K11a282

K11a281

K11a283

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a282 at Knotilus!

Knot presentations

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

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Chiral Unknotting number 1 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a282/ThurstonBennequinNumber Hyperbolic Volume 16.6494 A-Polynomial See Data:K11a282/A-polynomial

[edit Notes for K11a282'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 K11a282's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{4}+6t^{3}-16t^{2}+26t-29+26t^{-1}-16t^{-2}+6t^{-3}-t^{-4}$
Conway polynomial	$-z^{8}-2z^{6}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 127, -2 }
Jones polynomial	$-q^{4}+4q^{3}-8q^{2}+13q-17+20q^{-1}-20q^{-2}+18q^{-3}-13q^{-4}+8q^{-5}-4q^{-6}+q^{-7}$
HOMFLY-PT polynomial (db, data sources)	$-a^{2}z^{8}+a^{4}z^{6}-5a^{2}z^{6}+2z^{6}+3a^{4}z^{4}-9a^{2}z^{4}-z^{4}a^{-2}+7z^{4}+2a^{4}z^{2}-6a^{2}z^{2}-2z^{2}a^{-2}+6z^{2}+1$
Kauffman polynomial (db, data sources)	$3a^{2}z^{10}+3z^{10}+9a^{3}z^{9}+15az^{9}+6z^{9}a^{-1}+12a^{4}z^{8}+10a^{2}z^{8}+4z^{8}a^{-2}+2z^{8}+11a^{5}z^{7}-14a^{3}z^{7}-46az^{7}-20z^{7}a^{-1}+z^{7}a^{-3}+8a^{6}z^{6}-20a^{4}z^{6}-47a^{2}z^{6}-14z^{6}a^{-2}-33z^{6}+4a^{7}z^{5}-13a^{5}z^{5}+a^{3}z^{5}+39az^{5}+18z^{5}a^{-1}-3z^{5}a^{-3}+a^{8}z^{4}-7a^{6}z^{4}+10a^{4}z^{4}+45a^{2}z^{4}+14z^{4}a^{-2}+41z^{4}-2a^{7}z^{3}+3a^{5}z^{3}+3a^{3}z^{3}-9az^{3}-5z^{3}a^{-1}+2z^{3}a^{-3}+a^{6}z^{2}-2a^{4}z^{2}-12a^{2}z^{2}-5z^{2}a^{-2}-14z^{2}+1$
The A2 invariant	$q^{20}-2q^{18}+2q^{16}-2q^{14}-q^{12}+3q^{10}-3q^{8}+5q^{6}-2q^{4}+q^{2}+1-3q^{-2}+3q^{-4}-q^{-6}+q^{-8}+q^{-10}-q^{-12}$
The G2 invariant	Data:K11a282/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a282 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["K11a282"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{4}+6t^{3}-16t^{2}+26t-29+26t^{-1}-16t^{-2}+6t^{-3}-t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 127, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{4}+4q^{3}-8q^{2}+13q-17+20q^{-1}-20q^{-2}+18q^{-3}-13q^{-4}+8q^{-5}-4q^{-6}+q^{-7}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$3a^{2}z^{10}+3z^{10}+9a^{3}z^{9}+15az^{9}+6z^{9}a^{-1}+12a^{4}z^{8}+10a^{2}z^{8}+4z^{8}a^{-2}+2z^{8}+11a^{5}z^{7}-14a^{3}z^{7}-46az^{7}-20z^{7}a^{-1}+z^{7}a^{-3}+8a^{6}z^{6}-20a^{4}z^{6}-47a^{2}z^{6}-14z^{6}a^{-2}-33z^{6}+4a^{7}z^{5}-13a^{5}z^{5}+a^{3}z^{5}+39az^{5}+18z^{5}a^{-1}-3z^{5}a^{-3}+a^{8}z^{4}-7a^{6}z^{4}+10a^{4}z^{4}+45a^{2}z^{4}+14z^{4}a^{-2}+41z^{4}-2a^{7}z^{3}+3a^{5}z^{3}+3a^{3}z^{3}-9az^{3}-5z^{3}a^{-1}+2z^{3}a^{-3}+a^{6}z^{2}-2a^{4}z^{2}-12a^{2}z^{2}-5z^{2}a^{-2}-14z^{2}+1$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a81,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {K11a81,}

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["K11a282"];

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]=

{ $-t^{4}+6t^{3}-16t^{2}+26t-29+26t^{-1}-16t^{-2}+6t^{-3}-t^{-4}$, $-q^{4}+4q^{3}-8q^{2}+13q-17+20q^{-1}-20q^{-2}+18q^{-3}-13q^{-4}+8q^{-5}-4q^{-6}+q^{-7}$ }

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]=

{K11a81,}

In[6]:=

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

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

Out[6]=

{K11a81,}

Vassiliev invariants

V2 and V3:

(0, 0)

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 $0$ $0$ $0$ $0$ $0$ $0$ $0$ $-64$ $64$ $0$ $0$ $0$ $0$ $0$ $160$ $-192$ $32$ $-32$

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 K11a282. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 9                       1 -1 7                     3   3 5                   5 1   -4 3                 8 3     5 1               9 5       -4 -1             11 8         3 -3           10 10           0 -5         8 10             -2 -7       5 10               5 -9     3 8                 -5 -11   1 5                   4 -13   3                     -3 -15 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-3$ $i=-1$ $r=-6$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-3$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-2$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=-1$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=0$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{11}$ $r=1$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=2$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=3$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=4$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=5$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

K11a281 K11a283