K11a177

K11a176

K11a178

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a177 at Knotilus!

Knot presentations

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

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 4 Bridge index 2 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a177/ThurstonBennequinNumber Hyperbolic Volume 13.045 A-Polynomial See Data:K11a177/A-polynomial

[edit Notes for K11a177'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 K11a177's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ t^4-5 t^3+13 t^2-19 t+21-19 t^{-1} +13 t^{-2} -5 t^{-3} + t^{-4} }[/math]
Conway polynomial	[math]\displaystyle{ z^8+3 z^6+3 z^4+4 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 97, 4 }
Jones polynomial	[math]\displaystyle{ q^{10}-4 q^9+7 q^8-11 q^7+14 q^6-15 q^5+15 q^4-12 q^3+9 q^2-5 q+3- q^{-1} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ z^8 a^{-4} -z^6 a^{-2} +6 z^6 a^{-4} -2 z^6 a^{-6} -4 z^4 a^{-2} +14 z^4 a^{-4} -8 z^4 a^{-6} +z^4 a^{-8} -4 z^2 a^{-2} +15 z^2 a^{-4} -9 z^2 a^{-6} +2 z^2 a^{-8} - a^{-2} +5 a^{-4} -3 a^{-6} }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^{10} a^{-4} +z^{10} a^{-6} +3 z^9 a^{-3} +7 z^9 a^{-5} +4 z^9 a^{-7} +3 z^8 a^{-2} +6 z^8 a^{-4} +10 z^8 a^{-6} +7 z^8 a^{-8} +z^7 a^{-1} -9 z^7 a^{-3} -18 z^7 a^{-5} +8 z^7 a^{-9} -13 z^6 a^{-2} -34 z^6 a^{-4} -36 z^6 a^{-6} -8 z^6 a^{-8} +7 z^6 a^{-10} -4 z^5 a^{-1} +3 z^5 a^{-3} +7 z^5 a^{-5} -12 z^5 a^{-7} -8 z^5 a^{-9} +4 z^5 a^{-11} +17 z^4 a^{-2} +47 z^4 a^{-4} +39 z^4 a^{-6} +z^4 a^{-8} -7 z^4 a^{-10} +z^4 a^{-12} +4 z^3 a^{-1} +6 z^3 a^{-3} +8 z^3 a^{-5} +8 z^3 a^{-7} -z^3 a^{-9} -3 z^3 a^{-11} -8 z^2 a^{-2} -25 z^2 a^{-4} -18 z^2 a^{-6} +z^2 a^{-10} -z a^{-1} -3 z a^{-3} -5 z a^{-5} -z a^{-7} +2 z a^{-9} + a^{-2} +5 a^{-4} +3 a^{-6} }[/math]
The A2 invariant	[math]\displaystyle{ -q^2+1- q^{-2} + q^{-4} +2 q^{-6} - q^{-8} +4 q^{-10} -2 q^{-12} +2 q^{-14} + q^{-16} - q^{-18} +2 q^{-20} -3 q^{-22} - q^{-26} - q^{-28} + q^{-30} }[/math]
The G2 invariant	[math]\displaystyle{ q^{12}-2 q^{10}+5 q^8-9 q^6+10 q^4-11 q^2+3+14 q^{-2} -35 q^{-4} +56 q^{-6} -66 q^{-8} +50 q^{-10} -12 q^{-12} -51 q^{-14} +116 q^{-16} -156 q^{-18} +154 q^{-20} -93 q^{-22} -8 q^{-24} +123 q^{-26} -205 q^{-28} +225 q^{-30} -168 q^{-32} +57 q^{-34} +71 q^{-36} -167 q^{-38} +196 q^{-40} -142 q^{-42} +46 q^{-44} +69 q^{-46} -136 q^{-48} +131 q^{-50} -62 q^{-52} -50 q^{-54} +153 q^{-56} -199 q^{-58} +170 q^{-60} -60 q^{-62} -87 q^{-64} +224 q^{-66} -291 q^{-68} +264 q^{-70} -149 q^{-72} -16 q^{-74} +171 q^{-76} -265 q^{-78} +267 q^{-80} -178 q^{-82} +44 q^{-84} +84 q^{-86} -158 q^{-88} +150 q^{-90} -84 q^{-92} -12 q^{-94} +86 q^{-96} -112 q^{-98} +76 q^{-100} - q^{-102} -88 q^{-104} +148 q^{-106} -157 q^{-108} +113 q^{-110} -33 q^{-112} -62 q^{-114} +131 q^{-116} -166 q^{-118} +158 q^{-120} -107 q^{-122} +37 q^{-124} +37 q^{-126} -91 q^{-128} +114 q^{-130} -107 q^{-132} +78 q^{-134} -35 q^{-136} -6 q^{-138} +34 q^{-140} -50 q^{-142} +47 q^{-144} -32 q^{-146} +18 q^{-148} -2 q^{-150} -6 q^{-152} +9 q^{-154} -10 q^{-156} +6 q^{-158} -3 q^{-160} + q^{-162} }[/math]

Further Quantum Invariants

K11a177 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["K11a177"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ t^4-5 t^3+13 t^2-19 t+21-19 t^{-1} +13 t^{-2} -5 t^{-3} + t^{-4} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

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

{ 97, 4 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ q^{10}-4 q^9+7 q^8-11 q^7+14 q^6-15 q^5+15 q^4-12 q^3+9 q^2-5 q+3- 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} +6 z^6 a^{-4} -2 z^6 a^{-6} -4 z^4 a^{-2} +14 z^4 a^{-4} -8 z^4 a^{-6} +z^4 a^{-8} -4 z^2 a^{-2} +15 z^2 a^{-4} -9 z^2 a^{-6} +2 z^2 a^{-8} - a^{-2} +5 a^{-4} -3 a^{-6} }[/math]

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ z^{10} a^{-4} +z^{10} a^{-6} +3 z^9 a^{-3} +7 z^9 a^{-5} +4 z^9 a^{-7} +3 z^8 a^{-2} +6 z^8 a^{-4} +10 z^8 a^{-6} +7 z^8 a^{-8} +z^7 a^{-1} -9 z^7 a^{-3} -18 z^7 a^{-5} +8 z^7 a^{-9} -13 z^6 a^{-2} -34 z^6 a^{-4} -36 z^6 a^{-6} -8 z^6 a^{-8} +7 z^6 a^{-10} -4 z^5 a^{-1} +3 z^5 a^{-3} +7 z^5 a^{-5} -12 z^5 a^{-7} -8 z^5 a^{-9} +4 z^5 a^{-11} +17 z^4 a^{-2} +47 z^4 a^{-4} +39 z^4 a^{-6} +z^4 a^{-8} -7 z^4 a^{-10} +z^4 a^{-12} +4 z^3 a^{-1} +6 z^3 a^{-3} +8 z^3 a^{-5} +8 z^3 a^{-7} -z^3 a^{-9} -3 z^3 a^{-11} -8 z^2 a^{-2} -25 z^2 a^{-4} -18 z^2 a^{-6} +z^2 a^{-10} -z a^{-1} -3 z a^{-3} -5 z a^{-5} -z a^{-7} +2 z a^{-9} + a^{-2} +5 a^{-4} +3 a^{-6} }[/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["K11a177"];

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+13 t^2-19 t+21-19 t^{-1} +13 t^{-2} -5 t^{-3} + t^{-4} }[/math], [math]\displaystyle{ q^{10}-4 q^9+7 q^8-11 q^7+14 q^6-15 q^5+15 q^4-12 q^3+9 q^2-5 q+3- 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:

(4, 7)

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{ 56 }[/math] [math]\displaystyle{ 128 }[/math] [math]\displaystyle{ \frac{824}{3} }[/math] [math]\displaystyle{ \frac{112}{3} }[/math] [math]\displaystyle{ 896 }[/math] [math]\displaystyle{ \frac{4496}{3} }[/math] [math]\displaystyle{ \frac{896}{3} }[/math] [math]\displaystyle{ 152 }[/math] [math]\displaystyle{ \frac{2048}{3} }[/math] [math]\displaystyle{ 1568 }[/math] [math]\displaystyle{ \frac{13184}{3} }[/math] [math]\displaystyle{ \frac{1792}{3} }[/math] [math]\displaystyle{ \frac{124142}{15} }[/math] [math]\displaystyle{ \frac{8032}{15} }[/math] [math]\displaystyle{ \frac{129968}{45} }[/math] [math]\displaystyle{ \frac{34}{9} }[/math] [math]\displaystyle{ \frac{5582}{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 K11a177. 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                   4 1   3 15                 7 3     -4 13               7 4       3 11             8 7         -1 9           7 7           0 7         5 8             3 5       4 7               -3 3     2 6                 4 1   1 3                   -2 -1   2                     2 -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}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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.

K11a176 K11a178