K11a72

K11a71

K11a73

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a72 at Knotilus!

Knot presentations

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

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Chiral 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:K11a72/ThurstonBennequinNumber Hyperbolic Volume 17.173 A-Polynomial See Data:K11a72/A-polynomial

[edit Notes for K11a72's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus [math]\displaystyle{ [2,4] }[/math] Rasmussen s-Invariant 0

[edit Notes for K11a72's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ t^4-6 t^3+18 t^2-32 t+39-32 t^{-1} +18 t^{-2} -6 t^{-3} + t^{-4} }[/math]
Conway polynomial	[math]\displaystyle{ z^8+2 z^6+2 z^4+2 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 153, 0 }
Jones polynomial	[math]\displaystyle{ q^6-5 q^5+10 q^4-16 q^3+22 q^2-24 q+25-21 q^{-1} +15 q^{-2} -9 q^{-3} +4 q^{-4} - q^{-5} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ z^8-a^2 z^6-2 z^6 a^{-2} +5 z^6-3 a^2 z^4-6 z^4 a^{-2} +z^4 a^{-4} +10 z^4-3 a^2 z^2-4 z^2 a^{-2} +z^2 a^{-4} +8 z^2-a^2+ a^{-2} - a^{-4} +2 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ 2 z^{10} a^{-2} +2 z^{10}+7 a z^9+14 z^9 a^{-1} +7 z^9 a^{-3} +10 a^2 z^8+19 z^8 a^{-2} +9 z^8 a^{-4} +20 z^8+8 a^3 z^7+a z^7-16 z^7 a^{-1} -4 z^7 a^{-3} +5 z^7 a^{-5} +4 a^4 z^6-15 a^2 z^6-55 z^6 a^{-2} -20 z^6 a^{-4} +z^6 a^{-6} -53 z^6+a^5 z^5-12 a^3 z^5-20 a z^5-15 z^5 a^{-1} -18 z^5 a^{-3} -10 z^5 a^{-5} -5 a^4 z^4+11 a^2 z^4+44 z^4 a^{-2} +12 z^4 a^{-4} -z^4 a^{-6} +47 z^4-a^5 z^3+7 a^3 z^3+20 a z^3+22 z^3 a^{-1} +14 z^3 a^{-3} +4 z^3 a^{-5} +a^4 z^2-5 a^2 z^2-12 z^2 a^{-2} -2 z^2 a^{-4} -16 z^2-2 a^3 z-5 a z-5 z a^{-1} -z a^{-3} +z a^{-5} +a^2- a^{-2} - a^{-4} +2 }[/math]
The A2 invariant	[math]\displaystyle{ -q^{14}+2 q^{12}-3 q^{10}+2 q^8+q^6-4 q^4+5 q^2-4+4 q^{-2} + q^{-4} +5 q^{-8} -4 q^{-10} + q^{-12} - q^{-14} -2 q^{-16} + q^{-18} }[/math]
The G2 invariant	[math]\displaystyle{ q^{80}-3 q^{78}+7 q^{76}-13 q^{74}+16 q^{72}-16 q^{70}+7 q^{68}+16 q^{66}-45 q^{64}+83 q^{62}-114 q^{60}+115 q^{58}-80 q^{56}-9 q^{54}+139 q^{52}-278 q^{50}+389 q^{48}-410 q^{46}+295 q^{44}-41 q^{42}-306 q^{40}+643 q^{38}-839 q^{36}+789 q^{34}-473 q^{32}-53 q^{30}+605 q^{28}-976 q^{26}+1019 q^{24}-679 q^{22}+96 q^{20}+491 q^{18}-837 q^{16}+777 q^{14}-340 q^{12}-279 q^{10}+797 q^8-952 q^6+644 q^4+28 q^2-796+1345 q^{-2} -1425 q^{-4} +976 q^{-6} -156 q^{-8} -748 q^{-10} +1412 q^{-12} -1590 q^{-14} +1240 q^{-16} -491 q^{-18} -358 q^{-20} +997 q^{-22} -1192 q^{-24} +906 q^{-26} -281 q^{-28} -387 q^{-30} +813 q^{-32} -819 q^{-34} +412 q^{-36} +228 q^{-38} -798 q^{-40} +1054 q^{-42} -878 q^{-44} +334 q^{-46} +339 q^{-48} -892 q^{-50} +1112 q^{-52} -949 q^{-54} +500 q^{-56} +44 q^{-58} -492 q^{-60} +703 q^{-62} -662 q^{-64} +443 q^{-66} -155 q^{-68} -92 q^{-70} +226 q^{-72} -253 q^{-74} +196 q^{-76} -106 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

K11a72 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["K11a72"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ t^4-6 t^3+18 t^2-32 t+39-32 t^{-1} +18 t^{-2} -6 t^{-3} + t^{-4} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ z^8+2 z^6+2 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]=

{ 153, 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+22 q^2-24 q+25-21 q^{-1} +15 q^{-2} -9 q^{-3} +4 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} +5 z^6-3 a^2 z^4-6 z^4 a^{-2} +z^4 a^{-4} +10 z^4-3 a^2 z^2-4 z^2 a^{-2} +z^2 a^{-4} +8 z^2-a^2+ a^{-2} - a^{-4} +2 }[/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}+7 a z^9+14 z^9 a^{-1} +7 z^9 a^{-3} +10 a^2 z^8+19 z^8 a^{-2} +9 z^8 a^{-4} +20 z^8+8 a^3 z^7+a z^7-16 z^7 a^{-1} -4 z^7 a^{-3} +5 z^7 a^{-5} +4 a^4 z^6-15 a^2 z^6-55 z^6 a^{-2} -20 z^6 a^{-4} +z^6 a^{-6} -53 z^6+a^5 z^5-12 a^3 z^5-20 a z^5-15 z^5 a^{-1} -18 z^5 a^{-3} -10 z^5 a^{-5} -5 a^4 z^4+11 a^2 z^4+44 z^4 a^{-2} +12 z^4 a^{-4} -z^4 a^{-6} +47 z^4-a^5 z^3+7 a^3 z^3+20 a z^3+22 z^3 a^{-1} +14 z^3 a^{-3} +4 z^3 a^{-5} +a^4 z^2-5 a^2 z^2-12 z^2 a^{-2} -2 z^2 a^{-4} -16 z^2-2 a^3 z-5 a z-5 z a^{-1} -z a^{-3} +z a^{-5} +a^2- a^{-2} - a^{-4} +2 }[/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["K11a72"];

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+18 t^2-32 t+39-32 t^{-1} +18 t^{-2} -6 t^{-3} + t^{-4} }[/math], [math]\displaystyle{ q^6-5 q^5+10 q^4-16 q^3+22 q^2-24 q+25-21 q^{-1} +15 q^{-2} -9 q^{-3} +4 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:

(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{124}{3} }[/math] [math]\displaystyle{ -\frac{4}{3} }[/math] [math]\displaystyle{ 64 }[/math] [math]\displaystyle{ \frac{368}{3} }[/math] [math]\displaystyle{ \frac{128}{3} }[/math] [math]\displaystyle{ 8 }[/math] [math]\displaystyle{ \frac{256}{3} }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ \frac{992}{3} }[/math] [math]\displaystyle{ -\frac{32}{3} }[/math] [math]\displaystyle{ \frac{6751}{15} }[/math] [math]\displaystyle{ -\frac{164}{15} }[/math] [math]\displaystyle{ \frac{7684}{45} }[/math] [math]\displaystyle{ -\frac{31}{9} }[/math] [math]\displaystyle{ \frac{271}{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 K11a72. 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               12 6       6 3             12 10         -2 1           13 12           1 -1         9 13             4 -3       6 12               -6 -5     3 9                 6 -7   1 6                   -5 -9   3                     3 -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}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{12} }[/math] [math]\displaystyle{ {\mathbb Z}^{13} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\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^{12} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/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.

K11a71 K11a73