K11a68

K11a67

K11a69

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a68 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X8394 X16,5,17,6 X14,8,15,7 X2,9,3,10 X18,12,19,11 X20,14,21,13 X22,15,1,16 X10,18,11,17 X12,20,13,19 X6,21,7,22
Gauss code	1, -5, 2, -1, 3, -11, 4, -2, 5, -9, 6, -10, 7, -4, 8, -3, 9, -6, 10, -7, 11, -8
Dowker-Thistlethwaite code	4 8 16 14 2 18 20 22 10 12 6

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:K11a68/ThurstonBennequinNumber Hyperbolic Volume 14.7003 A-Polynomial See Data:K11a68/A-polynomial

[edit Notes for K11a68's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus [math]\displaystyle{ 4 }[/math] Rasmussen s-Invariant -2

[edit Notes for K11a68's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ -t^4+6 t^3-14 t^2+20 t-21+20 t^{-1} -14 t^{-2} +6 t^{-3} - t^{-4} }[/math]
Conway polynomial	[math]\displaystyle{ -z^8-2 z^6+2 z^4+2 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 103, 2 }
Jones polynomial	[math]\displaystyle{ q^7-4 q^6+7 q^5-11 q^4+15 q^3-16 q^2+16 q-13+10 q^{-1} -6 q^{-2} +3 q^{-3} - q^{-4} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ -z^8 a^{-2} -5 z^6 a^{-2} +z^6 a^{-4} +2 z^6-a^2 z^4-8 z^4 a^{-2} +3 z^4 a^{-4} +8 z^4-3 a^2 z^2-4 z^2 a^{-2} +z^2 a^{-4} +8 z^2-a^2+ a^{-2} - a^{-4} +2 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ 2 z^{10} a^{-2} +2 z^{10}+4 a z^9+10 z^9 a^{-1} +6 z^9 a^{-3} +3 a^2 z^8+6 z^8 a^{-2} +8 z^8 a^{-4} +z^8+a^3 z^7-14 a z^7-33 z^7 a^{-1} -10 z^7 a^{-3} +8 z^7 a^{-5} -12 a^2 z^6-30 z^6 a^{-2} -11 z^6 a^{-4} +7 z^6 a^{-6} -24 z^6-4 a^3 z^5+12 a z^5+32 z^5 a^{-1} +5 z^5 a^{-3} -7 z^5 a^{-5} +4 z^5 a^{-7} +14 a^2 z^4+30 z^4 a^{-2} +2 z^4 a^{-4} -7 z^4 a^{-6} +z^4 a^{-8} +34 z^4+4 a^3 z^3-2 a z^3-10 z^3 a^{-1} -4 z^3 a^{-3} -3 z^3 a^{-5} -3 z^3 a^{-7} -7 a^2 z^2-7 z^2 a^{-2} +2 z^2 a^{-4} +z^2 a^{-6} -15 z^2-a^3 z-a z+z a^{-1} +3 z a^{-3} +2 z a^{-5} +a^2- a^{-2} - a^{-4} +2 }[/math]
The A2 invariant	[math]\displaystyle{ -q^{12}+q^8-q^6+2 q^4-2 q^2+1+2 q^{-2} - q^{-4} +5 q^{-6} -2 q^{-8} +2 q^{-10} - q^{-12} -2 q^{-14} + q^{-16} -2 q^{-18} + q^{-20} }[/math]
The G2 invariant	[math]\displaystyle{ q^{60}-2 q^{58}+5 q^{56}-9 q^{54}+11 q^{52}-13 q^{50}+6 q^{48}+11 q^{46}-36 q^{44}+65 q^{42}-85 q^{40}+75 q^{38}-31 q^{36}-52 q^{34}+146 q^{32}-211 q^{30}+217 q^{28}-139 q^{26}-7 q^{24}+171 q^{22}-287 q^{20}+306 q^{18}-209 q^{16}+35 q^{14}+141 q^{12}-251 q^{10}+246 q^8-135 q^6-24 q^4+167 q^2-218+159 q^{-2} -20 q^{-4} -145 q^{-6} +262 q^{-8} -285 q^{-10} +203 q^{-12} -34 q^{-14} -159 q^{-16} +317 q^{-18} -370 q^{-20} +309 q^{-22} -137 q^{-24} -74 q^{-26} +250 q^{-28} -329 q^{-30} +286 q^{-32} -138 q^{-34} -38 q^{-36} +174 q^{-38} -207 q^{-40} +137 q^{-42} -7 q^{-44} -119 q^{-46} +176 q^{-48} -148 q^{-50} +46 q^{-52} +72 q^{-54} -166 q^{-56} +201 q^{-58} -166 q^{-60} +85 q^{-62} +10 q^{-64} -97 q^{-66} +141 q^{-68} -155 q^{-70} +134 q^{-72} -86 q^{-74} +29 q^{-76} +33 q^{-78} -81 q^{-80} +104 q^{-82} -100 q^{-84} +75 q^{-86} -37 q^{-88} -3 q^{-90} +32 q^{-92} -49 q^{-94} +47 q^{-96} -32 q^{-98} +18 q^{-100} -2 q^{-102} -6 q^{-104} +9 q^{-106} -10 q^{-108} +6 q^{-110} -3 q^{-112} + q^{-114} }[/math]

Further Quantum Invariants

K11a68 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["K11a68"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ -t^4+6 t^3-14 t^2+20 t-21+20 t^{-1} -14 t^{-2} +6 t^{-3} - t^{-4} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ -z^8-2 z^6+2 z^4+2 z^2+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]=

{ 103, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ q^7-4 q^6+7 q^5-11 q^4+15 q^3-16 q^2+16 q-13+10 q^{-1} -6 q^{-2} +3 q^{-3} - q^{-4} }[/math]

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ 2 z^{10} a^{-2} +2 z^{10}+4 a z^9+10 z^9 a^{-1} +6 z^9 a^{-3} +3 a^2 z^8+6 z^8 a^{-2} +8 z^8 a^{-4} +z^8+a^3 z^7-14 a z^7-33 z^7 a^{-1} -10 z^7 a^{-3} +8 z^7 a^{-5} -12 a^2 z^6-30 z^6 a^{-2} -11 z^6 a^{-4} +7 z^6 a^{-6} -24 z^6-4 a^3 z^5+12 a z^5+32 z^5 a^{-1} +5 z^5 a^{-3} -7 z^5 a^{-5} +4 z^5 a^{-7} +14 a^2 z^4+30 z^4 a^{-2} +2 z^4 a^{-4} -7 z^4 a^{-6} +z^4 a^{-8} +34 z^4+4 a^3 z^3-2 a z^3-10 z^3 a^{-1} -4 z^3 a^{-3} -3 z^3 a^{-5} -3 z^3 a^{-7} -7 a^2 z^2-7 z^2 a^{-2} +2 z^2 a^{-4} +z^2 a^{-6} -15 z^2-a^3 z-a z+z a^{-1} +3 z a^{-3} +2 z a^{-5} +a^2- a^{-2} - a^{-4} +2 }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, [math]\displaystyle{ q\leftrightarrow q^{-1} }[/math]): {K11a111,}

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["K11a68"];

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^3-14 t^2+20 t-21+20 t^{-1} -14 t^{-2} +6 t^{-3} - t^{-4} }[/math], [math]\displaystyle{ q^7-4 q^6+7 q^5-11 q^4+15 q^3-16 q^2+16 q-13+10 q^{-1} -6 q^{-2} +3 q^{-3} - q^{-4} }[/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]=

{}

In[6]:=

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

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

Out[6]=

{K11a111,}

Vassiliev invariants

V2 and V3:

(2, 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{ 8 }[/math] [math]\displaystyle{ 8 }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ \frac{124}{3} }[/math] [math]\displaystyle{ -\frac{4}{3} }[/math] [math]\displaystyle{ 64 }[/math] [math]\displaystyle{ \frac{272}{3} }[/math] [math]\displaystyle{ \frac{224}{3} }[/math] [math]\displaystyle{ -56 }[/math] [math]\displaystyle{ \frac{256}{3} }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ \frac{992}{3} }[/math] [math]\displaystyle{ -\frac{32}{3} }[/math] [math]\displaystyle{ \frac{6271}{15} }[/math] [math]\displaystyle{ \frac{3916}{15} }[/math] [math]\displaystyle{ -\frac{8156}{45} }[/math] [math]\displaystyle{ \frac{401}{9} }[/math] [math]\displaystyle{ -\frac{929}{15} }[/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]2 is the signature of K11a68. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 6 χ 15                       1 1 13                     3   -3 11                   4 1   3 9                 7 3     -4 7               8 4       4 5             8 7         -1 3           8 8           0 1         6 9             3 -1       4 7               -3 -3     2 6                 4 -5   1 4                   -3 -7   2                     2 -9 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=3 }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=6 }[/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.

K11a67 K11a69