K11n141

K11n140

K11n142

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n141 at Knotilus!
	K11n141 is also known as the pretzel knot P(5,-3,-3).

Knot presentations

Planar diagram presentation	X4251 X12,3,13,4 X5,16,6,17 X7,14,8,15 X9,21,10,20 X11,19,12,18 X2,13,3,14 X15,6,16,7 X17,22,18,1 X19,11,20,10 X21,9,22,8
Gauss code	1, -7, 2, -1, -3, 8, -4, 11, -5, 10, -6, -2, 7, 4, -8, 3, -9, 6, -10, 5, -11, 9
Dowker-Thistlethwaite code	4 12 -16 -14 -20 -18 2 -6 -22 -10 -8

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number $\{1,2,3\}$ 3-genus 1 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11n141/ThurstonBennequinNumber Hyperbolic Volume 7.80349 A-Polynomial See Data:K11n141/A-polynomial

[edit Notes for K11n141's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $1$ Rasmussen s-Invariant 0

[edit Notes for K11n141's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-5t+11-5t^{-1}$
Conway polynomial	$1-5z^{2}$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 21, 0 }
Jones polynomial	$2q^{2}-2q+3-4q^{-1}+3q^{-2}-3q^{-3}+2q^{-4}-q^{-5}+q^{-6}$
HOMFLY-PT polynomial (db, data sources)	$a^{6}-z^{2}a^{4}-2z^{2}a^{2}-a^{2}-2z^{2}-1+2a^{-2}$
Kauffman polynomial (db, data sources)	$a^{3}z^{9}+az^{9}+a^{4}z^{8}+2a^{2}z^{8}+z^{8}+a^{5}z^{7}-6a^{3}z^{7}-7az^{7}+a^{6}z^{6}-4a^{4}z^{6}-11a^{2}z^{6}-6z^{6}-4a^{5}z^{5}+15a^{3}z^{5}+20az^{5}+z^{5}a^{-1}-5a^{6}z^{4}+4a^{4}z^{4}+22a^{2}z^{4}+13z^{4}+3a^{5}z^{3}-19a^{3}z^{3}-24az^{3}-2z^{3}a^{-1}+6a^{6}z^{2}-2a^{4}z^{2}-16a^{2}z^{2}+2z^{2}a^{-2}-6z^{2}+8a^{3}z+11az+3za^{-1}-a^{6}+a^{2}-2a^{-2}-1$
The A2 invariant	Data:K11n141/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11n141/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n141 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["K11n141"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-5t+11-5t^{-1}$

In[5]:=

Conway[K][z]

Out[5]=

$1-5z^{2}$

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]=

$\{1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 21, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

$2q^{2}-2q+3-4q^{-1}+3q^{-2}-3q^{-3}+2q^{-4}-q^{-5}+q^{-6}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$a^{6}-z^{2}a^{4}-2z^{2}a^{2}-a^{2}-2z^{2}-1+2a^{-2}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$a^{3}z^{9}+az^{9}+a^{4}z^{8}+2a^{2}z^{8}+z^{8}+a^{5}z^{7}-6a^{3}z^{7}-7az^{7}+a^{6}z^{6}-4a^{4}z^{6}-11a^{2}z^{6}-6z^{6}-4a^{5}z^{5}+15a^{3}z^{5}+20az^{5}+z^{5}a^{-1}-5a^{6}z^{4}+4a^{4}z^{4}+22a^{2}z^{4}+13z^{4}+3a^{5}z^{3}-19a^{3}z^{3}-24az^{3}-2z^{3}a^{-1}+6a^{6}z^{2}-2a^{4}z^{2}-16a^{2}z^{2}+2z^{2}a^{-2}-6z^{2}+8a^{3}z+11az+3za^{-1}-a^{6}+a^{2}-2a^{-2}-1$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {}

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["K11n141"];

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]=

{ $-5t+11-5t^{-1}$, $2q^{2}-2q+3-4q^{-1}+3q^{-2}-3q^{-3}+2q^{-4}-q^{-5}+q^{-6}$ }

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:

(-5, 4)

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 $-20$ $32$ $200$ ${\frac {746}{3}}$ ${\frac {310}{3}}$ $-640$ $-{\frac {3040}{3}}$ $-{\frac {640}{3}}$ $-224$ $-{\frac {4000}{3}}$ $512$ $-{\frac {14920}{3}}$ $-{\frac {6200}{3}}$ $-{\frac {19279}{6}}$ $778$ $-{\frac {30782}{9}}$ ${\frac {9995}{18}}$ $-{\frac {5071}{6}}$

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 $t^{r}q^{j}$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$). The squares with yellow highlighting are those on the "critical diagonals", where $j-2r=s+1$ or $j-2r=s-1$, where $s=$0 is the signature of K11n141. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-1$ $i=1$ $r=-6$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{2}$ $r=-3$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-2$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-1$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=0$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{3}$ $r=1$ ${\mathbb {Z} }^{2}$ $r=2$ ${\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$

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.

K11n140 K11n142