K11a160

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a160 at Knotilus!
	[edit K11a160 Quick Notes]

[edit K11a160 Further Notes and Views]

Knot presentations

Planar diagram presentation	X₄₂₅₁ X_10,4,11,3 X_18,5,19,6 X_14,7,15,8 X_2,10,3,9 X_22,11,1,12 X_20,14,21,13 X_8,15,9,16 X_12,18,13,17 X_6,19,7,20 X_16,22,17,21
Gauss code	1, -5, 2, -1, 3, -10, 4, -8, 5, -2, 6, -9, 7, -4, 8, -11, 9, -3, 10, -7, 11, -6
Dowker-Thistlethwaite code	4 10 18 14 2 22 20 8 12 6 16

A Braid Representative

A Morse Link Presentation

Three dimensional invariants

Symmetry type	Reversible
Unknotting number	1
3-genus	4
Bridge index	3
Super bridge index	Missing
Nakanishi index	Missing
Maximal Thurston-Bennequin number	Data:K11a160/ThurstonBennequinNumber
Hyperbolic Volume	17.0985
A-Polynomial	See Data:K11a160/A-polynomial

[edit Notes for K11a160's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus	Missing
Topological 4 genus	Missing
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 4}
Rasmussen s-Invariant	0

[edit Notes for K11a160'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^4-6 t^3+17 t^2-30 t+37-30 t^{-1} +17 t^{-2} -6 t^{-3} + t^{-4} }
Conway polynomial	$z^{8}+2z^{6}+z^{4}+1$
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	{ 145, 0 }
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^6-4 q^5+9 q^4-15 q^3+20 q^2-23 q+24-20 q^{-1} +15 q^{-2} -9 q^{-3} +4 q^{-4} - q^{-5} }
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^8-a^2 z^6-2 z^6 a^{-2} +5 z^6-3 a^2 z^4-7 z^4 a^{-2} +z^4 a^{-4} +10 z^4-3 a^2 z^2-8 z^2 a^{-2} +2 z^2 a^{-4} +9 z^2-a^2-3 a^{-2} + a^{-4} +4}
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 2 z^{10} a^{-2} +2 z^{10}+7 a z^9+13 z^9 a^{-1} +6 z^9 a^{-3} +10 a^2 z^8+15 z^8 a^{-2} +7 z^8 a^{-4} +18 z^8+8 a^3 z^7-a z^7-17 z^7 a^{-1} -4 z^7 a^{-3} +4 z^7 a^{-5} +4 a^4 z^6-16 a^2 z^6-47 z^6 a^{-2} -15 z^6 a^{-4} +z^6 a^{-6} -51 z^6+a^5 z^5-12 a^3 z^5-17 a z^5-11 z^5 a^{-1} -16 z^5 a^{-3} -9 z^5 a^{-5} -5 a^4 z^4+12 a^2 z^4+44 z^4 a^{-2} +9 z^4 a^{-4} -2 z^4 a^{-6} +50 z^4-a^5 z^3+6 a^3 z^3+19 a z^3+24 z^3 a^{-1} +18 z^3 a^{-3} +6 z^3 a^{-5} +a^4 z^2-6 a^2 z^2-18 z^2 a^{-2} -4 z^2 a^{-4} +z^2 a^{-6} -20 z^2-2 a^3 z-6 a z-8 z a^{-1} -6 z a^{-3} -2 z a^{-5} +a^2+3 a^{-2} + a^{-4} +4}
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^{14}+2 q^{12}-3 q^{10}+2 q^8+q^6-3 q^4+6 q^2-3+4 q^{-2} - q^{-4} -2 q^{-6} +3 q^{-8} -4 q^{-10} +2 q^{-12} - q^{-16} + q^{-18} }
The G2 invariant	$q^{80}-3q^{78}+7q^{76}-13q^{74}+16q^{72}-16q^{70}+7q^{68}+16q^{66}-46q^{64}+84q^{62}-113q^{60}+111q^{58}-73q^{56}-15q^{54}+143q^{52}-275q^{50}+376q^{48}-386q^{46}+258q^{44}-5q^{42}-327q^{40}+628q^{38}-780q^{36}+692q^{34}-355q^{32}-146q^{30}+635q^{28}-907q^{26}+852q^{24}-467q^{22}-87q^{20}+565q^{18}-770q^{16}+591q^{14}-104q^{12}-456q^{10}+848q^{8}-849q^{6}+439q^{4}+234q^{2}-898+1280q^{-2}-1218q^{-4}+713q^{-6}+58q^{-8}-812q^{-10}+1298q^{-12}-1328q^{-14}+916q^{-16}-225q^{-18}-485q^{-20}+928q^{-22}-967q^{-24}+608q^{-26}-22q^{-28}-518q^{-30}+780q^{-32}-647q^{-34}+179q^{-36}+404q^{-38}-847q^{-40}+945q^{-42}-667q^{-44}+124q^{-46}+452q^{-48}-845q^{-50}+935q^{-52}-705q^{-54}+288q^{-56}+144q^{-58}-455q^{-60}+556q^{-62}-473q^{-64}+285q^{-66}-69q^{-68}-91q^{-70}+170q^{-72}-173q^{-74}+127q^{-76}-66q^{-78}+18q^{-80}+13q^{-82}-25q^{-84}+22q^{-86}-16q^{-88}+8q^{-90}-3q^{-92}+q^{-94}$

Further Quantum Invariants

K11a160 Quantum Invariants.

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["K11a160"];

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^4-6 t^3+17 t^2-30 t+37-30 t^{-1} +17 t^{-2} -6 t^{-3} + t^{-4} }

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

Out[5]= $z^{8}+2z^{6}+z^{4}+1$

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]= { 145, 0 }

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^6-4 q^5+9 q^4-15 q^3+20 q^2-23 q+24-20 q^{-1} +15 q^{-2} -9 q^{-3} +4 q^{-4} - q^{-5} }

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^8-a^2 z^6-2 z^6 a^{-2} +5 z^6-3 a^2 z^4-7 z^4 a^{-2} +z^4 a^{-4} +10 z^4-3 a^2 z^2-8 z^2 a^{-2} +2 z^2 a^{-4} +9 z^2-a^2-3 a^{-2} + a^{-4} +4}

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 2 z^{10} a^{-2} +2 z^{10}+7 a z^9+13 z^9 a^{-1} +6 z^9 a^{-3} +10 a^2 z^8+15 z^8 a^{-2} +7 z^8 a^{-4} +18 z^8+8 a^3 z^7-a z^7-17 z^7 a^{-1} -4 z^7 a^{-3} +4 z^7 a^{-5} +4 a^4 z^6-16 a^2 z^6-47 z^6 a^{-2} -15 z^6 a^{-4} +z^6 a^{-6} -51 z^6+a^5 z^5-12 a^3 z^5-17 a z^5-11 z^5 a^{-1} -16 z^5 a^{-3} -9 z^5 a^{-5} -5 a^4 z^4+12 a^2 z^4+44 z^4 a^{-2} +9 z^4 a^{-4} -2 z^4 a^{-6} +50 z^4-a^5 z^3+6 a^3 z^3+19 a z^3+24 z^3 a^{-1} +18 z^3 a^{-3} +6 z^3 a^{-5} +a^4 z^2-6 a^2 z^2-18 z^2 a^{-2} -4 z^2 a^{-4} +z^2 a^{-6} -20 z^2-2 a^3 z-6 a z-8 z a^{-1} -6 z a^{-3} -2 z a^{-5} +a^2+3 a^{-2} + a^{-4} +4}

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a76, K11a289,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$ ): {K11a76, K11a289,}

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["K11a160"];

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]= { $t^{4}-6t^{3}+17t^{2}-30t+37-30t^{-1}+17t^{-2}-6t^{-3}+t^{-4}$ , $q^{6}-4q^{5}+9q^{4}-15q^{3}+20q^{2}-23q+24-20q^{-1}+15q^{-2}-9q^{-3}+4q^{-4}-q^{-5}$ }

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]= {K11a76, K11a289,}

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

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

Out[6]= {K11a76, K11a289,}

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