K11a214

K11a213

K11a215

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a214 at Knotilus!

Knot presentations

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

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number $\{1,2\}$ 3-genus 2 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a214/ThurstonBennequinNumber Hyperbolic Volume 11.7947 A-Polynomial See Data:K11a214/A-polynomial

[edit Notes for K11a214's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $2$ Rasmussen s-Invariant 0

[edit Notes for K11a214's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$3t^{2}-17t+29-17t^{-1}+3t^{-2}$
Conway polynomial	$3z^{4}-5z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 69, 0 }
Jones polynomial	$-q^{7}+3q^{6}-4q^{5}+7q^{4}-9q^{3}+10q^{2}-11q+9-7q^{-1}+5q^{-2}-2q^{-3}+q^{-4}$
HOMFLY-PT polynomial (db, data sources)	$a^{4}-2z^{2}a^{2}+z^{4}-z^{2}+z^{4}a^{-2}-2z^{2}a^{-2}-2a^{-2}+z^{4}a^{-4}+z^{2}a^{-4}+2a^{-4}-z^{2}a^{-6}$
Kauffman polynomial (db, data sources)	$z^{10}a^{-2}+z^{10}a^{-4}+3z^{9}a^{-1}+6z^{9}a^{-3}+3z^{9}a^{-5}+3z^{8}a^{-2}+3z^{8}a^{-4}+3z^{8}a^{-6}+3z^{8}+3az^{7}-9z^{7}a^{-1}-25z^{7}a^{-3}-12z^{7}a^{-5}+z^{7}a^{-7}+3a^{2}z^{6}-21z^{6}a^{-2}-28z^{6}a^{-4}-14z^{6}a^{-6}-4z^{6}+2a^{3}z^{5}-2az^{5}+17z^{5}a^{-1}+37z^{5}a^{-3}+12z^{5}a^{-5}-4z^{5}a^{-7}+a^{4}z^{4}-3a^{2}z^{4}+34z^{4}a^{-2}+45z^{4}a^{-4}+18z^{4}a^{-6}+3z^{4}-2a^{3}z^{3}-19z^{3}a^{-1}-29z^{3}a^{-3}-5z^{3}a^{-5}+3z^{3}a^{-7}-2a^{4}z^{2}+a^{2}z^{2}-21z^{2}a^{-2}-24z^{2}a^{-4}-7z^{2}a^{-6}-z^{2}+8za^{-1}+11za^{-3}+3za^{-5}+a^{4}+2a^{-2}+2a^{-4}$
The A2 invariant	Data:K11a214/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a214/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a214 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["K11a214"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$3t^{2}-17t+29-17t^{-1}+3t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

$3z^{4}-5z^{2}+1$

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]=

{ 69, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{7}+3q^{6}-4q^{5}+7q^{4}-9q^{3}+10q^{2}-11q+9-7q^{-1}+5q^{-2}-2q^{-3}+q^{-4}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{10}a^{-2}+z^{10}a^{-4}+3z^{9}a^{-1}+6z^{9}a^{-3}+3z^{9}a^{-5}+3z^{8}a^{-2}+3z^{8}a^{-4}+3z^{8}a^{-6}+3z^{8}+3az^{7}-9z^{7}a^{-1}-25z^{7}a^{-3}-12z^{7}a^{-5}+z^{7}a^{-7}+3a^{2}z^{6}-21z^{6}a^{-2}-28z^{6}a^{-4}-14z^{6}a^{-6}-4z^{6}+2a^{3}z^{5}-2az^{5}+17z^{5}a^{-1}+37z^{5}a^{-3}+12z^{5}a^{-5}-4z^{5}a^{-7}+a^{4}z^{4}-3a^{2}z^{4}+34z^{4}a^{-2}+45z^{4}a^{-4}+18z^{4}a^{-6}+3z^{4}-2a^{3}z^{3}-19z^{3}a^{-1}-29z^{3}a^{-3}-5z^{3}a^{-5}+3z^{3}a^{-7}-2a^{4}z^{2}+a^{2}z^{2}-21z^{2}a^{-2}-24z^{2}a^{-4}-7z^{2}a^{-6}-z^{2}+8za^{-1}+11za^{-3}+3za^{-5}+a^{4}+2a^{-2}+2a^{-4}$

"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["K11a214"];

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]=

{ $3t^{2}-17t+29-17t^{-1}+3t^{-2}$, $-q^{7}+3q^{6}-4q^{5}+7q^{4}-9q^{3}+10q^{2}-11q+9-7q^{-1}+5q^{-2}-2q^{-3}+q^{-4}$ }

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:

(-5, -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 $-20$ $-8$ $200$ ${\frac {746}{3}}$ ${\frac {238}{3}}$ $160$ ${\frac {592}{3}}$ ${\frac {64}{3}}$ $24$ $-{\frac {4000}{3}}$ $32$ $-{\frac {14920}{3}}$ $-{\frac {4760}{3}}$ $-{\frac {27055}{6}}$ $258$ $-{\frac {27254}{9}}$ ${\frac {5387}{18}}$ $-{\frac {3631}{6}}$

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 K11a214. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

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

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.

K11a213 K11a215