K11n75

K11n74

K11n76

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n75 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X8493 X5,14,6,15 X2837 X9,20,10,21 X11,16,12,17 X13,6,14,7 X15,18,16,19 X17,12,18,13 X19,22,20,1 X21,10,22,11
Gauss code	1, -4, 2, -1, -3, 7, 4, -2, -5, 11, -6, 9, -7, 3, -8, 6, -9, 8, -10, 5, -11, 10
Dowker-Thistlethwaite code	4 8 -14 2 -20 -16 -6 -18 -12 -22 -10

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number [math]\displaystyle{ \{1,2\} }[/math] 3-genus 3 Bridge index 4 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11n75/ThurstonBennequinNumber Hyperbolic Volume 13.5931 A-Polynomial See Data:K11n75/A-polynomial

[edit Notes for K11n75's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus [math]\displaystyle{ 1 }[/math] Rasmussen s-Invariant 2

[edit Notes for K11n75's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ 2 t^3-7 t^2+14 t-17+14 t^{-1} -7 t^{-2} +2 t^{-3} }[/math]
Conway polynomial	[math]\displaystyle{ 2 z^6+5 z^4+4 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \left\{t^2-t+1\right\} }[/math]
Determinant and Signature	{ 63, -2 }
Jones polynomial	[math]\displaystyle{ -2+5 q^{-1} -7 q^{-2} +11 q^{-3} -10 q^{-4} +10 q^{-5} -9 q^{-6} +5 q^{-7} -3 q^{-8} + q^{-9} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ z^2 a^8+2 a^8-3 z^4 a^6-10 z^2 a^6-9 a^6+2 z^6 a^4+10 z^4 a^4+18 z^2 a^4+11 a^4-2 z^4 a^2-5 z^2 a^2-3 a^2 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^6 a^{10}-3 z^4 a^{10}+2 z^2 a^{10}+3 z^7 a^9-10 z^5 a^9+10 z^3 a^9-4 z a^9+3 z^8 a^8-6 z^6 a^8-z^4 a^8+z^2 a^8+2 a^8+z^9 a^7+7 z^7 a^7-31 z^5 a^7+36 z^3 a^7-17 z a^7+7 z^8 a^6-18 z^6 a^6+17 z^4 a^6-15 z^2 a^6+9 a^6+z^9 a^5+8 z^7 a^5-31 z^5 a^5+42 z^3 a^5-21 z a^5+4 z^8 a^4-10 z^6 a^4+19 z^4 a^4-20 z^2 a^4+11 a^4+4 z^7 a^3-10 z^5 a^3+19 z^3 a^3-11 z a^3+z^6 a^2+4 z^4 a^2-6 z^2 a^2+3 a^2+3 z^3 a-3 z a }[/math]
The A2 invariant	Data:K11n75/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11n75/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n75 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["K11n75"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ 2 t^3-7 t^2+14 t-17+14 t^{-1} -7 t^{-2} +2 t^{-3} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ 2 z^6+5 z^4+4 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]=

{ 63, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ -2+5 q^{-1} -7 q^{-2} +11 q^{-3} -10 q^{-4} +10 q^{-5} -9 q^{-6} +5 q^{-7} -3 q^{-8} + q^{-9} }[/math]

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

[math]\displaystyle{ z^2 a^8+2 a^8-3 z^4 a^6-10 z^2 a^6-9 a^6+2 z^6 a^4+10 z^4 a^4+18 z^2 a^4+11 a^4-2 z^4 a^2-5 z^2 a^2-3 a^2 }[/math]

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ z^6 a^{10}-3 z^4 a^{10}+2 z^2 a^{10}+3 z^7 a^9-10 z^5 a^9+10 z^3 a^9-4 z a^9+3 z^8 a^8-6 z^6 a^8-z^4 a^8+z^2 a^8+2 a^8+z^9 a^7+7 z^7 a^7-31 z^5 a^7+36 z^3 a^7-17 z a^7+7 z^8 a^6-18 z^6 a^6+17 z^4 a^6-15 z^2 a^6+9 a^6+z^9 a^5+8 z^7 a^5-31 z^5 a^5+42 z^3 a^5-21 z a^5+4 z^8 a^4-10 z^6 a^4+19 z^4 a^4-20 z^2 a^4+11 a^4+4 z^7 a^3-10 z^5 a^3+19 z^3 a^3-11 z a^3+z^6 a^2+4 z^4 a^2-6 z^2 a^2+3 a^2+3 z^3 a-3 z a }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_65, 10_77, K11n71,}

Same Jones Polynomial (up to mirroring, [math]\displaystyle{ q\leftrightarrow q^{-1} }[/math]): {K11n71,}

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["K11n75"];

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{ 2 t^3-7 t^2+14 t-17+14 t^{-1} -7 t^{-2} +2 t^{-3} }[/math], [math]\displaystyle{ -2+5 q^{-1} -7 q^{-2} +11 q^{-3} -10 q^{-4} +10 q^{-5} -9 q^{-6} +5 q^{-7} -3 q^{-8} + q^{-9} }[/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]=

{10_65, 10_77, K11n71,}

In[6]:=

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

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

Out[6]=

{K11n71,}

Vassiliev invariants

V2 and V3:

(4, -5)

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{ 16 }[/math] [math]\displaystyle{ -40 }[/math] [math]\displaystyle{ 128 }[/math] [math]\displaystyle{ \frac{536}{3} }[/math] [math]\displaystyle{ \frac{64}{3} }[/math] [math]\displaystyle{ -640 }[/math] [math]\displaystyle{ -\frac{2416}{3} }[/math] [math]\displaystyle{ -\frac{352}{3} }[/math] [math]\displaystyle{ -104 }[/math] [math]\displaystyle{ \frac{2048}{3} }[/math] [math]\displaystyle{ 800 }[/math] [math]\displaystyle{ \frac{8576}{3} }[/math] [math]\displaystyle{ \frac{1024}{3} }[/math] [math]\displaystyle{ \frac{58622}{15} }[/math] [math]\displaystyle{ \frac{3632}{15} }[/math] [math]\displaystyle{ \frac{53888}{45} }[/math] [math]\displaystyle{ \frac{370}{9} }[/math] [math]\displaystyle{ \frac{1982}{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 K11n75. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

K11n74 K11n76