K11a55

K11a54

K11a56

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a55 at Knotilus!

Knot presentations

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

A Braid Representative
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:K11a55/ThurstonBennequinNumber Hyperbolic Volume 11.6846 A-Polynomial See Data:K11a55/A-polynomial

[edit Notes for K11a55'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 K11a55's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ -t^4+5 t^3-10 t^2+13 t-13+13 t^{-1} -10 t^{-2} +5 t^{-3} - t^{-4} }[/math]
Conway polynomial	[math]\displaystyle{ -z^8-3 z^6+2 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 71, -2 }
Jones polynomial	[math]\displaystyle{ -q^4+2 q^3-4 q^2+7 q-8+11 q^{-1} -11 q^{-2} +10 q^{-3} -8 q^{-4} +5 q^{-5} -3 q^{-6} + q^{-7} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ -a^2 z^8+a^4 z^6-6 a^2 z^6+2 z^6+4 a^4 z^4-13 a^2 z^4-z^4 a^{-2} +10 z^4+4 a^4 z^2-13 a^2 z^2-4 z^2 a^{-2} +15 z^2+a^4-5 a^2-3 a^{-2} +8 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ a^2 z^{10}+z^{10}+3 a^3 z^9+5 a z^9+2 z^9 a^{-1} +4 a^4 z^8+3 a^2 z^8+2 z^8 a^{-2} +z^8+4 a^5 z^7-6 a^3 z^7-17 a z^7-6 z^7 a^{-1} +z^7 a^{-3} +4 a^6 z^6-7 a^4 z^6-19 a^2 z^6-9 z^6 a^{-2} -17 z^6+3 a^7 z^5-3 a^5 z^5+4 a^3 z^5+14 a z^5-z^5 a^{-1} -5 z^5 a^{-3} +a^8 z^4-4 a^6 z^4+7 a^4 z^4+29 a^2 z^4+12 z^4 a^{-2} +29 z^4-4 a^7 z^3-2 a^5 z^3+a^3 z^3+2 a z^3+10 z^3 a^{-1} +7 z^3 a^{-3} -a^8 z^2-4 a^4 z^2-19 a^2 z^2-8 z^2 a^{-2} -22 z^2+a^7 z+2 a^5 z-a^3 z-5 a z-6 z a^{-1} -3 z a^{-3} +a^4+5 a^2+3 a^{-2} +8 }[/math]
The A2 invariant	[math]\displaystyle{ q^{20}-q^{18}+q^{16}-q^{14}-q^{12}-3 q^8+2 q^6-q^4+3 q^2+3+ q^{-2} +2 q^{-4} - q^{-6} - q^{-10} - q^{-12} }[/math]
The G2 invariant	[math]\displaystyle{ q^{114}-2 q^{112}+4 q^{110}-6 q^{108}+4 q^{106}-2 q^{104}-4 q^{102}+12 q^{100}-18 q^{98}+22 q^{96}-19 q^{94}+8 q^{92}+7 q^{90}-23 q^{88}+37 q^{86}-40 q^{84}+34 q^{82}-19 q^{80}-2 q^{78}+23 q^{76}-36 q^{74}+46 q^{72}-44 q^{70}+33 q^{68}-16 q^{66}-8 q^{64}+30 q^{62}-44 q^{60}+48 q^{58}-35 q^{56}+11 q^{54}+15 q^{52}-37 q^{50}+35 q^{48}-16 q^{46}-16 q^{44}+41 q^{42}-49 q^{40}+26 q^{38}+11 q^{36}-55 q^{34}+81 q^{32}-86 q^{30}+51 q^{28}-2 q^{26}-53 q^{24}+93 q^{22}-101 q^{20}+80 q^{18}-33 q^{16}-17 q^{14}+59 q^{12}-77 q^{10}+73 q^8-34 q^6-7 q^4+46 q^2-52+39 q^{-2} +6 q^{-4} -44 q^{-6} +68 q^{-8} -59 q^{-10} +26 q^{-12} +25 q^{-14} -68 q^{-16} +92 q^{-18} -78 q^{-20} +42 q^{-22} +3 q^{-24} -47 q^{-26} +66 q^{-28} -65 q^{-30} +45 q^{-32} -19 q^{-34} -7 q^{-36} +22 q^{-38} -29 q^{-40} +23 q^{-42} -16 q^{-44} +6 q^{-46} -5 q^{-50} +4 q^{-52} -4 q^{-54} +3 q^{-56} - q^{-58} + q^{-60} }[/math]

Further Quantum Invariants

K11a55 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["K11a55"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ -t^4+5 t^3-10 t^2+13 t-13+13 t^{-1} -10 t^{-2} +5 t^{-3} - t^{-4} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

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

{ 71, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ -q^4+2 q^3-4 q^2+7 q-8+11 q^{-1} -11 q^{-2} +10 q^{-3} -8 q^{-4} +5 q^{-5} -3 q^{-6} + q^{-7} }[/math]

In[9]:=

HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]=

[math]\displaystyle{ -a^2 z^8+a^4 z^6-6 a^2 z^6+2 z^6+4 a^4 z^4-13 a^2 z^4-z^4 a^{-2} +10 z^4+4 a^4 z^2-13 a^2 z^2-4 z^2 a^{-2} +15 z^2+a^4-5 a^2-3 a^{-2} +8 }[/math]

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ a^2 z^{10}+z^{10}+3 a^3 z^9+5 a z^9+2 z^9 a^{-1} +4 a^4 z^8+3 a^2 z^8+2 z^8 a^{-2} +z^8+4 a^5 z^7-6 a^3 z^7-17 a z^7-6 z^7 a^{-1} +z^7 a^{-3} +4 a^6 z^6-7 a^4 z^6-19 a^2 z^6-9 z^6 a^{-2} -17 z^6+3 a^7 z^5-3 a^5 z^5+4 a^3 z^5+14 a z^5-z^5 a^{-1} -5 z^5 a^{-3} +a^8 z^4-4 a^6 z^4+7 a^4 z^4+29 a^2 z^4+12 z^4 a^{-2} +29 z^4-4 a^7 z^3-2 a^5 z^3+a^3 z^3+2 a z^3+10 z^3 a^{-1} +7 z^3 a^{-3} -a^8 z^2-4 a^4 z^2-19 a^2 z^2-8 z^2 a^{-2} -22 z^2+a^7 z+2 a^5 z-a^3 z-5 a z-6 z a^{-1} -3 z a^{-3} +a^4+5 a^2+3 a^{-2} +8 }[/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["K11a55"];

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^3-10 t^2+13 t-13+13 t^{-1} -10 t^{-2} +5 t^{-3} - t^{-4} }[/math], [math]\displaystyle{ -q^4+2 q^3-4 q^2+7 q-8+11 q^{-1} -11 q^{-2} +10 q^{-3} -8 q^{-4} +5 q^{-5} -3 q^{-6} + q^{-7} }[/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{44}{3} }[/math] [math]\displaystyle{ 64 }[/math] [math]\displaystyle{ \frac{176}{3} }[/math] [math]\displaystyle{ -\frac{160}{3} }[/math] [math]\displaystyle{ 40 }[/math] [math]\displaystyle{ \frac{256}{3} }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ \frac{992}{3} }[/math] [math]\displaystyle{ \frac{352}{3} }[/math] [math]\displaystyle{ \frac{7231}{15} }[/math] [math]\displaystyle{ \frac{292}{5} }[/math] [math]\displaystyle{ \frac{10324}{45} }[/math] [math]\displaystyle{ \frac{353}{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]-2 is the signature of K11a55. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 9                       1 -1 7                     1   1 5                   3 1   -2 3                 4 1     3 1               4 3       -1 -1             7 4         3 -3           5 5           0 -5         5 6             -1 -7       3 5               2 -9     2 5                 -3 -11   1 3                   2 -13   2                     -2 -15 1                       1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-3 }[/math] [math]\displaystyle{ i=-1 }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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}^{5}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/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}^{5}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=5 }[/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.

K11a54 K11a56