K11a127

K11a126

K11a128

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a127 at Knotilus!

Knot presentations

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

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Chiral Unknotting number 2 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a127/ThurstonBennequinNumber Hyperbolic Volume 16.6559 A-Polynomial See Data:K11a127/A-polynomial

[edit Notes for K11a127's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a127's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ t^4-6 t^3+17 t^2-28 t+33-28 t^{-1} +17 t^{-2} -6 t^{-3} + t^{-4} }[/math]
Conway polynomial	[math]\displaystyle{ z^8+2 z^6+z^4+2 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 137, 4 }
Jones polynomial	[math]\displaystyle{ q^{10}-4 q^9+9 q^8-15 q^7+19 q^6-22 q^5+22 q^4-18 q^3+14 q^2-8 q+4- q^{-1} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ z^8 a^{-4} -z^6 a^{-2} +5 z^6 a^{-4} -2 z^6 a^{-6} -3 z^4 a^{-2} +10 z^4 a^{-4} -7 z^4 a^{-6} +z^4 a^{-8} -2 z^2 a^{-2} +10 z^2 a^{-4} -8 z^2 a^{-6} +2 z^2 a^{-8} +4 a^{-4} -4 a^{-6} + a^{-8} }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ 2 z^{10} a^{-4} +2 z^{10} a^{-6} +5 z^9 a^{-3} +13 z^9 a^{-5} +8 z^9 a^{-7} +4 z^8 a^{-2} +10 z^8 a^{-4} +20 z^8 a^{-6} +14 z^8 a^{-8} +z^7 a^{-1} -12 z^7 a^{-3} -25 z^7 a^{-5} +2 z^7 a^{-7} +14 z^7 a^{-9} -14 z^6 a^{-2} -48 z^6 a^{-4} -64 z^6 a^{-6} -21 z^6 a^{-8} +9 z^6 a^{-10} -3 z^5 a^{-1} +z^5 a^{-3} -8 z^5 a^{-5} -36 z^5 a^{-7} -20 z^5 a^{-9} +4 z^5 a^{-11} +16 z^4 a^{-2} +53 z^4 a^{-4} +54 z^4 a^{-6} +9 z^4 a^{-8} -7 z^4 a^{-10} +z^4 a^{-12} +3 z^3 a^{-1} +11 z^3 a^{-3} +27 z^3 a^{-5} +32 z^3 a^{-7} +12 z^3 a^{-9} -z^3 a^{-11} -6 z^2 a^{-2} -21 z^2 a^{-4} -20 z^2 a^{-6} -3 z^2 a^{-8} +2 z^2 a^{-10} -z a^{-1} -4 z a^{-3} -10 z a^{-5} -10 z a^{-7} -3 z a^{-9} +4 a^{-4} +4 a^{-6} + a^{-8} }[/math]
The A2 invariant	[math]\displaystyle{ -q^2+2-2 q^{-2} +2 q^{-4} +2 q^{-6} -2 q^{-8} +6 q^{-10} -3 q^{-12} +3 q^{-14} - q^{-16} -3 q^{-18} +2 q^{-20} -4 q^{-22} +2 q^{-24} - q^{-28} + q^{-30} }[/math]
The G2 invariant	[math]\displaystyle{ q^{12}-3 q^{10}+9 q^8-19 q^6+28 q^4-33 q^2+17+26 q^{-2} -91 q^{-4} +164 q^{-6} -204 q^{-8} +168 q^{-10} -41 q^{-12} -167 q^{-14} +393 q^{-16} -528 q^{-18} +497 q^{-20} -263 q^{-22} -119 q^{-24} +510 q^{-26} -760 q^{-28} +752 q^{-30} -457 q^{-32} -4 q^{-34} +460 q^{-36} -713 q^{-38} +662 q^{-40} -333 q^{-42} -116 q^{-44} +492 q^{-46} -624 q^{-48} +450 q^{-50} -37 q^{-52} -434 q^{-54} +775 q^{-56} -811 q^{-58} +524 q^{-60} + q^{-62} -577 q^{-64} +976 q^{-66} -1060 q^{-68} +779 q^{-70} -227 q^{-72} -388 q^{-74} +840 q^{-76} -967 q^{-78} +728 q^{-80} -250 q^{-82} -270 q^{-84} +594 q^{-86} -624 q^{-88} +356 q^{-90} +62 q^{-92} -428 q^{-94} +588 q^{-96} -465 q^{-98} +124 q^{-100} +274 q^{-102} -578 q^{-104} +662 q^{-106} -511 q^{-108} +211 q^{-110} +135 q^{-112} -399 q^{-114} +517 q^{-116} -473 q^{-118} +312 q^{-120} -101 q^{-122} -88 q^{-124} +207 q^{-126} -253 q^{-128} +225 q^{-130} -153 q^{-132} +72 q^{-134} +6 q^{-136} -54 q^{-138} +72 q^{-140} -73 q^{-142} +54 q^{-144} -31 q^{-146} +12 q^{-148} +4 q^{-150} -10 q^{-152} +11 q^{-154} -10 q^{-156} +6 q^{-158} -3 q^{-160} + q^{-162} }[/math]

Further Quantum Invariants

K11a127 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["K11a127"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ t^4-6 t^3+17 t^2-28 t+33-28 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+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]=

{ 137, 4 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ q^{10}-4 q^9+9 q^8-15 q^7+19 q^6-22 q^5+22 q^4-18 q^3+14 q^2-8 q+4- q^{-1} }[/math]

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ 2 z^{10} a^{-4} +2 z^{10} a^{-6} +5 z^9 a^{-3} +13 z^9 a^{-5} +8 z^9 a^{-7} +4 z^8 a^{-2} +10 z^8 a^{-4} +20 z^8 a^{-6} +14 z^8 a^{-8} +z^7 a^{-1} -12 z^7 a^{-3} -25 z^7 a^{-5} +2 z^7 a^{-7} +14 z^7 a^{-9} -14 z^6 a^{-2} -48 z^6 a^{-4} -64 z^6 a^{-6} -21 z^6 a^{-8} +9 z^6 a^{-10} -3 z^5 a^{-1} +z^5 a^{-3} -8 z^5 a^{-5} -36 z^5 a^{-7} -20 z^5 a^{-9} +4 z^5 a^{-11} +16 z^4 a^{-2} +53 z^4 a^{-4} +54 z^4 a^{-6} +9 z^4 a^{-8} -7 z^4 a^{-10} +z^4 a^{-12} +3 z^3 a^{-1} +11 z^3 a^{-3} +27 z^3 a^{-5} +32 z^3 a^{-7} +12 z^3 a^{-9} -z^3 a^{-11} -6 z^2 a^{-2} -21 z^2 a^{-4} -20 z^2 a^{-6} -3 z^2 a^{-8} +2 z^2 a^{-10} -z a^{-1} -4 z a^{-3} -10 z a^{-5} -10 z a^{-7} -3 z a^{-9} +4 a^{-4} +4 a^{-6} + a^{-8} }[/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["K11a127"];

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^2-28 t+33-28 t^{-1} +17 t^{-2} -6 t^{-3} + t^{-4} }[/math], [math]\displaystyle{ q^{10}-4 q^9+9 q^8-15 q^7+19 q^6-22 q^5+22 q^4-18 q^3+14 q^2-8 q+4- q^{-1} }[/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:

(2, 2)

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{ 16 }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ \frac{124}{3} }[/math] [math]\displaystyle{ \frac{20}{3} }[/math] [math]\displaystyle{ 128 }[/math] [math]\displaystyle{ \frac{448}{3} }[/math] [math]\displaystyle{ \frac{160}{3} }[/math] [math]\displaystyle{ 16 }[/math] [math]\displaystyle{ \frac{256}{3} }[/math] [math]\displaystyle{ 128 }[/math] [math]\displaystyle{ \frac{992}{3} }[/math] [math]\displaystyle{ \frac{160}{3} }[/math] [math]\displaystyle{ \frac{7471}{15} }[/math] [math]\displaystyle{ \frac{356}{15} }[/math] [math]\displaystyle{ \frac{13324}{45} }[/math] [math]\displaystyle{ -\frac{127}{9} }[/math] [math]\displaystyle{ \frac{991}{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]4 is the signature of K11a127. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -3 -2 -1 0 1 2 3 4 5 6 7 8 χ 21                       1 1 19                     3   -3 17                   6 1   5 15                 9 3     -6 13               10 6       4 11             12 9         -3 9           10 10           0 7         8 12             4 5       6 10               -4 3     3 9                 6 1   1 5                   -4 -1   3                     3 -3 1                       -1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=3 }[/math] [math]\displaystyle{ i=5 }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{12} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=7 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=8 }[/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.

K11a126 K11a128