K11n77

K11n76

K11n78

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n77 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X8493 X5,15,6,14 X2837 X20,10,21,9 X11,17,12,16 X13,19,14,18 X15,7,16,6 X17,13,18,12 X22,20,1,19 X10,22,11,21
Gauss code	1, -4, 2, -1, -3, 8, 4, -2, 5, -11, -6, 9, -7, 3, -8, 6, -9, 7, 10, -5, 11, -10
Dowker-Thistlethwaite code	4 8 -14 2 20 -16 -18 -6 -12 22 10

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

Three dimensional invariants

Symmetry type Reversible Unknotting number 4 3-genus 4 Bridge index 4 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11n77/ThurstonBennequinNumber Hyperbolic Volume 11.9604 A-Polynomial See Data:K11n77/A-polynomial

[edit Notes for K11n77's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus [math]\displaystyle{ 4 }[/math] Rasmussen s-Invariant 8

[edit Notes for K11n77's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ t^4-t^3-2 t^2+8 t-11+8 t^{-1} -2 t^{-2} - t^{-3} + t^{-4} }[/math]
Conway polynomial	[math]\displaystyle{ z^8+7 z^6+12 z^4+7 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \left\{t^2-t+1\right\} }[/math]
Determinant and Signature	{ 27, 6 }
Jones polynomial	[math]\displaystyle{ -q^{14}+3 q^{13}-4 q^{12}+5 q^{11}-6 q^{10}+4 q^9-4 q^8+2 q^7+q^6+q^4 }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ z^8 a^{-8} +8 z^6 a^{-8} -z^6 a^{-10} +21 z^4 a^{-8} -9 z^4 a^{-10} +23 z^2 a^{-8} -20 z^2 a^{-10} +4 z^2 a^{-12} +9 a^{-8} -13 a^{-10} +6 a^{-12} - a^{-14} }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^8 a^{-8} +z^8 a^{-10} +z^8 a^{-12} +z^8 a^{-14} +z^7 a^{-9} +2 z^7 a^{-11} +4 z^7 a^{-13} +3 z^7 a^{-15} -8 z^6 a^{-8} -10 z^6 a^{-10} -4 z^6 a^{-12} +z^6 a^{-14} +3 z^6 a^{-16} -9 z^5 a^{-9} -15 z^5 a^{-11} -15 z^5 a^{-13} -8 z^5 a^{-15} +z^5 a^{-17} +21 z^4 a^{-8} +29 z^4 a^{-10} +8 z^4 a^{-12} -8 z^4 a^{-14} -8 z^4 a^{-16} +20 z^3 a^{-9} +36 z^3 a^{-11} +23 z^3 a^{-13} +5 z^3 a^{-15} -2 z^3 a^{-17} -23 z^2 a^{-8} -31 z^2 a^{-10} -6 z^2 a^{-12} +5 z^2 a^{-14} +3 z^2 a^{-16} -13 z a^{-9} -22 z a^{-11} -12 z a^{-13} -3 z a^{-15} +9 a^{-8} +13 a^{-10} +6 a^{-12} + a^{-14} }[/math]
The A2 invariant	[math]\displaystyle{ q^{-14} + q^{-16} +2 q^{-18} +4 q^{-20} +2 q^{-22} + q^{-24} -5 q^{-28} -3 q^{-30} -4 q^{-32} +2 q^{-36} + q^{-38} +3 q^{-40} - q^{-42} - q^{-44} }[/math]
The G2 invariant	Data:K11n77/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n77 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["K11n77"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ t^4-t^3-2 t^2+8 t-11+8 t^{-1} -2 t^{-2} - t^{-3} + t^{-4} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ z^8+7 z^6+12 z^4+7 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{ \left\{t^2-t+1\right\} }[/math]

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 27, 6 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ -q^{14}+3 q^{13}-4 q^{12}+5 q^{11}-6 q^{10}+4 q^9-4 q^8+2 q^7+q^6+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^{-8} +8 z^6 a^{-8} -z^6 a^{-10} +21 z^4 a^{-8} -9 z^4 a^{-10} +23 z^2 a^{-8} -20 z^2 a^{-10} +4 z^2 a^{-12} +9 a^{-8} -13 a^{-10} +6 a^{-12} - a^{-14} }[/math]

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ z^8 a^{-8} +z^8 a^{-10} +z^8 a^{-12} +z^8 a^{-14} +z^7 a^{-9} +2 z^7 a^{-11} +4 z^7 a^{-13} +3 z^7 a^{-15} -8 z^6 a^{-8} -10 z^6 a^{-10} -4 z^6 a^{-12} +z^6 a^{-14} +3 z^6 a^{-16} -9 z^5 a^{-9} -15 z^5 a^{-11} -15 z^5 a^{-13} -8 z^5 a^{-15} +z^5 a^{-17} +21 z^4 a^{-8} +29 z^4 a^{-10} +8 z^4 a^{-12} -8 z^4 a^{-14} -8 z^4 a^{-16} +20 z^3 a^{-9} +36 z^3 a^{-11} +23 z^3 a^{-13} +5 z^3 a^{-15} -2 z^3 a^{-17} -23 z^2 a^{-8} -31 z^2 a^{-10} -6 z^2 a^{-12} +5 z^2 a^{-14} +3 z^2 a^{-16} -13 z a^{-9} -22 z a^{-11} -12 z a^{-13} -3 z a^{-15} +9 a^{-8} +13 a^{-10} +6 a^{-12} + a^{-14} }[/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["K11n77"];

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

{}

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:

(7, 16)

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{ 28 }[/math] [math]\displaystyle{ 128 }[/math] [math]\displaystyle{ 392 }[/math] [math]\displaystyle{ \frac{2354}{3} }[/math] [math]\displaystyle{ \frac{286}{3} }[/math] [math]\displaystyle{ 3584 }[/math] [math]\displaystyle{ \frac{16256}{3} }[/math] [math]\displaystyle{ \frac{2624}{3} }[/math] [math]\displaystyle{ 544 }[/math] [math]\displaystyle{ \frac{10976}{3} }[/math] [math]\displaystyle{ 8192 }[/math] [math]\displaystyle{ \frac{65912}{3} }[/math] [math]\displaystyle{ \frac{8008}{3} }[/math] [math]\displaystyle{ \frac{1164937}{30} }[/math] [math]\displaystyle{ \frac{12922}{5} }[/math] [math]\displaystyle{ \frac{531554}{45} }[/math] [math]\displaystyle{ \frac{4439}{18} }[/math] [math]\displaystyle{ \frac{41257}{30} }[/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]6 is the signature of K11n77. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ 0 1 2 3 4 5 6 7 8 9 10 11 χ 29                       1 -1 27                     2   2 25                   2 1   -1 23                 3 2     1 21               3 2       -1 19           1 2 3         -2 17           3 3           0 15       1 1 2             -2 13         3               3 11     1                   1 9 1                       1 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=5 }[/math] [math]\displaystyle{ i=7 }[/math] [math]\displaystyle{ i=9 }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=7 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=8 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=9 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=10 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=11 }[/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.

K11n76 K11n78