K11n26

K11n25

K11n27

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n26 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X8493 X5,13,6,12 X2837 X14,9,15,10 X18,12,19,11 X13,7,14,6 X20,15,21,16 X22,17,1,18 X10,20,11,19 X16,21,17,22
Gauss code	1, -4, 2, -1, -3, 7, 4, -2, 5, -10, 6, 3, -7, -5, 8, -11, 9, -6, 10, -8, 11, -9
Dowker-Thistlethwaite code	4 8 -12 2 14 18 -6 20 22 10 16

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Chiral Unknotting number 1 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11n26/ThurstonBennequinNumber Hyperbolic Volume 11.7289 A-Polynomial See Data:K11n26/A-polynomial

[edit Notes for K11n26's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $3$ Rasmussen s-Invariant 0

[edit Notes for K11n26's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{3}+5t^{2}-9t+11-9t^{-1}+5t^{-2}-t^{-3}$
Conway polynomial	$-z^{6}-z^{4}+2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 41, 0 }
Jones polynomial	$-q^{7}+2q^{6}-4q^{5}+6q^{4}-6q^{3}+7q^{2}-6q+5-3q^{-1}+q^{-2}$
HOMFLY-PT polynomial (db, data sources)	$-z^{6}a^{-2}-4z^{4}a^{-2}+2z^{4}a^{-4}+z^{4}-5z^{2}a^{-2}+6z^{2}a^{-4}-z^{2}a^{-6}+2z^{2}-2a^{-2}+4a^{-4}-2a^{-6}+1$
Kauffman polynomial (db, data sources)	$z^{9}a^{-3}+z^{9}a^{-5}+3z^{8}a^{-2}+5z^{8}a^{-4}+2z^{8}a^{-6}+3z^{7}a^{-1}+2z^{7}a^{-3}+z^{7}a^{-7}-9z^{6}a^{-2}-19z^{6}a^{-4}-9z^{6}a^{-6}+z^{6}-8z^{5}a^{-1}-16z^{5}a^{-3}-13z^{5}a^{-5}-5z^{5}a^{-7}+9z^{4}a^{-2}+20z^{4}a^{-4}+12z^{4}a^{-6}+z^{4}+3az^{3}+8z^{3}a^{-1}+16z^{3}a^{-3}+19z^{3}a^{-5}+8z^{3}a^{-7}+a^{2}z^{2}-7z^{2}a^{-2}-11z^{2}a^{-4}-6z^{2}a^{-6}-z^{2}-az-3za^{-1}-5za^{-3}-7za^{-5}-4za^{-7}+2a^{-2}+4a^{-4}+2a^{-6}+1$
The A2 invariant	$q^{6}-q^{4}+q^{2}-q^{-2}+q^{-4}-q^{-6}+2q^{-8}+q^{-10}+q^{-12}+2q^{-14}-q^{-16}-q^{-20}-q^{-22}$
The G2 invariant	Data:K11n26/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n26 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["K11n26"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{3}+5t^{2}-9t+11-9t^{-1}+5t^{-2}-t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 41, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-z^{6}a^{-2}-4z^{4}a^{-2}+2z^{4}a^{-4}+z^{4}-5z^{2}a^{-2}+6z^{2}a^{-4}-z^{2}a^{-6}+2z^{2}-2a^{-2}+4a^{-4}-2a^{-6}+1$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{9}a^{-3}+z^{9}a^{-5}+3z^{8}a^{-2}+5z^{8}a^{-4}+2z^{8}a^{-6}+3z^{7}a^{-1}+2z^{7}a^{-3}+z^{7}a^{-7}-9z^{6}a^{-2}-19z^{6}a^{-4}-9z^{6}a^{-6}+z^{6}-8z^{5}a^{-1}-16z^{5}a^{-3}-13z^{5}a^{-5}-5z^{5}a^{-7}+9z^{4}a^{-2}+20z^{4}a^{-4}+12z^{4}a^{-6}+z^{4}+3az^{3}+8z^{3}a^{-1}+16z^{3}a^{-3}+19z^{3}a^{-5}+8z^{3}a^{-7}+a^{2}z^{2}-7z^{2}a^{-2}-11z^{2}a^{-4}-6z^{2}a^{-6}-z^{2}-az-3za^{-1}-5za^{-3}-7za^{-5}-4za^{-7}+2a^{-2}+4a^{-4}+2a^{-6}+1$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {9_20, 10_149,}

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["K11n26"];

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]=

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

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_20, 10_149,}

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, 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 $8$ $32$ $32$ ${\frac {364}{3}}$ ${\frac {68}{3}}$ $256$ ${\frac {1568}{3}}$ ${\frac {224}{3}}$ $96$ ${\frac {256}{3}}$ $512$ ${\frac {2912}{3}}$ ${\frac {544}{3}}$ ${\frac {35911}{15}}$ $-{\frac {1084}{15}}$ ${\frac {49804}{45}}$ ${\frac {329}{9}}$ ${\frac {2071}{15}}$

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=$0 is the signature of K11n26. 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 15                   1 -1 13                 1   1 11               3 1   -2 9             3 1     2 7           3 3       0 5         4 3         1 3       2 3           1 1     3 4             -1 -1   1 3               2 -3   2                 -2 -5 1                   1

Integral Khovanov Homology (db, data source)

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

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.

K11n25 K11n27