K11a269

K11a268

K11a270

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a269 at Knotilus!

Knot presentations

Planar diagram presentation	X6271 X10,3,11,4 X12,6,13,5 X20,8,21,7 X16,10,17,9 X18,11,19,12 X22,13,1,14 X8,16,9,15 X4,18,5,17 X2,19,3,20 X14,21,15,22
Gauss code	1, -10, 2, -9, 3, -1, 4, -8, 5, -2, 6, -3, 7, -11, 8, -5, 9, -6, 10, -4, 11, -7
Dowker-Thistlethwaite code	6 10 12 20 16 18 22 8 4 2 14

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Chiral Unknotting number $\{1,2\}$ 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a269/ThurstonBennequinNumber Hyperbolic Volume 18.0173 A-Polynomial See Data:K11a269/A-polynomial

[edit Notes for K11a269's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a269's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{4}+6t^{3}-18t^{2}+32t-37+32t^{-1}-18t^{-2}+6t^{-3}-t^{-4}$
Conway polynomial	$-z^{8}-2z^{6}-2z^{4}-2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 151, 2 }
Jones polynomial	$q^{7}-4q^{6}+9q^{5}-15q^{4}+21q^{3}-24q^{2}+24q-21+16q^{-1}-10q^{-2}+5q^{-3}-q^{-4}$
HOMFLY-PT polynomial (db, data sources)	$-z^{8}a^{-2}-5z^{6}a^{-2}+z^{6}a^{-4}+2z^{6}-a^{2}z^{4}-10z^{4}a^{-2}+3z^{4}a^{-4}+6z^{4}-a^{2}z^{2}-8z^{2}a^{-2}+3z^{2}a^{-4}+4z^{2}+a^{2}-a^{-2}+a^{-4}$
Kauffman polynomial (db, data sources)	$4z^{10}a^{-2}+4z^{10}+8az^{9}+20z^{9}a^{-1}+12z^{9}a^{-3}+5a^{2}z^{8}+15z^{8}a^{-2}+16z^{8}a^{-4}+4z^{8}+a^{3}z^{7}-24az^{7}-55z^{7}a^{-1}-16z^{7}a^{-3}+14z^{7}a^{-5}-15a^{2}z^{6}-63z^{6}a^{-2}-27z^{6}a^{-4}+9z^{6}a^{-6}-42z^{6}-2a^{3}z^{5}+18az^{5}+37z^{5}a^{-1}-5z^{5}a^{-3}-18z^{5}a^{-5}+4z^{5}a^{-7}+12a^{2}z^{4}+55z^{4}a^{-2}+15z^{4}a^{-4}-7z^{4}a^{-6}+z^{4}a^{-8}+44z^{4}+a^{3}z^{3}-az^{3}-4z^{3}a^{-1}+7z^{3}a^{-3}+8z^{3}a^{-5}-z^{3}a^{-7}-a^{2}z^{2}-16z^{2}a^{-2}-5z^{2}a^{-4}+2z^{2}a^{-6}-10z^{2}-az-za^{-1}-za^{-3}-za^{-5}-a^{2}+a^{-2}+a^{-4}$
The A2 invariant	$-q^{12}+2q^{10}+q^{8}-q^{6}+4q^{4}-4q^{2}+1-3q^{-4}+5q^{-6}-4q^{-8}+4q^{-10}-q^{-12}-2q^{-14}+3q^{-16}-2q^{-18}+q^{-20}$
The G2 invariant	Data:K11a269/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a269 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["K11a269"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{4}+6t^{3}-18t^{2}+32t-37+32t^{-1}-18t^{-2}+6t^{-3}-t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

$-z^{8}-2z^{6}-2z^{4}-2z^{2}+1$

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]=

{ 151, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{7}-4q^{6}+9q^{5}-15q^{4}+21q^{3}-24q^{2}+24q-21+16q^{-1}-10q^{-2}+5q^{-3}-q^{-4}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-z^{8}a^{-2}-5z^{6}a^{-2}+z^{6}a^{-4}+2z^{6}-a^{2}z^{4}-10z^{4}a^{-2}+3z^{4}a^{-4}+6z^{4}-a^{2}z^{2}-8z^{2}a^{-2}+3z^{2}a^{-4}+4z^{2}+a^{2}-a^{-2}+a^{-4}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$4z^{10}a^{-2}+4z^{10}+8az^{9}+20z^{9}a^{-1}+12z^{9}a^{-3}+5a^{2}z^{8}+15z^{8}a^{-2}+16z^{8}a^{-4}+4z^{8}+a^{3}z^{7}-24az^{7}-55z^{7}a^{-1}-16z^{7}a^{-3}+14z^{7}a^{-5}-15a^{2}z^{6}-63z^{6}a^{-2}-27z^{6}a^{-4}+9z^{6}a^{-6}-42z^{6}-2a^{3}z^{5}+18az^{5}+37z^{5}a^{-1}-5z^{5}a^{-3}-18z^{5}a^{-5}+4z^{5}a^{-7}+12a^{2}z^{4}+55z^{4}a^{-2}+15z^{4}a^{-4}-7z^{4}a^{-6}+z^{4}a^{-8}+44z^{4}+a^{3}z^{3}-az^{3}-4z^{3}a^{-1}+7z^{3}a^{-3}+8z^{3}a^{-5}-z^{3}a^{-7}-a^{2}z^{2}-16z^{2}a^{-2}-5z^{2}a^{-4}+2z^{2}a^{-6}-10z^{2}-az-za^{-1}-za^{-3}-za^{-5}-a^{2}+a^{-2}+a^{-4}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

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["K11a269"];

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]=

{ $-t^{4}+6t^{3}-18t^{2}+32t-37+32t^{-1}-18t^{-2}+6t^{-3}-t^{-4}$, $q^{7}-4q^{6}+9q^{5}-15q^{4}+21q^{3}-24q^{2}+24q-21+16q^{-1}-10q^{-2}+5q^{-3}-q^{-4}$ }

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]=

{}

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, -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 $-8$ $-8$ $32$ ${\frac {164}{3}}$ ${\frac {100}{3}}$ $64$ ${\frac {400}{3}}$ ${\frac {256}{3}}$ $-8$ $-{\frac {256}{3}}$ $32$ $-{\frac {1312}{3}}$ $-{\frac {800}{3}}$ $-{\frac {4111}{15}}$ ${\frac {1548}{5}}$ $-{\frac {29764}{45}}$ ${\frac {1231}{9}}$ $-{\frac {2431}{15}}$

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=$2 is the signature of K11a269. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 6 χ 15                       1 1 13                     3   -3 11                   6 1   5 9                 9 3     -6 7               12 6       6 5             12 9         -3 3           12 12           0 1         10 13             3 -1       6 11               -5 -3     4 10                 6 -5   1 6                   -5 -7   4                     4 -9 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=1$ $i=3$ $r=-5$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-2$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=-1$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=0$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{12}$ $r=1$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=2$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=3$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=5$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=6$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

K11a268 K11a270