-q^{12}+2q^{8}-2q^{4}-q^{2}+1-q^{-4}+2q^{-10}+q^{-12}+3q^{-18}+2q^{-20}-2q^{-24}+q^{-26}+q^{-28}-q^{-30}-2q^{-32}+q^{-36}-q^{-40}-q^{-42}+q^{-44}$

G2 Invariants.

Weight

Invariant

1,0

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{46}-q^{44}+2 q^{42}-2 q^{40}+q^{38}-2 q^{34}+6 q^{32}-4 q^{30}+3 q^{28}-2 q^{24}+3 q^{22}-2 q^{20}-q^{18}+3 q^{16}-3 q^{14}+3 q^{10}-6 q^8+6 q^6-7 q^4+3-5 q^{-2} +4 q^{-4} -3 q^{-6} +3 q^{-8} + q^{-12} - q^{-14} +2 q^{-18} + q^{-20} + q^{-24} +3 q^{-26} +4 q^{-30} -6 q^{-32} +6 q^{-34} -2 q^{-36} +4 q^{-40} -7 q^{-42} +6 q^{-44} -2 q^{-48} + q^{-50} -2 q^{-52} -2 q^{-54} +3 q^{-56} -3 q^{-58} + q^{-60} + q^{-62} -3 q^{-64} +3 q^{-66} -3 q^{-68} + q^{-70} + q^{-72} -2 q^{-74} + q^{-76} }

.

Computer Talk

The above data is available with the Mathematica package KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:= AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of August 31, 2006, 11:25:27.5625.

In[3]:= K = Knot["10 136"];

In[4]:= Alexander[K][t]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]= Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -t^2+4 t-5+4 t^{-1} - t^{-2} }

In[5]:= Conway[K][z]

Out[5]= Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1-z^4}

In[6]:= Alexander[K, 2][t]

KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.

Out[6]= Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{1\}}

In[7]:= {KnotDet[K], KnotSignature[K]}

Out[7]= { 15, 2 }

In[8]:= Jones[K][q]

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[8]= Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^4+2 q^3-2 q^2+3 q-2+2 q^{-1} -2 q^{-2} + q^{-3} }

In[9]:= HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]= Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -z^4+a^2 z^2+2 z^2 a^{-2} -3 z^2+a^2+3 a^{-2} - a^{-4} -2}

In[10]:= Kauffman[K][a, z]

KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.

Out[10]= Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^8 a^{-2} +z^8+2 a z^7+3 z^7 a^{-1} +z^7 a^{-3} +a^2 z^6-4 z^6 a^{-2} -3 z^6-9 a z^5-14 z^5 a^{-1} -5 z^5 a^{-3} -4 a^2 z^4+2 z^4 a^{-2} -2 z^4+9 a z^3+16 z^3 a^{-1} +7 z^3 a^{-3} +3 a^2 z^2+4 z^2 a^{-2} +z^2 a^{-4} +6 z^2-2 a z-4 z a^{-1} -2 z a^{-3} -a^2-3 a^{-2} - a^{-4} -2}

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {8_21,}

Same Jones Polynomial (up to mirroring, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q\leftrightarrow q^{-1}} ): {K11n92,}

Computer Talk

The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:= AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of May 31, 2006, 14:15:20.091.

In[3]:= K = Knot["10 136"];

In[4]:= {A = Alexander[K][t], J = Jones[K][q]}

KnotTheory::loading: Loading precomputed data in PD4Knots`.

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[4]= { Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -t^2+4 t-5+4 t^{-1} - t^{-2} } , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^4+2 q^3-2 q^2+3 q-2+2 q^{-1} -2 q^{-2} + q^{-3} } }

In[5]:= DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]

KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.

KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.

Out[5]= {8_21,}

In[6]:= DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]

KnotTheory::loading: Loading precomputed data in Jones4Knots11`.

Out[6]= {K11n92,}

Vassiliev invariants

V₂ and V₃:

(0, 1)

V_2,1 through V_6,9:

V_2,1

V_3,1

V_4,1

V_4,2

V_4,3

V_5,1

V_5,2

V_5,3

V_5,4

V_6,1

V_6,2

V_6,3

V_6,4

V_6,5

V_6,6

V_6,7

V_6,8

V_6,9