K11a248

K11a247

K11a249

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a248 at Knotilus!

Knot presentations

Planar diagram presentation	X6271 X8394 X10,6,11,5 X18,8,19,7 X16,9,17,10 X20,11,21,12 X22,13,1,14 X4,16,5,15 X2,17,3,18 X14,19,15,20 X12,21,13,22
Gauss code	1, -9, 2, -8, 3, -1, 4, -2, 5, -3, 6, -11, 7, -10, 8, -5, 9, -4, 10, -6, 11, -7
Dowker-Thistlethwaite code	6 8 10 18 16 20 22 4 2 14 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:K11a248/ThurstonBennequinNumber Hyperbolic Volume 17.7145 A-Polynomial See Data:K11a248/A-polynomial

[edit Notes for K11a248'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 K11a248's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ -t^4+6 t^3-18 t^2+34 t-41+34 t^{-1} -18 t^{-2} +6 t^{-3} - t^{-4} }[/math]
Conway polynomial	[math]\displaystyle{ -z^8-2 z^6-2 z^4+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 159, -2 }
Jones polynomial	[math]\displaystyle{ q^3-4 q^2+9 q-15+22 q^{-1} -25 q^{-2} +26 q^{-3} -23 q^{-4} +17 q^{-5} -11 q^{-6} +5 q^{-7} - q^{-8} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ -a^2 z^8+2 a^4 z^6-5 a^2 z^6+z^6-a^6 z^4+6 a^4 z^4-10 a^2 z^4+3 z^4-a^6 z^2+5 a^4 z^2-7 a^2 z^2+3 z^2+1 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ 3 a^4 z^{10}+3 a^2 z^{10}+9 a^5 z^9+17 a^3 z^9+8 a z^9+13 a^6 z^8+17 a^4 z^8+12 a^2 z^8+8 z^8+11 a^7 z^7-2 a^5 z^7-33 a^3 z^7-16 a z^7+4 z^7 a^{-1} +5 a^8 z^6-17 a^6 z^6-44 a^4 z^6-44 a^2 z^6+z^6 a^{-2} -21 z^6+a^9 z^5-15 a^7 z^5-14 a^5 z^5+21 a^3 z^5+10 a z^5-9 z^5 a^{-1} -4 a^8 z^4+4 a^6 z^4+32 a^4 z^4+45 a^2 z^4-2 z^4 a^{-2} +19 z^4+4 a^7 z^3+3 a^5 z^3-7 a^3 z^3-2 a z^3+4 z^3 a^{-1} -a^6 z^2-9 a^4 z^2-15 a^2 z^2-7 z^2+a^7 z+3 a^5 z+3 a^3 z+a z+1 }[/math]
The A2 invariant	[math]\displaystyle{ -q^{24}+2 q^{22}-2 q^{18}+4 q^{16}-5 q^{14}+q^{12}-2 q^8+6 q^6-4 q^4+5 q^2-1-2 q^{-2} +3 q^{-4} -2 q^{-6} + q^{-8} }[/math]
The G2 invariant	Data:K11a248/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a248 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["K11a248"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ -t^4+6 t^3-18 t^2+34 t-41+34 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+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]=

{ 159, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ q^3-4 q^2+9 q-15+22 q^{-1} -25 q^{-2} +26 q^{-3} -23 q^{-4} +17 q^{-5} -11 q^{-6} +5 q^{-7} - q^{-8} }[/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+2 a^4 z^6-5 a^2 z^6+z^6-a^6 z^4+6 a^4 z^4-10 a^2 z^4+3 z^4-a^6 z^2+5 a^4 z^2-7 a^2 z^2+3 z^2+1 }[/math]

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ 3 a^4 z^{10}+3 a^2 z^{10}+9 a^5 z^9+17 a^3 z^9+8 a z^9+13 a^6 z^8+17 a^4 z^8+12 a^2 z^8+8 z^8+11 a^7 z^7-2 a^5 z^7-33 a^3 z^7-16 a z^7+4 z^7 a^{-1} +5 a^8 z^6-17 a^6 z^6-44 a^4 z^6-44 a^2 z^6+z^6 a^{-2} -21 z^6+a^9 z^5-15 a^7 z^5-14 a^5 z^5+21 a^3 z^5+10 a z^5-9 z^5 a^{-1} -4 a^8 z^4+4 a^6 z^4+32 a^4 z^4+45 a^2 z^4-2 z^4 a^{-2} +19 z^4+4 a^7 z^3+3 a^5 z^3-7 a^3 z^3-2 a z^3+4 z^3 a^{-1} -a^6 z^2-9 a^4 z^2-15 a^2 z^2-7 z^2+a^7 z+3 a^5 z+3 a^3 z+a z+1 }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a71,}

Same Jones Polynomial (up to mirroring, [math]\displaystyle{ q\leftrightarrow q^{-1} }[/math]): {K11a71,}

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["K11a248"];

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+34 t-41+34 t^{-1} -18 t^{-2} +6 t^{-3} - t^{-4} }[/math], [math]\displaystyle{ q^3-4 q^2+9 q-15+22 q^{-1} -25 q^{-2} +26 q^{-3} -23 q^{-4} +17 q^{-5} -11 q^{-6} +5 q^{-7} - q^{-8} }[/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]=

{K11a71,}

In[6]:=

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

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

Out[6]=

{K11a71,}

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{ 16 }[/math] [math]\displaystyle{ 16 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ -64 }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ -96 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 136 }[/math] [math]\displaystyle{ -\frac{832}{3} }[/math] [math]\displaystyle{ \frac{896}{3} }[/math] [math]\displaystyle{ \frac{344}{3} }[/math] [math]\displaystyle{ 40 }[/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 K11a248. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 7                       1 1 5                     3   -3 3                   6 1   5 1                 9 3     -6 -1               13 6       7 -3             13 10         -3 -5           13 12           1 -7         10 13             3 -9       7 13               -6 -11     4 10                 6 -13   1 7                   -6 -15   4                     4 -17 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=-7 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{13} }[/math] [math]\displaystyle{ {\mathbb Z}^{13} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{13} }[/math] [math]\displaystyle{ {\mathbb Z}^{13} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{12} }[/math] [math]\displaystyle{ {\mathbb Z}^{13} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/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}_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.

K11a247 K11a249