K11a215

K11a214

K11a216

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a215 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X12,4,13,3 X18,6,19,5 X14,7,15,8 X16,10,17,9 X2,12,3,11 X22,13,1,14 X20,16,21,15 X10,18,11,17 X6,20,7,19 X8,21,9,22
Gauss code	1, -6, 2, -1, 3, -10, 4, -11, 5, -9, 6, -2, 7, -4, 8, -5, 9, -3, 10, -8, 11, -7
Dowker-Thistlethwaite code	4 12 18 14 16 2 22 20 10 6 8

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a215/ThurstonBennequinNumber Hyperbolic Volume 16.7322 A-Polynomial See Data:K11a215/A-polynomial

[edit Notes for K11a215's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a215's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{4}-6t^{3}+17t^{2}-27t+31-27t^{-1}+17t^{-2}-6t^{-3}+t^{-4}$
Conway polynomial	$z^{8}+2z^{6}+z^{4}+3z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 133, 4 }
Jones polynomial	$q^{10}-4q^{9}+8q^{8}-14q^{7}+19q^{6}-21q^{5}+21q^{4}-18q^{3}+14q^{2}-8q+4-q^{-1}$
HOMFLY-PT polynomial (db, data sources)	$z^{8}a^{-4}-z^{6}a^{-2}+5z^{6}a^{-4}-2z^{6}a^{-6}-3z^{4}a^{-2}+10z^{4}a^{-4}-7z^{4}a^{-6}+z^{4}a^{-8}-2z^{2}a^{-2}+10z^{2}a^{-4}-7z^{2}a^{-6}+2z^{2}a^{-8}+3a^{-4}-2a^{-6}$
Kauffman polynomial (db, data sources)	$2z^{10}a^{-4}+2z^{10}a^{-6}+5z^{9}a^{-3}+13z^{9}a^{-5}+8z^{9}a^{-7}+4z^{8}a^{-2}+10z^{8}a^{-4}+19z^{8}a^{-6}+13z^{8}a^{-8}+z^{7}a^{-1}-12z^{7}a^{-3}-27z^{7}a^{-5}-2z^{7}a^{-7}+12z^{7}a^{-9}-14z^{6}a^{-2}-49z^{6}a^{-4}-64z^{6}a^{-6}-21z^{6}a^{-8}+8z^{6}a^{-10}-3z^{5}a^{-1}+z^{5}a^{-3}-3z^{5}a^{-5}-26z^{5}a^{-7}-15z^{5}a^{-9}+4z^{5}a^{-11}+16z^{4}a^{-2}+54z^{4}a^{-4}+56z^{4}a^{-6}+11z^{4}a^{-8}-6z^{4}a^{-10}+z^{4}a^{-12}+3z^{3}a^{-1}+10z^{3}a^{-3}+22z^{3}a^{-5}+23z^{3}a^{-7}+6z^{3}a^{-9}-2z^{3}a^{-11}-6z^{2}a^{-2}-21z^{2}a^{-4}-18z^{2}a^{-6}-2z^{2}a^{-8}+z^{2}a^{-10}-za^{-1}-4za^{-3}-8za^{-5}-5za^{-7}+3a^{-4}+2a^{-6}$
The A2 invariant	$-q^{2}+2-2q^{-2}+2q^{-4}+2q^{-6}-2q^{-8}+5q^{-10}-4q^{-12}+3q^{-14}-q^{-18}+3q^{-20}-4q^{-22}+q^{-24}-q^{-26}-q^{-28}+q^{-30}$
The G2 invariant	$q^{12}-3q^{10}+9q^{8}-19q^{6}+28q^{4}-33q^{2}+17+26q^{-2}-91q^{-4}+165q^{-6}-205q^{-8}+166q^{-10}-35q^{-12}-176q^{-14}+395q^{-16}-518q^{-18}+476q^{-20}-232q^{-22}-146q^{-24}+522q^{-26}-735q^{-28}+685q^{-30}-371q^{-32}-82q^{-34}+483q^{-36}-670q^{-38}+562q^{-40}-208q^{-42}-215q^{-44}+535q^{-46}-584q^{-48}+345q^{-50}+76q^{-52}-513q^{-54}+774q^{-56}-745q^{-58}+424q^{-60}+89q^{-62}-602q^{-64}+937q^{-66}-952q^{-68}+640q^{-70}-115q^{-72}-433q^{-74}+789q^{-76}-837q^{-78}+567q^{-80}-103q^{-82}-338q^{-84}+580q^{-86}-525q^{-88}+214q^{-90}+182q^{-92}-488q^{-94}+556q^{-96}-372q^{-98}+23q^{-100}+335q^{-102}-564q^{-104}+587q^{-106}-407q^{-108}+116q^{-110}+170q^{-112}-372q^{-114}+426q^{-116}-361q^{-118}+225q^{-120}-58q^{-122}-76q^{-124}+164q^{-126}-192q^{-128}+168q^{-130}-118q^{-132}+59q^{-134}-5q^{-136}-35q^{-138}+55q^{-140}-60q^{-142}+48q^{-144}-29q^{-146}+14q^{-148}+q^{-150}-8q^{-152}+10q^{-154}-10q^{-156}+6q^{-158}-3q^{-160}+q^{-162}$

Further Quantum Invariants

K11a215 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["K11a215"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{4}-6t^{3}+17t^{2}-27t+31-27t^{-1}+17t^{-2}-6t^{-3}+t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

$z^{8}+2z^{6}+z^{4}+3z^{2}+1$

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]=

{ 133, 4 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{10}-4q^{9}+8q^{8}-14q^{7}+19q^{6}-21q^{5}+21q^{4}-18q^{3}+14q^{2}-8q+4-q^{-1}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$2z^{10}a^{-4}+2z^{10}a^{-6}+5z^{9}a^{-3}+13z^{9}a^{-5}+8z^{9}a^{-7}+4z^{8}a^{-2}+10z^{8}a^{-4}+19z^{8}a^{-6}+13z^{8}a^{-8}+z^{7}a^{-1}-12z^{7}a^{-3}-27z^{7}a^{-5}-2z^{7}a^{-7}+12z^{7}a^{-9}-14z^{6}a^{-2}-49z^{6}a^{-4}-64z^{6}a^{-6}-21z^{6}a^{-8}+8z^{6}a^{-10}-3z^{5}a^{-1}+z^{5}a^{-3}-3z^{5}a^{-5}-26z^{5}a^{-7}-15z^{5}a^{-9}+4z^{5}a^{-11}+16z^{4}a^{-2}+54z^{4}a^{-4}+56z^{4}a^{-6}+11z^{4}a^{-8}-6z^{4}a^{-10}+z^{4}a^{-12}+3z^{3}a^{-1}+10z^{3}a^{-3}+22z^{3}a^{-5}+23z^{3}a^{-7}+6z^{3}a^{-9}-2z^{3}a^{-11}-6z^{2}a^{-2}-21z^{2}a^{-4}-18z^{2}a^{-6}-2z^{2}a^{-8}+z^{2}a^{-10}-za^{-1}-4za^{-3}-8za^{-5}-5za^{-7}+3a^{-4}+2a^{-6}$

"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["K11a215"];

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]=

{ $t^{4}-6t^{3}+17t^{2}-27t+31-27t^{-1}+17t^{-2}-6t^{-3}+t^{-4}$, $q^{10}-4q^{9}+8q^{8}-14q^{7}+19q^{6}-21q^{5}+21q^{4}-18q^{3}+14q^{2}-8q+4-q^{-1}$ }

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]=

{}

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, 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 $12$ $40$ $72$ $174$ $26$ $480$ ${\frac {2608}{3}}$ ${\frac {544}{3}}$ $104$ $288$ $800$ $2088$ $312$ ${\frac {42751}{10}}$ ${\frac {2734}{15}}$ ${\frac {8434}{5}}$ ${\frac {11}{2}}$ ${\frac {2431}{10}}$

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=$4 is the signature of K11a215. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -3 -2 -1 0 1 2 3 4 5 6 7 8 χ 21                       1 1 19                     3   -3 17                   5 1   4 15                 9 3     -6 13               10 5       5 11             11 9         -2 9           10 10           0 7         8 11             3 5       6 10               -4 3     3 9                 6 1   1 5                   -4 -1   3                     3 -3 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=3$ $i=5$ $r=-3$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-1$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=0$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{6}$ $r=1$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=2$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=3$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=4$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=5$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=6$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=7$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=8$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

K11a214 K11a216