K11a126

K11a125

K11a127

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a126 at Knotilus!

Knot presentations

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

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number [math]\displaystyle{ \{1,2,3\} }[/math] 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a126/ThurstonBennequinNumber Hyperbolic Volume 17.2092 A-Polynomial See Data:K11a126/A-polynomial

[edit Notes for K11a126'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 K11a126's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ t^4-5 t^3+16 t^2-31 t+39-31 t^{-1} +16 t^{-2} -5 t^{-3} + t^{-4} }[/math]
Conway polynomial	[math]\displaystyle{ z^8+3 z^6+6 z^4+4 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 145, 0 }
Jones polynomial	[math]\displaystyle{ q^6-5 q^5+10 q^4-16 q^3+21 q^2-23 q+24-19 q^{-1} +14 q^{-2} -8 q^{-3} +3 q^{-4} - q^{-5} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ z^8-a^2 z^6-2 z^6 a^{-2} +6 z^6-4 a^2 z^4-7 z^4 a^{-2} +z^4 a^{-4} +16 z^4-7 a^2 z^2-9 z^2 a^{-2} +z^2 a^{-4} +19 z^2-4 a^2-4 a^{-2} +9 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ 2 z^{10} a^{-2} +2 z^{10}+6 a z^9+13 z^9 a^{-1} +7 z^9 a^{-3} +8 a^2 z^8+17 z^8 a^{-2} +9 z^8 a^{-4} +16 z^8+6 a^3 z^7-16 z^7 a^{-1} -5 z^7 a^{-3} +5 z^7 a^{-5} +3 a^4 z^6-13 a^2 z^6-50 z^6 a^{-2} -20 z^6 a^{-4} +z^6 a^{-6} -45 z^6+a^5 z^5-9 a^3 z^5-16 a z^5-13 z^5 a^{-1} -17 z^5 a^{-3} -10 z^5 a^{-5} -4 a^4 z^4+15 a^2 z^4+41 z^4 a^{-2} +11 z^4 a^{-4} -z^4 a^{-6} +48 z^4-2 a^5 z^3+7 a^3 z^3+23 a z^3+25 z^3 a^{-1} +15 z^3 a^{-3} +4 z^3 a^{-5} +a^4 z^2-11 a^2 z^2-17 z^2 a^{-2} -2 z^2 a^{-4} -27 z^2+a^5 z-3 a^3 z-10 a z-10 z a^{-1} -3 z a^{-3} +z a^{-5} +4 a^2+4 a^{-2} +9 }[/math]
The A2 invariant	[math]\displaystyle{ -q^{14}+q^{12}-4 q^{10}+q^8+q^6-3 q^4+7 q^2-1+5 q^{-2} + q^{-4} -2 q^{-6} +3 q^{-8} -5 q^{-10} + q^{-12} -2 q^{-16} + q^{-18} }[/math]
The G2 invariant	[math]\displaystyle{ q^{80}-2 q^{78}+5 q^{76}-8 q^{74}+10 q^{72}-10 q^{70}+4 q^{68}+10 q^{66}-29 q^{64}+52 q^{62}-73 q^{60}+75 q^{58}-57 q^{56}+5 q^{54}+81 q^{52}-176 q^{50}+264 q^{48}-301 q^{46}+243 q^{44}-90 q^{42}-162 q^{40}+435 q^{38}-642 q^{36}+679 q^{34}-495 q^{32}+97 q^{30}+391 q^{28}-790 q^{26}+940 q^{24}-756 q^{22}+286 q^{20}+280 q^{18}-717 q^{16}+822 q^{14}-537 q^{12}+9 q^{10}+548 q^8-837 q^6+717 q^4-214 q^2-464+1042 q^{-2} -1252 q^{-4} +998 q^{-6} -336 q^{-8} -472 q^{-10} +1152 q^{-12} -1438 q^{-14} +1242 q^{-16} -639 q^{-18} -146 q^{-20} +810 q^{-22} -1127 q^{-24} +1002 q^{-26} -494 q^{-28} -155 q^{-30} +664 q^{-32} -825 q^{-34} +565 q^{-36} -24 q^{-38} -560 q^{-40} +909 q^{-42} -872 q^{-44} +456 q^{-46} +157 q^{-48} -722 q^{-50} +1015 q^{-52} -936 q^{-54} +549 q^{-56} -32 q^{-58} -426 q^{-60} +665 q^{-62} -654 q^{-64} +456 q^{-66} -172 q^{-68} -78 q^{-70} +221 q^{-72} -254 q^{-74} +198 q^{-76} -107 q^{-78} +32 q^{-80} +21 q^{-82} -39 q^{-84} +35 q^{-86} -24 q^{-88} +11 q^{-90} -4 q^{-92} + q^{-94} }[/math]

Further Quantum Invariants

K11a126 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["K11a126"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ t^4-5 t^3+16 t^2-31 t+39-31 t^{-1} +16 t^{-2} -5 t^{-3} + t^{-4} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ z^8+3 z^6+6 z^4+4 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]=

{ 145, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ q^6-5 q^5+10 q^4-16 q^3+21 q^2-23 q+24-19 q^{-1} +14 q^{-2} -8 q^{-3} +3 q^{-4} - q^{-5} }[/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 z^6-2 z^6 a^{-2} +6 z^6-4 a^2 z^4-7 z^4 a^{-2} +z^4 a^{-4} +16 z^4-7 a^2 z^2-9 z^2 a^{-2} +z^2 a^{-4} +19 z^2-4 a^2-4 a^{-2} +9 }[/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}+6 a z^9+13 z^9 a^{-1} +7 z^9 a^{-3} +8 a^2 z^8+17 z^8 a^{-2} +9 z^8 a^{-4} +16 z^8+6 a^3 z^7-16 z^7 a^{-1} -5 z^7 a^{-3} +5 z^7 a^{-5} +3 a^4 z^6-13 a^2 z^6-50 z^6 a^{-2} -20 z^6 a^{-4} +z^6 a^{-6} -45 z^6+a^5 z^5-9 a^3 z^5-16 a z^5-13 z^5 a^{-1} -17 z^5 a^{-3} -10 z^5 a^{-5} -4 a^4 z^4+15 a^2 z^4+41 z^4 a^{-2} +11 z^4 a^{-4} -z^4 a^{-6} +48 z^4-2 a^5 z^3+7 a^3 z^3+23 a z^3+25 z^3 a^{-1} +15 z^3 a^{-3} +4 z^3 a^{-5} +a^4 z^2-11 a^2 z^2-17 z^2 a^{-2} -2 z^2 a^{-4} -27 z^2+a^5 z-3 a^3 z-10 a z-10 z a^{-1} -3 z a^{-3} +z a^{-5} +4 a^2+4 a^{-2} +9 }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

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

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["K11a126"];

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^4-5 t^3+16 t^2-31 t+39-31 t^{-1} +16 t^{-2} -5 t^{-3} + t^{-4} }[/math], [math]\displaystyle{ q^6-5 q^5+10 q^4-16 q^3+21 q^2-23 q+24-19 q^{-1} +14 q^{-2} -8 q^{-3} +3 q^{-4} - q^{-5} }[/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]=

{}

Vassiliev invariants

V2 and V3:

(4, 0)

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{ 16 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 128 }[/math] [math]\displaystyle{ \frac{440}{3} }[/math] [math]\displaystyle{ \frac{40}{3} }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ -32 }[/math] [math]\displaystyle{ \frac{2048}{3} }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ \frac{7040}{3} }[/math] [math]\displaystyle{ \frac{640}{3} }[/math] [math]\displaystyle{ \frac{34622}{15} }[/math] [math]\displaystyle{ \frac{1144}{5} }[/math] [math]\displaystyle{ \frac{26648}{45} }[/math] [math]\displaystyle{ \frac{322}{9} }[/math] [math]\displaystyle{ \frac{542}{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]0 is the signature of K11a126. 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 13                       1 1 11                     4   -4 9                   6 1   5 7                 10 4     -6 5               11 6       5 3             12 10         -2 1           12 11           1 -1         8 13             5 -3       6 11               -5 -5     2 8                 6 -7   1 6                   -5 -9   2                     2 -11 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=-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}^{6}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{12} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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.

K11a125 K11a127