K11a81

K11a80

K11a82

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a81 at Knotilus!

Knot presentations

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

A Braid Representative	{{{braid_table}}}
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:K11a81/ThurstonBennequinNumber Hyperbolic Volume 16.051 A-Polynomial See Data:K11a81/A-polynomial

[edit Notes for K11a81'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 K11a81's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ -t^4+6 t^3-16 t^2+26 t-29+26 t^{-1} -16 t^{-2} +6 t^{-3} - t^{-4} }[/math]
Conway polynomial	[math]\displaystyle{ -z^8-2 z^6+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 127, 2 }
Jones polynomial	[math]\displaystyle{ q^7-4 q^6+8 q^5-13 q^4+18 q^3-20 q^2+20 q-17+13 q^{-1} -8 q^{-2} +4 q^{-3} - q^{-4} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ -z^8 a^{-2} -5 z^6 a^{-2} +z^6 a^{-4} +2 z^6-a^2 z^4-9 z^4 a^{-2} +3 z^4 a^{-4} +7 z^4-2 a^2 z^2-6 z^2 a^{-2} +2 z^2 a^{-4} +6 z^2+1 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ 2 z^{10} a^{-2} +2 z^{10}+5 a z^9+12 z^9 a^{-1} +7 z^9 a^{-3} +4 a^2 z^8+15 z^8 a^{-2} +11 z^8 a^{-4} +8 z^8+a^3 z^7-13 a z^7-26 z^7 a^{-1} -z^7 a^{-3} +11 z^7 a^{-5} -14 a^2 z^6-50 z^6 a^{-2} -14 z^6 a^{-4} +8 z^6 a^{-6} -42 z^6-3 a^3 z^5+3 a z^5-23 z^5 a^{-3} -13 z^5 a^{-5} +4 z^5 a^{-7} +15 a^2 z^4+39 z^4 a^{-2} +2 z^4 a^{-4} -7 z^4 a^{-6} +z^4 a^{-8} +44 z^4+3 a^3 z^3+9 a z^3+18 z^3 a^{-1} +19 z^3 a^{-3} +5 z^3 a^{-5} -2 z^3 a^{-7} -5 a^2 z^2-9 z^2 a^{-2} +z^2 a^{-4} +2 z^2 a^{-6} -13 z^2-a^3 z-4 a z-6 z a^{-1} -4 z a^{-3} -z a^{-5} +1 }[/math]
The A2 invariant	[math]\displaystyle{ -q^{12}+q^{10}+q^8-q^6+3 q^4-3 q^2+1+ q^{-2} -2 q^{-4} +5 q^{-6} -3 q^{-8} +3 q^{-10} - q^{-12} -2 q^{-14} +2 q^{-16} -2 q^{-18} + q^{-20} }[/math]
The G2 invariant	[math]\displaystyle{ q^{60}-3 q^{58}+9 q^{56}-19 q^{54}+28 q^{52}-32 q^{50}+15 q^{48}+29 q^{46}-92 q^{44}+157 q^{42}-186 q^{40}+140 q^{38}-15 q^{36}-174 q^{34}+360 q^{32}-449 q^{30}+392 q^{28}-169 q^{26}-152 q^{24}+452 q^{22}-607 q^{20}+545 q^{18}-277 q^{16}-87 q^{14}+395 q^{12}-526 q^{10}+425 q^8-144 q^6-183 q^4+416 q^2-444+250 q^{-2} +81 q^{-4} -420 q^{-6} +618 q^{-8} -586 q^{-10} +326 q^{-12} +84 q^{-14} -495 q^{-16} +758 q^{-18} -764 q^{-20} +514 q^{-22} -89 q^{-24} -345 q^{-26} +628 q^{-28} -662 q^{-30} +448 q^{-32} -89 q^{-34} -249 q^{-36} +431 q^{-38} -389 q^{-40} +158 q^{-42} +136 q^{-44} -356 q^{-46} +405 q^{-48} -270 q^{-50} +10 q^{-52} +254 q^{-54} -425 q^{-56} +448 q^{-58} -321 q^{-60} +111 q^{-62} +110 q^{-64} -276 q^{-66} +338 q^{-68} -310 q^{-70} +215 q^{-72} -84 q^{-74} -35 q^{-76} +125 q^{-78} -169 q^{-80} +165 q^{-82} -127 q^{-84} +72 q^{-86} -16 q^{-88} -29 q^{-90} +53 q^{-92} -62 q^{-94} +51 q^{-96} -31 q^{-98} +15 q^{-100} + q^{-102} -8 q^{-104} +10 q^{-106} -10 q^{-108} +6 q^{-110} -3 q^{-112} + q^{-114} }[/math]

Further Quantum Invariants

K11a81 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["K11a81"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ -t^4+6 t^3-16 t^2+26 t-29+26 t^{-1} -16 t^{-2} +6 t^{-3} - t^{-4} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

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

{ 127, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ q^7-4 q^6+8 q^5-13 q^4+18 q^3-20 q^2+20 q-17+13 q^{-1} -8 q^{-2} +4 q^{-3} - q^{-4} }[/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} -5 z^6 a^{-2} +z^6 a^{-4} +2 z^6-a^2 z^4-9 z^4 a^{-2} +3 z^4 a^{-4} +7 z^4-2 a^2 z^2-6 z^2 a^{-2} +2 z^2 a^{-4} +6 z^2+1 }[/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}+5 a z^9+12 z^9 a^{-1} +7 z^9 a^{-3} +4 a^2 z^8+15 z^8 a^{-2} +11 z^8 a^{-4} +8 z^8+a^3 z^7-13 a z^7-26 z^7 a^{-1} -z^7 a^{-3} +11 z^7 a^{-5} -14 a^2 z^6-50 z^6 a^{-2} -14 z^6 a^{-4} +8 z^6 a^{-6} -42 z^6-3 a^3 z^5+3 a z^5-23 z^5 a^{-3} -13 z^5 a^{-5} +4 z^5 a^{-7} +15 a^2 z^4+39 z^4 a^{-2} +2 z^4 a^{-4} -7 z^4 a^{-6} +z^4 a^{-8} +44 z^4+3 a^3 z^3+9 a z^3+18 z^3 a^{-1} +19 z^3 a^{-3} +5 z^3 a^{-5} -2 z^3 a^{-7} -5 a^2 z^2-9 z^2 a^{-2} +z^2 a^{-4} +2 z^2 a^{-6} -13 z^2-a^3 z-4 a z-6 z a^{-1} -4 z a^{-3} -z a^{-5} +1 }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a282,}

Same Jones Polynomial (up to mirroring, [math]\displaystyle{ q\leftrightarrow q^{-1} }[/math]): {K11a282,}

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["K11a81"];

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^3-16 t^2+26 t-29+26 t^{-1} -16 t^{-2} +6 t^{-3} - t^{-4} }[/math], [math]\displaystyle{ q^7-4 q^6+8 q^5-13 q^4+18 q^3-20 q^2+20 q-17+13 q^{-1} -8 q^{-2} +4 q^{-3} - q^{-4} }[/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]=

{K11a282,}

In[6]:=

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

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

Out[6]=

{K11a282,}

Vassiliev invariants

V2 and V3:

(0, 0)

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{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ 176 }[/math] [math]\displaystyle{ -160 }[/math] [math]\displaystyle{ 16 }[/math] [math]\displaystyle{ -48 }[/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 K11a81. 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 15                       1 1 13                     3   -3 11                   5 1   4 9                 8 3     -5 7               10 5       5 5             10 8         -2 3           10 10           0 1         8 11             3 -1       5 9               -4 -3     3 8                 5 -5   1 5                   -4 -7   3                     3 -9 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=-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}^{5}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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.

K11a80 K11a82