K11a34

K11a33

K11a35

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a34 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X8493 X14,5,15,6 X2837 X16,9,17,10 X18,12,19,11 X20,14,21,13 X6,15,7,16 X22,17,1,18 X12,20,13,19 X10,22,11,21
Gauss code	1, -4, 2, -1, 3, -8, 4, -2, 5, -11, 6, -10, 7, -3, 8, -5, 9, -6, 10, -7, 11, -9
Dowker-Thistlethwaite code	4 8 14 2 16 18 20 6 22 12 10

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a34/ThurstonBennequinNumber Hyperbolic Volume 14.9742 A-Polynomial See Data:K11a34/A-polynomial

[edit Notes for K11a34'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 K11a34's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ -t^4+5 t^3-14 t^2+25 t-29+25 t^{-1} -14 t^{-2} +5 t^{-3} - t^{-4} }[/math]
Conway polynomial	[math]\displaystyle{ -z^8-3 z^6-4 z^4-2 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 119, 2 }
Jones polynomial	[math]\displaystyle{ -q^8+4 q^7-8 q^6+13 q^5-17 q^4+19 q^3-19 q^2+16 q-11+7 q^{-1} -3 q^{-2} + q^{-3} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ -z^8 a^{-2} -6 z^6 a^{-2} +2 z^6 a^{-4} +z^6-15 z^4 a^{-2} +8 z^4 a^{-4} -z^4 a^{-6} +4 z^4-17 z^2 a^{-2} +11 z^2 a^{-4} -2 z^2 a^{-6} +6 z^2-7 a^{-2} +5 a^{-4} - a^{-6} +4 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^{10} a^{-2} +z^{10} a^{-4} +4 z^9 a^{-1} +8 z^9 a^{-3} +4 z^9 a^{-5} +12 z^8 a^{-2} +14 z^8 a^{-4} +7 z^8 a^{-6} +5 z^8+3 a z^7-5 z^7 a^{-1} -9 z^7 a^{-3} +6 z^7 a^{-5} +7 z^7 a^{-7} +a^2 z^6-41 z^6 a^{-2} -37 z^6 a^{-4} -7 z^6 a^{-6} +4 z^6 a^{-8} -14 z^6-8 a z^5-4 z^5 a^{-1} -9 z^5 a^{-3} -25 z^5 a^{-5} -11 z^5 a^{-7} +z^5 a^{-9} -3 a^2 z^4+53 z^4 a^{-2} +39 z^4 a^{-4} -2 z^4 a^{-6} -6 z^4 a^{-8} +15 z^4+5 a z^3+7 z^3 a^{-1} +19 z^3 a^{-3} +23 z^3 a^{-5} +5 z^3 a^{-7} -z^3 a^{-9} +2 a^2 z^2-32 z^2 a^{-2} -20 z^2 a^{-4} +z^2 a^{-6} +2 z^2 a^{-8} -11 z^2-a z-3 z a^{-1} -7 z a^{-3} -7 z a^{-5} -2 z a^{-7} +7 a^{-2} +5 a^{-4} + a^{-6} +4 }[/math]
The A2 invariant	[math]\displaystyle{ q^8-q^6+3 q^4+3 q^{-2} -5 q^{-4} +2 q^{-6} -3 q^{-8} +2 q^{-12} -2 q^{-14} +4 q^{-16} - q^{-18} + q^{-22} - q^{-24} }[/math]
The G2 invariant	[math]\displaystyle{ q^{46}-2 q^{44}+5 q^{42}-9 q^{40}+11 q^{38}-12 q^{36}+5 q^{34}+12 q^{32}-34 q^{30}+62 q^{28}-83 q^{26}+79 q^{24}-44 q^{22}-30 q^{20}+134 q^{18}-222 q^{16}+269 q^{14}-226 q^{12}+92 q^{10}+111 q^8-317 q^6+451 q^4-442 q^2+284-18 q^{-2} -260 q^{-4} +446 q^{-6} -463 q^{-8} +316 q^{-10} -56 q^{-12} -204 q^{-14} +343 q^{-16} -319 q^{-18} +124 q^{-20} +141 q^{-22} -360 q^{-24} +430 q^{-26} -308 q^{-28} +18 q^{-30} +321 q^{-32} -592 q^{-34} +670 q^{-36} -522 q^{-38} +179 q^{-40} +230 q^{-42} -564 q^{-44} +705 q^{-46} -601 q^{-48} +312 q^{-50} +52 q^{-52} -347 q^{-54} +467 q^{-56} -377 q^{-58} +147 q^{-60} +125 q^{-62} -297 q^{-64} +311 q^{-66} -157 q^{-68} -85 q^{-70} +313 q^{-72} -425 q^{-74} +382 q^{-76} -198 q^{-78} -62 q^{-80} +292 q^{-82} -425 q^{-84} +423 q^{-86} -298 q^{-88} +108 q^{-90} +81 q^{-92} -221 q^{-94} +271 q^{-96} -242 q^{-98} +159 q^{-100} -55 q^{-102} -32 q^{-104} +82 q^{-106} -98 q^{-108} +82 q^{-110} -49 q^{-112} +21 q^{-114} +3 q^{-116} -14 q^{-118} +15 q^{-120} -13 q^{-122} +7 q^{-124} -3 q^{-126} + q^{-128} }[/math]

Further Quantum Invariants

K11a34 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["K11a34"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ -t^4+5 t^3-14 t^2+25 t-29+25 t^{-1} -14 t^{-2} +5 t^{-3} - t^{-4} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

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

{ 119, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ -q^8+4 q^7-8 q^6+13 q^5-17 q^4+19 q^3-19 q^2+16 q-11+7 q^{-1} -3 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} -6 z^6 a^{-2} +2 z^6 a^{-4} +z^6-15 z^4 a^{-2} +8 z^4 a^{-4} -z^4 a^{-6} +4 z^4-17 z^2 a^{-2} +11 z^2 a^{-4} -2 z^2 a^{-6} +6 z^2-7 a^{-2} +5 a^{-4} - a^{-6} +4 }[/math]

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ z^{10} a^{-2} +z^{10} a^{-4} +4 z^9 a^{-1} +8 z^9 a^{-3} +4 z^9 a^{-5} +12 z^8 a^{-2} +14 z^8 a^{-4} +7 z^8 a^{-6} +5 z^8+3 a z^7-5 z^7 a^{-1} -9 z^7 a^{-3} +6 z^7 a^{-5} +7 z^7 a^{-7} +a^2 z^6-41 z^6 a^{-2} -37 z^6 a^{-4} -7 z^6 a^{-6} +4 z^6 a^{-8} -14 z^6-8 a z^5-4 z^5 a^{-1} -9 z^5 a^{-3} -25 z^5 a^{-5} -11 z^5 a^{-7} +z^5 a^{-9} -3 a^2 z^4+53 z^4 a^{-2} +39 z^4 a^{-4} -2 z^4 a^{-6} -6 z^4 a^{-8} +15 z^4+5 a z^3+7 z^3 a^{-1} +19 z^3 a^{-3} +23 z^3 a^{-5} +5 z^3 a^{-7} -z^3 a^{-9} +2 a^2 z^2-32 z^2 a^{-2} -20 z^2 a^{-4} +z^2 a^{-6} +2 z^2 a^{-8} -11 z^2-a z-3 z a^{-1} -7 z a^{-3} -7 z a^{-5} -2 z a^{-7} +7 a^{-2} +5 a^{-4} + a^{-6} +4 }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a158,}

Same Jones Polynomial (up to mirroring, [math]\displaystyle{ q\leftrightarrow q^{-1} }[/math]): {K11a89,}

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["K11a34"];

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^3-14 t^2+25 t-29+25 t^{-1} -14 t^{-2} +5 t^{-3} - t^{-4} }[/math], [math]\displaystyle{ -q^8+4 q^7-8 q^6+13 q^5-17 q^4+19 q^3-19 q^2+16 q-11+7 q^{-1} -3 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]=

{K11a158,}

In[6]:=

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

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

Out[6]=

{K11a89,}

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{260}{3} }[/math] [math]\displaystyle{ \frac{148}{3} }[/math] [math]\displaystyle{ 64 }[/math] [math]\displaystyle{ \frac{496}{3} }[/math] [math]\displaystyle{ \frac{160}{3} }[/math] [math]\displaystyle{ 24 }[/math] [math]\displaystyle{ -\frac{256}{3} }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ -\frac{2080}{3} }[/math] [math]\displaystyle{ -\frac{1184}{3} }[/math] [math]\displaystyle{ -\frac{10111}{15} }[/math] [math]\displaystyle{ \frac{2588}{5} }[/math] [math]\displaystyle{ -\frac{54484}{45} }[/math] [math]\displaystyle{ \frac{1471}{9} }[/math] [math]\displaystyle{ -\frac{4591}{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 K11a34. 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                     3   3 13                   5 1   -4 11                 8 3     5 9               9 5       -4 7             10 8         2 5           9 9           0 3         7 10             -3 1       5 10               5 -1     2 6                 -4 -3   1 5                   4 -5   2                     -2 -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}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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.

K11a33 K11a35