K11n1

K11a367

K11n2

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n1 at Knotilus!

Knot K11n1.

A graph, knot K11n1.

A part of a knot and a part of a graph.

Knot presentations

Planar diagram presentation	X4251 X8394 X10,6,11,5 X14,7,15,8 X2,9,3,10 X11,16,12,17 X13,20,14,21 X6,15,7,16 X17,22,18,1 X19,12,20,13 X21,18,22,19
Gauss code	1, -5, 2, -1, 3, -8, 4, -2, 5, -3, -6, 10, -7, -4, 8, 6, -9, 11, -10, 7, -11, 9
Dowker-Thistlethwaite code	4 8 10 14 2 -16 -20 6 -22 -12 -18

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 2 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11n1/ThurstonBennequinNumber Hyperbolic Volume 9.70048 A-Polynomial See Data:K11n1/A-polynomial

[edit Notes for K11n1's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11n1's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{2}+7t-11+7t^{-1}-t^{-2}$
Conway polynomial	$-z^{4}+3z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 27, -2 }
Jones polynomial	$q^{-1}-q^{-2}+3q^{-3}-4q^{-4}+4q^{-5}-4q^{-6}+4q^{-7}-3q^{-8}+2q^{-9}-q^{-10}$
HOMFLY-PT polynomial (db, data sources)	$-a^{10}+2z^{2}a^{8}+2a^{8}-z^{4}a^{6}-2z^{2}a^{6}-2a^{6}+2z^{2}a^{4}+a^{4}+z^{2}a^{2}+a^{2}$
Kauffman polynomial (db, data sources)	$z^{7}a^{11}-5z^{5}a^{11}+7z^{3}a^{11}-3za^{11}+2z^{8}a^{10}-10z^{6}a^{10}+14z^{4}a^{10}-7z^{2}a^{10}+a^{10}+z^{9}a^{9}-2z^{7}a^{9}-7z^{5}a^{9}+13z^{3}a^{9}-4za^{9}+4z^{8}a^{8}-19z^{6}a^{8}+25z^{4}a^{8}-12z^{2}a^{8}+2a^{8}+z^{9}a^{7}-2z^{7}a^{7}-5z^{5}a^{7}+8z^{3}a^{7}-2za^{7}+2z^{8}a^{6}-9z^{6}a^{6}+13z^{4}a^{6}-9z^{2}a^{6}+2a^{6}+z^{7}a^{5}-3z^{5}a^{5}+3z^{3}a^{5}-za^{5}+2z^{4}a^{4}-3z^{2}a^{4}+a^{4}+z^{3}a^{3}+z^{2}a^{2}-a^{2}$
The A2 invariant	Data:K11n1/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11n1/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n1 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["K11n1"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{2}+7t-11+7t^{-1}-t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 27, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{-1}-q^{-2}+3q^{-3}-4q^{-4}+4q^{-5}-4q^{-6}+4q^{-7}-3q^{-8}+2q^{-9}-q^{-10}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-a^{10}+2z^{2}a^{8}+2a^{8}-z^{4}a^{6}-2z^{2}a^{6}-2a^{6}+2z^{2}a^{4}+a^{4}+z^{2}a^{2}+a^{2}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{7}a^{11}-5z^{5}a^{11}+7z^{3}a^{11}-3za^{11}+2z^{8}a^{10}-10z^{6}a^{10}+14z^{4}a^{10}-7z^{2}a^{10}+a^{10}+z^{9}a^{9}-2z^{7}a^{9}-7z^{5}a^{9}+13z^{3}a^{9}-4za^{9}+4z^{8}a^{8}-19z^{6}a^{8}+25z^{4}a^{8}-12z^{2}a^{8}+2a^{8}+z^{9}a^{7}-2z^{7}a^{7}-5z^{5}a^{7}+8z^{3}a^{7}-2za^{7}+2z^{8}a^{6}-9z^{6}a^{6}+13z^{4}a^{6}-9z^{2}a^{6}+2a^{6}+z^{7}a^{5}-3z^{5}a^{5}+3z^{3}a^{5}-za^{5}+2z^{4}a^{4}-3z^{2}a^{4}+a^{4}+z^{3}a^{3}+z^{2}a^{2}-a^{2}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {9_48,}

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["K11n1"];

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]=

{ $-t^{2}+7t-11+7t^{-1}-t^{-2}$, $q^{-1}-q^{-2}+3q^{-3}-4q^{-4}+4q^{-5}-4q^{-6}+4q^{-7}-3q^{-8}+2q^{-9}-q^{-10}$ }

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]=

{9_48,}

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:

(3, -7)

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 $12$ $-56$ $72$ $286$ $42$ $-672$ $-{\frac {4688}{3}}$ $-{\frac {800}{3}}$ $-216$ $288$ $1568$ $3432$ $504$ ${\frac {87151}{10}}$ ${\frac {1694}{15}}$ ${\frac {52822}{15}}$ ${\frac {529}{6}}$ ${\frac {4431}{10}}$

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 $s=$-2 is the signature of K11n1. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -1                   1 1 -3                 1 1 0 -5               2     2 -7             2 1     -1 -9           2 2       0 -11         2 2         0 -13       2 2           0 -15     1 2             1 -17   1 2               -1 -19   1                 1 -21 1                   -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-3$ $i=-1$ $r=-9$ ${\mathbb {Z} }$ $r=-8$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-7$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-5$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-4$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-3$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-2$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-1$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=0$ ${\mathbb {Z} }$ ${\mathbb {Z} }$

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.

K11a367 K11n2