K11n114

K11n113

K11n115

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n114 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X10,3,11,4 X14,6,15,5 X18,7,19,8 X16,9,17,10 X2,11,3,12 X13,21,14,20 X22,16,1,15 X8,17,9,18 X19,13,20,12 X6,21,7,22
Gauss code	1, -6, 2, -1, 3, -11, 4, -9, 5, -2, 6, 10, -7, -3, 8, -5, 9, -4, -10, 7, 11, -8
Dowker-Thistlethwaite code	4 10 14 18 16 2 -20 22 8 -12 6

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Chiral Unknotting number 1 3-genus 2 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11n114/ThurstonBennequinNumber Hyperbolic Volume 12.8914 A-Polynomial See Data:K11n114/A-polynomial

[edit Notes for K11n114's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $2$ Rasmussen s-Invariant 0

[edit Notes for K11n114's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$3t^{2}-13t+21-13t^{-1}+3t^{-2}$
Conway polynomial	$3z^{4}-z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 53, 0 }
Jones polynomial	$-q^{5}+3q^{4}-5q^{3}+8q^{2}-9q+9-8q^{-1}+6q^{-2}-3q^{-3}+q^{-4}$
HOMFLY-PT polynomial (db, data sources)	$a^{4}-3z^{2}a^{2}-2a^{2}+2z^{4}+3z^{2}+2+z^{4}a^{-2}-z^{2}a^{-4}$
Kauffman polynomial (db, data sources)	$az^{9}+z^{9}a^{-1}+a^{2}z^{8}+3z^{8}a^{-2}+4z^{8}-2az^{7}+2z^{7}a^{-1}+4z^{7}a^{-3}-2a^{2}z^{6}-5z^{6}a^{-2}+3z^{6}a^{-4}-10z^{6}+3a^{3}z^{5}+7az^{5}-6z^{5}a^{-1}-9z^{5}a^{-3}+z^{5}a^{-5}+a^{4}z^{4}+8a^{2}z^{4}+2z^{4}a^{-2}-7z^{4}a^{-4}+16z^{4}-4a^{3}z^{3}-5az^{3}+6z^{3}a^{-1}+5z^{3}a^{-3}-2z^{3}a^{-5}-2a^{4}z^{2}-9a^{2}z^{2}+3z^{2}a^{-4}-10z^{2}+a^{3}z-2za^{-1}-za^{-3}+a^{4}+2a^{2}+2$
The A2 invariant	Data:K11n114/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11n114/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n114 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["K11n114"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$3t^{2}-13t+21-13t^{-1}+3t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

$3z^{4}-z^{2}+1$

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]=

{ 53, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{5}+3q^{4}-5q^{3}+8q^{2}-9q+9-8q^{-1}+6q^{-2}-3q^{-3}+q^{-4}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$a^{4}-3z^{2}a^{2}-2a^{2}+2z^{4}+3z^{2}+2+z^{4}a^{-2}-z^{2}a^{-4}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$az^{9}+z^{9}a^{-1}+a^{2}z^{8}+3z^{8}a^{-2}+4z^{8}-2az^{7}+2z^{7}a^{-1}+4z^{7}a^{-3}-2a^{2}z^{6}-5z^{6}a^{-2}+3z^{6}a^{-4}-10z^{6}+3a^{3}z^{5}+7az^{5}-6z^{5}a^{-1}-9z^{5}a^{-3}+z^{5}a^{-5}+a^{4}z^{4}+8a^{2}z^{4}+2z^{4}a^{-2}-7z^{4}a^{-4}+16z^{4}-4a^{3}z^{3}-5az^{3}+6z^{3}a^{-1}+5z^{3}a^{-3}-2z^{3}a^{-5}-2a^{4}z^{2}-9a^{2}z^{2}+3z^{2}a^{-4}-10z^{2}+a^{3}z-2za^{-1}-za^{-3}+a^{4}+2a^{2}+2$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a195,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {9_30,}

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["K11n114"];

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]=

{ $3t^{2}-13t+21-13t^{-1}+3t^{-2}$, $-q^{5}+3q^{4}-5q^{3}+8q^{2}-9q+9-8q^{-1}+6q^{-2}-3q^{-3}+q^{-4}$ }

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]=

{K11a195,}

In[6]:=

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

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

Out[6]=

{9_30,}

Vassiliev invariants

V2 and V3:

(-1, 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 $-4$ $8$ $8$ $-{\frac {62}{3}}$ $-{\frac {58}{3}}$ $-32$ $-{\frac {112}{3}}$ $-{\frac {160}{3}}$ $40$ $-{\frac {32}{3}}$ $32$ ${\frac {248}{3}}$ ${\frac {232}{3}}$ ${\frac {4529}{30}}$ $-{\frac {286}{5}}$ ${\frac {10138}{45}}$ $-{\frac {1073}{18}}$ ${\frac {1169}{30}}$

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 K11n114. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

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

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.

K11n113 K11n115