K11n2

K11n1

K11n3

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n2 at Knotilus!

Knot K11n2.

A graph, knot K11n2.

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

Knot presentations

Planar diagram presentation	X4251 X8394 X10,6,11,5 X7,15,8,14 X2,9,3,10 X16,12,17,11 X20,14,21,13 X15,7,16,6 X22,18,1,17 X12,20,13,19 X18,22,19,21
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 2 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11n2/ThurstonBennequinNumber Hyperbolic Volume 12.5932 A-Polynomial See Data:K11n2/A-polynomial

[edit Notes for K11n2's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $3$ Rasmussen s-Invariant -4

[edit Notes for K11n2's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-2t^{3}+8t^{2}-12t+13-12t^{-1}+8t^{-2}-2t^{-3}$
Conway polynomial	$-2z^{6}-4z^{4}+2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 57, 4 }
Jones polynomial	$-2q^{9}+5q^{8}-7q^{7}+9q^{6}-10q^{5}+9q^{4}-7q^{3}+5q^{2}-2q+1$
HOMFLY-PT polynomial (db, data sources)	$-z^{6}a^{-4}-z^{6}a^{-6}+z^{4}a^{-2}-3z^{4}a^{-4}-3z^{4}a^{-6}+z^{4}a^{-8}+3z^{2}a^{-2}-2z^{2}a^{-4}-2z^{2}a^{-6}+3z^{2}a^{-8}+2a^{-2}-a^{-4}-a^{-6}+2a^{-8}-a^{-10}$
Kauffman polynomial (db, data sources)	$z^{9}a^{-5}+z^{9}a^{-7}+2z^{8}a^{-4}+5z^{8}a^{-6}+3z^{8}a^{-8}+2z^{7}a^{-3}+2z^{7}a^{-5}+3z^{7}a^{-7}+3z^{7}a^{-9}+z^{6}a^{-2}-3z^{6}a^{-4}-11z^{6}a^{-6}-6z^{6}a^{-8}+z^{6}a^{-10}-6z^{5}a^{-3}-9z^{5}a^{-5}-7z^{5}a^{-7}-4z^{5}a^{-9}-4z^{4}a^{-2}-4z^{4}a^{-4}+8z^{4}a^{-6}+12z^{4}a^{-8}+4z^{4}a^{-10}+4z^{3}a^{-3}+4z^{3}a^{-5}+3z^{3}a^{-7}+6z^{3}a^{-9}+3z^{3}a^{-11}+5z^{2}a^{-2}+5z^{2}a^{-4}-5z^{2}a^{-6}-10z^{2}a^{-8}-5z^{2}a^{-10}-2za^{-9}-2za^{-11}-2a^{-2}-a^{-4}+a^{-6}+2a^{-8}+a^{-10}$
The A2 invariant	Data:K11n2/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11n2/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n2 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["K11n2"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-2t^{3}+8t^{2}-12t+13-12t^{-1}+8t^{-2}-2t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 57, 4 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-z^{6}a^{-4}-z^{6}a^{-6}+z^{4}a^{-2}-3z^{4}a^{-4}-3z^{4}a^{-6}+z^{4}a^{-8}+3z^{2}a^{-2}-2z^{2}a^{-4}-2z^{2}a^{-6}+3z^{2}a^{-8}+2a^{-2}-a^{-4}-a^{-6}+2a^{-8}-a^{-10}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

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

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_14, K11a161,}

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["K11n2"];

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]=

{ $-2t^{3}+8t^{2}-12t+13-12t^{-1}+8t^{-2}-2t^{-3}$, $-2q^{9}+5q^{8}-7q^{7}+9q^{6}-10q^{5}+9q^{4}-7q^{3}+5q^{2}-2q+1$ }

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]=

{10_14, K11a161,}

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 $8$ $40$ $32$ ${\frac {652}{3}}$ ${\frac {140}{3}}$ $320$ ${\frac {3280}{3}}$ ${\frac {640}{3}}$ $200$ ${\frac {256}{3}}$ $800$ ${\frac {5216}{3}}$ ${\frac {1120}{3}}$ ${\frac {81511}{15}}$ $-{\frac {1684}{15}}$ ${\frac {118564}{45}}$ ${\frac {905}{9}}$ ${\frac {5191}{15}}$

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 $s=$4 is the signature of K11n2. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

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

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.

K11n1 K11n3