K11n139

K11n138

K11n140

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n139 at Knotilus!
	K11n139 is also known as the pretzel knot P(5,3,-3).

Knot presentations

Planar diagram presentation	X4251 X12,3,13,4 X16,6,17,5 X14,8,15,7 X9,21,10,20 X11,19,12,18 X2,13,3,14 X6,16,7,15 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 Chiral Unknotting number $\{1,2\}$ 3-genus 1 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11n139/ThurstonBennequinNumber Hyperbolic Volume 5.56972 A-Polynomial See Data:K11n139/A-polynomial

[edit Notes for K11n139's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11n139's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-2t+5-2t^{-1}$
Conway polynomial	$1-2z^{2}$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 9, 0 }
Jones polynomial	$q^{8}-q^{7}+q^{6}-2q^{5}+q^{4}-q^{3}+q^{2}+1$
HOMFLY-PT polynomial (db, data sources)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -z^2 a^{-4} -z^2 a^{-6} + a^{-2} - a^{-4} - a^{-6} + a^{-8} +1}
Kauffman polynomial (db, data sources)	$z^{9}a^{-5}+z^{9}a^{-7}+z^{8}a^{-4}+2z^{8}a^{-6}+z^{8}a^{-8}-7z^{7}a^{-5}-7z^{7}a^{-7}-6z^{6}a^{-4}-13z^{6}a^{-6}-7z^{6}a^{-8}+z^{5}a^{-3}+17z^{5}a^{-5}+16z^{5}a^{-7}+10z^{4}a^{-4}+25z^{4}a^{-6}+15z^{4}a^{-8}-5z^{3}a^{-3}-20z^{3}a^{-5}-15z^{3}a^{-7}-5z^{2}a^{-4}-15z^{2}a^{-6}-10z^{2}a^{-8}+4za^{-3}+10za^{-5}+6za^{-7}-a^{-2}-a^{-4}+a^{-6}+a^{-8}+1$
The A2 invariant	Data:K11n139/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11n139/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n139 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["K11n139"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-2t+5-2t^{-1}$

In[5]:=

Conway[K][z]

Out[5]=

$1-2z^{2}$

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]=

{ 9, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{8}-q^{7}+q^{6}-2q^{5}+q^{4}-q^{3}+q^{2}+1$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -z^2 a^{-4} -z^2 a^{-6} + a^{-2} - a^{-4} - a^{-6} + a^{-8} +1}

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{9}a^{-5}+z^{9}a^{-7}+z^{8}a^{-4}+2z^{8}a^{-6}+z^{8}a^{-8}-7z^{7}a^{-5}-7z^{7}a^{-7}-6z^{6}a^{-4}-13z^{6}a^{-6}-7z^{6}a^{-8}+z^{5}a^{-3}+17z^{5}a^{-5}+16z^{5}a^{-7}+10z^{4}a^{-4}+25z^{4}a^{-6}+15z^{4}a^{-8}-5z^{3}a^{-3}-20z^{3}a^{-5}-15z^{3}a^{-7}-5z^{2}a^{-4}-15z^{2}a^{-6}-10z^{2}a^{-8}+4za^{-3}+10za^{-5}+6za^{-7}-a^{-2}-a^{-4}+a^{-6}+a^{-8}+1$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {6_1, 9_46, K11n67, K11n97,}

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["K11n139"];

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]=

{ $-2t+5-2t^{-1}$, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^8-q^7+q^6-2 q^5+q^4-q^3+q^2+1} }

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]=

{6_1, 9_46, K11n67, K11n97,}

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, -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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -8} $-40$ $32$ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{172}{3}} ${\frac {52}{3}}$ $320$ ${\frac {1040}{3}}$ ${\frac {320}{3}}$ $120$ $-{\frac {256}{3}}$ $800$ ${\frac {1376}{3}}$ $-{\frac {416}{3}}$ ${\frac {34889}{15}}$ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{1036}{15}} ${\frac {46076}{45}}$ ${\frac {1639}{9}}$ ${\frac {1289}{15}}$

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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s=} 0 is the signature of K11n139. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ 0 1 2 3 4 5 6 7 8 χ 17                 1 1 15                   0 13             1 1   0 11           1       -1 9           1       -1 7       1 1         0 5                   0 3     1             1 1 1                 1 -1 1                 1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i=-1} $i=1$ $r=0$ ${\mathbb {Z} }$ ${\mathbb {Z} }$ $r=1$ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=2} ${\mathbb {Z} }$ $r=3$ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}_2} ${\mathbb {Z} }$ $r=4$ ${\mathbb {Z} }$ $r=5$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=6$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=7$ ${\mathbb {Z} }$ $r=8$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

K11n138 K11n140