K11a171

K11a170

K11a172

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a171 at Knotilus!

Knot presentations

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

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible 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:K11a171/ThurstonBennequinNumber Hyperbolic Volume 18.6712 A-Polynomial See Data:K11a171/A-polynomial

[edit Notes for K11a171'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 K11a171's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ -t^4+7 t^3-21 t^2+39 t-47+39 t^{-1} -21 t^{-2} +7 t^{-3} - t^{-4} }[/math]
Conway polynomial	[math]\displaystyle{ -z^8-z^6+z^4+2 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 183, 2 }
Jones polynomial	[math]\displaystyle{ -q^8+5 q^7-12 q^6+19 q^5-26 q^4+30 q^3-29 q^2+26 q-18+11 q^{-1} -5 q^{-2} + q^{-3} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ -z^8 a^{-2} -4 z^6 a^{-2} +2 z^6 a^{-4} +z^6-5 z^4 a^{-2} +5 z^4 a^{-4} -z^4 a^{-6} +2 z^4+z^2 a^{-2} +2 z^2 a^{-4} -z^2 a^{-6} +4 a^{-2} -2 a^{-4} -1 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ 3 z^{10} a^{-2} +3 z^{10} a^{-4} +9 z^9 a^{-1} +20 z^9 a^{-3} +11 z^9 a^{-5} +23 z^8 a^{-2} +29 z^8 a^{-4} +16 z^8 a^{-6} +10 z^8+5 a z^7-7 z^7 a^{-1} -23 z^7 a^{-3} +z^7 a^{-5} +12 z^7 a^{-7} +a^2 z^6-65 z^6 a^{-2} -71 z^6 a^{-4} -23 z^6 a^{-6} +5 z^6 a^{-8} -21 z^6-9 a z^5-15 z^5 a^{-1} -18 z^5 a^{-3} -28 z^5 a^{-5} -15 z^5 a^{-7} +z^5 a^{-9} -a^2 z^4+47 z^4 a^{-2} +47 z^4 a^{-4} +12 z^4 a^{-6} -3 z^4 a^{-8} +14 z^4+4 a z^3+14 z^3 a^{-1} +21 z^3 a^{-3} +17 z^3 a^{-5} +6 z^3 a^{-7} -6 z^2 a^{-2} -8 z^2 a^{-4} -4 z^2 a^{-6} -2 z^2-z a^{-1} -z a^{-3} -z a^{-5} -z a^{-7} -4 a^{-2} -2 a^{-4} -1 }[/math]
The A2 invariant	[math]\displaystyle{ q^8-3 q^6+3 q^4-3 q^2-1+6 q^{-2} -4 q^{-4} +8 q^{-6} -2 q^{-8} + q^{-10} + q^{-12} -6 q^{-14} +4 q^{-16} -3 q^{-18} +2 q^{-22} - q^{-24} }[/math]
The G2 invariant	[math]\displaystyle{ q^{46}-4 q^{44}+11 q^{42}-24 q^{40}+36 q^{38}-43 q^{36}+30 q^{34}+20 q^{32}-100 q^{30}+210 q^{28}-303 q^{26}+316 q^{24}-204 q^{22}-78 q^{20}+476 q^{18}-859 q^{16}+1062 q^{14}-921 q^{12}+376 q^{10}+457 q^8-1316 q^6+1850 q^4-1784 q^2+1060+97 q^{-2} -1270 q^{-4} +1969 q^{-6} -1893 q^{-8} +1078 q^{-10} +148 q^{-12} -1230 q^{-14} +1686 q^{-16} -1303 q^{-18} +249 q^{-20} +1016 q^{-22} -1900 q^{-24} +2001 q^{-26} -1197 q^{-28} -222 q^{-30} +1715 q^{-32} -2695 q^{-34} +2772 q^{-36} -1870 q^{-38} +304 q^{-40} +1366 q^{-42} -2538 q^{-44} +2797 q^{-46} -2076 q^{-48} +684 q^{-50} +801 q^{-52} -1800 q^{-54} +1931 q^{-56} -1212 q^{-58} +9 q^{-60} +1116 q^{-62} -1648 q^{-64} +1352 q^{-66} -394 q^{-68} -811 q^{-70} +1726 q^{-72} -1967 q^{-74} +1468 q^{-76} -447 q^{-78} -674 q^{-80} +1480 q^{-82} -1735 q^{-84} +1427 q^{-86} -747 q^{-88} -13 q^{-90} +601 q^{-92} -883 q^{-94} +847 q^{-96} -591 q^{-98} +270 q^{-100} +17 q^{-102} -196 q^{-104} +252 q^{-106} -230 q^{-108} +152 q^{-110} -71 q^{-112} +14 q^{-114} +22 q^{-116} -31 q^{-118} +28 q^{-120} -20 q^{-122} +10 q^{-124} -4 q^{-126} + q^{-128} }[/math]

Further Quantum Invariants

K11a171 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["K11a171"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ -t^4+7 t^3-21 t^2+39 t-47+39 t^{-1} -21 t^{-2} +7 t^{-3} - t^{-4} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ -z^8-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]=

{ 183, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ -q^8+5 q^7-12 q^6+19 q^5-26 q^4+30 q^3-29 q^2+26 q-18+11 q^{-1} -5 q^{-2} + q^{-3} }[/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} -4 z^6 a^{-2} +2 z^6 a^{-4} +z^6-5 z^4 a^{-2} +5 z^4 a^{-4} -z^4 a^{-6} +2 z^4+z^2 a^{-2} +2 z^2 a^{-4} -z^2 a^{-6} +4 a^{-2} -2 a^{-4} -1 }[/math]

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ 3 z^{10} a^{-2} +3 z^{10} a^{-4} +9 z^9 a^{-1} +20 z^9 a^{-3} +11 z^9 a^{-5} +23 z^8 a^{-2} +29 z^8 a^{-4} +16 z^8 a^{-6} +10 z^8+5 a z^7-7 z^7 a^{-1} -23 z^7 a^{-3} +z^7 a^{-5} +12 z^7 a^{-7} +a^2 z^6-65 z^6 a^{-2} -71 z^6 a^{-4} -23 z^6 a^{-6} +5 z^6 a^{-8} -21 z^6-9 a z^5-15 z^5 a^{-1} -18 z^5 a^{-3} -28 z^5 a^{-5} -15 z^5 a^{-7} +z^5 a^{-9} -a^2 z^4+47 z^4 a^{-2} +47 z^4 a^{-4} +12 z^4 a^{-6} -3 z^4 a^{-8} +14 z^4+4 a z^3+14 z^3 a^{-1} +21 z^3 a^{-3} +17 z^3 a^{-5} +6 z^3 a^{-7} -6 z^2 a^{-2} -8 z^2 a^{-4} -4 z^2 a^{-6} -2 z^2-z a^{-1} -z a^{-3} -z a^{-5} -z a^{-7} -4 a^{-2} -2 a^{-4} -1 }[/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["K11a171"];

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+7 t^3-21 t^2+39 t-47+39 t^{-1} -21 t^{-2} +7 t^{-3} - t^{-4} }[/math], [math]\displaystyle{ -q^8+5 q^7-12 q^6+19 q^5-26 q^4+30 q^3-29 q^2+26 q-18+11 q^{-1} -5 q^{-2} + q^{-3} }[/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{172}{3} }[/math] [math]\displaystyle{ \frac{20}{3} }[/math] [math]\displaystyle{ 128 }[/math] [math]\displaystyle{ \frac{544}{3} }[/math] [math]\displaystyle{ \frac{64}{3} }[/math] [math]\displaystyle{ 16 }[/math] [math]\displaystyle{ \frac{256}{3} }[/math] [math]\displaystyle{ 128 }[/math] [math]\displaystyle{ \frac{1376}{3} }[/math] [math]\displaystyle{ \frac{160}{3} }[/math] [math]\displaystyle{ \frac{10471}{15} }[/math] [math]\displaystyle{ \frac{1636}{15} }[/math] [math]\displaystyle{ \frac{5404}{45} }[/math] [math]\displaystyle{ \frac{89}{9} }[/math] [math]\displaystyle{ -\frac{89}{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 K11a171. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -4 -3 -2 -1 0 1 2 3 4 5 6 7 χ 17                       1 -1 15                     4   4 13                   8 1   -7 11                 11 4     7 9               15 8       -7 7             15 11         4 5           14 15           1 3         12 15             -3 1       7 15               8 -1     4 11                 -7 -3   1 7                   6 -5   4                     -4 -7 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=-4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{15}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{15}\oplus{\mathbb Z}_2^{14} }[/math] [math]\displaystyle{ {\mathbb Z}^{14} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{15}\oplus{\mathbb Z}_2^{15} }[/math] [math]\displaystyle{ {\mathbb Z}^{15} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{15} }[/math] [math]\displaystyle{ {\mathbb Z}^{15} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=7 }[/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.

K11a170 K11a172