K11a76

K11a75

K11a77

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a76 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X10,3,11,4 X12,6,13,5 X14,7,15,8 X18,9,19,10 X2,11,3,12 X20,14,21,13 X8,15,9,16 X22,18,1,17 X6,19,7,20 X16,22,17,21
Gauss code	1, -6, 2, -1, 3, -10, 4, -8, 5, -2, 6, -3, 7, -4, 8, -11, 9, -5, 10, -7, 11, -9
Dowker-Thistlethwaite code	4 10 12 14 18 2 20 8 22 6 16

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Chiral Unknotting number [math]\displaystyle{ \{1,2\} }[/math] 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a76/ThurstonBennequinNumber Hyperbolic Volume 16.9805 A-Polynomial See Data:K11a76/A-polynomial

[edit Notes for K11a76's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus [math]\displaystyle{ 4 }[/math] Rasmussen s-Invariant 0

[edit Notes for K11a76's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ t^4-6 t^3+17 t^2-30 t+37-30 t^{-1} +17 t^{-2} -6 t^{-3} + t^{-4} }[/math]
Conway polynomial	[math]\displaystyle{ z^8+2 z^6+z^4+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 145, 0 }
Jones polynomial	[math]\displaystyle{ -q^5+4 q^4-9 q^3+15 q^2-20 q+24-23 q^{-1} +20 q^{-2} -15 q^{-3} +9 q^{-4} -4 q^{-5} + q^{-6} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ z^8-2 a^2 z^6-z^6 a^{-2} +5 z^6+a^4 z^4-7 a^2 z^4-3 z^4 a^{-2} +10 z^4+2 a^4 z^2-8 a^2 z^2-3 z^2 a^{-2} +9 z^2+a^4-3 a^2- a^{-2} +4 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ 2 a^2 z^{10}+2 z^{10}+6 a^3 z^9+13 a z^9+7 z^9 a^{-1} +7 a^4 z^8+14 a^2 z^8+10 z^8 a^{-2} +17 z^8+4 a^5 z^7-5 a^3 z^7-19 a z^7-2 z^7 a^{-1} +8 z^7 a^{-3} +a^6 z^6-15 a^4 z^6-41 a^2 z^6-16 z^6 a^{-2} +4 z^6 a^{-4} -45 z^6-9 a^5 z^5-10 a^3 z^5+a z^5-11 z^5 a^{-1} -12 z^5 a^{-3} +z^5 a^{-5} -2 a^6 z^4+9 a^4 z^4+35 a^2 z^4+12 z^4 a^{-2} -5 z^4 a^{-4} +41 z^4+6 a^5 z^3+9 a^3 z^3+6 a z^3+10 z^3 a^{-1} +6 z^3 a^{-3} -z^3 a^{-5} +a^6 z^2-3 a^4 z^2-15 a^2 z^2-5 z^2 a^{-2} +z^2 a^{-4} -17 z^2-a^5 z-2 a^3 z-2 a z-2 z a^{-1} -z a^{-3} +a^4+3 a^2+ a^{-2} +4 }[/math]
The A2 invariant	[math]\displaystyle{ q^{18}-q^{16}+2 q^{12}-4 q^{10}+3 q^8-2 q^6-q^4+4 q^2-3+6 q^{-2} -3 q^{-4} + q^{-6} +2 q^{-8} -3 q^{-10} +2 q^{-12} - q^{-14} }[/math]
The G2 invariant	[math]\displaystyle{ q^{94}-3 q^{92}+8 q^{90}-16 q^{88}+22 q^{86}-25 q^{84}+14 q^{82}+18 q^{80}-67 q^{78}+130 q^{76}-176 q^{74}+169 q^{72}-88 q^{70}-77 q^{68}+294 q^{66}-480 q^{64}+556 q^{62}-446 q^{60}+129 q^{58}+307 q^{56}-717 q^{54}+935 q^{52}-836 q^{50}+440 q^{48}+134 q^{46}-670 q^{44}+941 q^{42}-842 q^{40}+406 q^{38}+170 q^{36}-633 q^{34}+764 q^{32}-509 q^{30}-20 q^{28}+597 q^{26}-952 q^{24}+914 q^{22}-475 q^{20}-231 q^{18}+920 q^{16}-1333 q^{14}+1302 q^{12}-812 q^{10}+53 q^8+721 q^6-1226 q^4+1288 q^2-905+241 q^{-2} +429 q^{-4} -839 q^{-6} +843 q^{-8} -458 q^{-10} -93 q^{-12} +573 q^{-14} -751 q^{-16} +553 q^{-18} -85 q^{-20} -462 q^{-22} +844 q^{-24} -903 q^{-26} +640 q^{-28} -155 q^{-30} -349 q^{-32} +694 q^{-34} -793 q^{-36} +646 q^{-38} -344 q^{-40} +3 q^{-42} +259 q^{-44} -394 q^{-46} +389 q^{-48} -285 q^{-50} +149 q^{-52} -15 q^{-54} -76 q^{-56} +114 q^{-58} -115 q^{-60} +84 q^{-62} -46 q^{-64} +16 q^{-66} +7 q^{-68} -16 q^{-70} +16 q^{-72} -13 q^{-74} +7 q^{-76} -3 q^{-78} + q^{-80} }[/math]

Further Quantum Invariants

K11a76 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. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["K11a76"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ t^4-6 t^3+17 t^2-30 t+37-30 t^{-1} +17 t^{-2} -6 t^{-3} + t^{-4} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ z^8+2 z^6+z^4+1 }[/math]

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]=

[math]\displaystyle{ \{1\} }[/math]

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 145, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ -q^5+4 q^4-9 q^3+15 q^2-20 q+24-23 q^{-1} +20 q^{-2} -15 q^{-3} +9 q^{-4} -4 q^{-5} + q^{-6} }[/math]

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

[math]\displaystyle{ z^8-2 a^2 z^6-z^6 a^{-2} +5 z^6+a^4 z^4-7 a^2 z^4-3 z^4 a^{-2} +10 z^4+2 a^4 z^2-8 a^2 z^2-3 z^2 a^{-2} +9 z^2+a^4-3 a^2- a^{-2} +4 }[/math]

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ 2 a^2 z^{10}+2 z^{10}+6 a^3 z^9+13 a z^9+7 z^9 a^{-1} +7 a^4 z^8+14 a^2 z^8+10 z^8 a^{-2} +17 z^8+4 a^5 z^7-5 a^3 z^7-19 a z^7-2 z^7 a^{-1} +8 z^7 a^{-3} +a^6 z^6-15 a^4 z^6-41 a^2 z^6-16 z^6 a^{-2} +4 z^6 a^{-4} -45 z^6-9 a^5 z^5-10 a^3 z^5+a z^5-11 z^5 a^{-1} -12 z^5 a^{-3} +z^5 a^{-5} -2 a^6 z^4+9 a^4 z^4+35 a^2 z^4+12 z^4 a^{-2} -5 z^4 a^{-4} +41 z^4+6 a^5 z^3+9 a^3 z^3+6 a z^3+10 z^3 a^{-1} +6 z^3 a^{-3} -z^3 a^{-5} +a^6 z^2-3 a^4 z^2-15 a^2 z^2-5 z^2 a^{-2} +z^2 a^{-4} -17 z^2-a^5 z-2 a^3 z-2 a z-2 z a^{-1} -z a^{-3} +a^4+3 a^2+ a^{-2} +4 }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a160, K11a289,}

Same Jones Polynomial (up to mirroring, [math]\displaystyle{ q\leftrightarrow q^{-1} }[/math]): {K11a160, 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. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["K11a76"];

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]=

{ [math]\displaystyle{ t^4-6 t^3+17 t^2-30 t+37-30 t^{-1} +17 t^{-2} -6 t^{-3} + t^{-4} }[/math], [math]\displaystyle{ -q^5+4 q^4-9 q^3+15 q^2-20 q+24-23 q^{-1} +20 q^{-2} -15 q^{-3} +9 q^{-4} -4 q^{-5} + q^{-6} }[/math] }

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]=

{K11a160, 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]=

{K11a160, K11a289,}

Vassiliev invariants

V2 and V3:

(0, 1)

V2,1 through V6,9:

V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9 [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 8 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ -16 }[/math] [math]\displaystyle{ -8 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ -\frac{16}{3} }[/math] [math]\displaystyle{ -\frac{160}{3} }[/math] [math]\displaystyle{ 40 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 72 }[/math] [math]\displaystyle{ -24 }[/math] [math]\displaystyle{ \frac{344}{3} }[/math] [math]\displaystyle{ -\frac{136}{3} }[/math] [math]\displaystyle{ 24 }[/math]

V_2,1 through V_6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V₂ and V₃.

Khovanov Homology

The coefficients of the monomials [math]\displaystyle{ t^rq^j }[/math] are shown, along with their alternating sums [math]\displaystyle{ \chi }[/math] (fixed [math]\displaystyle{ j }[/math], alternation over [math]\displaystyle{ r }[/math]). The squares with yellow highlighting are those on the "critical diagonals", where [math]\displaystyle{ j-2r=s+1 }[/math] or [math]\displaystyle{ j-2r=s-1 }[/math], where [math]\displaystyle{ s= }[/math]0 is the signature of K11a76. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 11                       1 -1 9                     3   3 7                   6 1   -5 5                 9 3     6 3               11 6       -5 1             13 9         4 -1           11 12           1 -3         9 12             -3 -5       6 11               5 -7     3 9                 -6 -9   1 6                   5 -11   3                     -3 -13 1                       1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-1 }[/math] [math]\displaystyle{ i=1 }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{12} }[/math] [math]\displaystyle{ {\mathbb Z}^{13} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

Modifying This Page

Read me first: Modifying Knot Pages. See/edit the Hoste-Thistlethwaite Knot Page master template (intermediate). See/edit the Hoste-Thistlethwaite_Splice_Base (expert). Back to the top.

K11a75 K11a77