K11a2

K11a1

K11a3

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a2 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X8394 X10,6,11,5 X14,8,15,7 X2,9,3,10 X18,12,19,11 X6,14,7,13 X20,16,21,15 X12,18,13,17 X22,20,1,19 X16,22,17,21
Gauss code	1, -5, 2, -1, 3, -7, 4, -2, 5, -3, 6, -9, 7, -4, 8, -11, 9, -6, 10, -8, 11, -10
Dowker-Thistlethwaite code	4 8 10 14 2 18 6 20 12 22 16

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Chiral Unknotting number 2 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a2/ThurstonBennequinNumber Hyperbolic Volume 16.5881 A-Polynomial See Data:K11a2/A-polynomial

[edit Notes for K11a2's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a2's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-3t^{3}+15t^{2}-31t+39-31t^{-1}+15t^{-2}-3t^{-3}$
Conway polynomial	$-3z^{6}-3z^{4}+2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 137, 4 }
Jones polynomial	$-q^{11}+4q^{10}-9q^{9}+15q^{8}-20q^{7}+22q^{6}-22q^{5}+19q^{4}-13q^{3}+8q^{2}-3q+1$
HOMFLY-PT polynomial (db, data sources)	$-z^{6}a^{-4}-2z^{6}a^{-6}+z^{4}a^{-2}-z^{4}a^{-4}-6z^{4}a^{-6}+3z^{4}a^{-8}+2z^{2}a^{-2}+2z^{2}a^{-4}-7z^{2}a^{-6}+6z^{2}a^{-8}-z^{2}a^{-10}+a^{-2}+2a^{-4}-4a^{-6}+3a^{-8}-a^{-10}$
Kauffman polynomial (db, data sources)	$z^{10}a^{-6}+z^{10}a^{-8}+4z^{9}a^{-5}+9z^{9}a^{-7}+5z^{9}a^{-9}+5z^{8}a^{-4}+16z^{8}a^{-6}+20z^{8}a^{-8}+9z^{8}a^{-10}+3z^{7}a^{-3}+z^{7}a^{-5}+10z^{7}a^{-9}+8z^{7}a^{-11}+z^{6}a^{-2}-10z^{6}a^{-4}-42z^{6}a^{-6}-45z^{6}a^{-8}-10z^{6}a^{-10}+4z^{6}a^{-12}-7z^{5}a^{-3}-17z^{5}a^{-5}-33z^{5}a^{-7}-36z^{5}a^{-9}-12z^{5}a^{-11}+z^{5}a^{-13}-3z^{4}a^{-2}+7z^{4}a^{-4}+38z^{4}a^{-6}+35z^{4}a^{-8}+2z^{4}a^{-10}-5z^{4}a^{-12}+5z^{3}a^{-3}+17z^{3}a^{-5}+33z^{3}a^{-7}+30z^{3}a^{-9}+8z^{3}a^{-11}-z^{3}a^{-13}+3z^{2}a^{-2}-4z^{2}a^{-4}-18z^{2}a^{-6}-14z^{2}a^{-8}-z^{2}a^{-10}+2z^{2}a^{-12}-za^{-3}-6za^{-5}-10za^{-7}-8za^{-9}-3za^{-11}-a^{-2}+2a^{-4}+4a^{-6}+3a^{-8}+a^{-10}$
The A2 invariant	Data:K11a2/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a2/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a2 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["K11a2"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-3t^{3}+15t^{2}-31t+39-31t^{-1}+15t^{-2}-3t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 137, 4 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{11}+4q^{10}-9q^{9}+15q^{8}-20q^{7}+22q^{6}-22q^{5}+19q^{4}-13q^{3}+8q^{2}-3q+1$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{10}a^{-6}+z^{10}a^{-8}+4z^{9}a^{-5}+9z^{9}a^{-7}+5z^{9}a^{-9}+5z^{8}a^{-4}+16z^{8}a^{-6}+20z^{8}a^{-8}+9z^{8}a^{-10}+3z^{7}a^{-3}+z^{7}a^{-5}+10z^{7}a^{-9}+8z^{7}a^{-11}+z^{6}a^{-2}-10z^{6}a^{-4}-42z^{6}a^{-6}-45z^{6}a^{-8}-10z^{6}a^{-10}+4z^{6}a^{-12}-7z^{5}a^{-3}-17z^{5}a^{-5}-33z^{5}a^{-7}-36z^{5}a^{-9}-12z^{5}a^{-11}+z^{5}a^{-13}-3z^{4}a^{-2}+7z^{4}a^{-4}+38z^{4}a^{-6}+35z^{4}a^{-8}+2z^{4}a^{-10}-5z^{4}a^{-12}+5z^{3}a^{-3}+17z^{3}a^{-5}+33z^{3}a^{-7}+30z^{3}a^{-9}+8z^{3}a^{-11}-z^{3}a^{-13}+3z^{2}a^{-2}-4z^{2}a^{-4}-18z^{2}a^{-6}-14z^{2}a^{-8}-z^{2}a^{-10}+2z^{2}a^{-12}-za^{-3}-6za^{-5}-10za^{-7}-8za^{-9}-3za^{-11}-a^{-2}+2a^{-4}+4a^{-6}+3a^{-8}+a^{-10}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a116,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {K11a116,}

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["K11a2"];

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]=

{ $-3t^{3}+15t^{2}-31t+39-31t^{-1}+15t^{-2}-3t^{-3}$, $-q^{11}+4q^{10}-9q^{9}+15q^{8}-20q^{7}+22q^{6}-22q^{5}+19q^{4}-13q^{3}+8q^{2}-3q+1$ }

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]=

{K11a116,}

In[6]:=

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

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

Out[6]=

{K11a116,}

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 $8$ $32$ $32$ ${\frac {508}{3}}$ ${\frac {116}{3}}$ $256$ ${\frac {2432}{3}}$ ${\frac {512}{3}}$ $128$ ${\frac {256}{3}}$ $512$ ${\frac {4064}{3}}$ ${\frac {928}{3}}$ ${\frac {56431}{15}}$ ${\frac {2116}{15}}$ ${\frac {70444}{45}}$ ${\frac {353}{9}}$ ${\frac {2911}{15}}$

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=$4 is the signature of K11a2. 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 8 9 χ 23                       1 -1 21                     3   3 19                   6 1   -5 17                 9 3     6 15               11 6       -5 13             11 9         2 11           11 11           0 9         8 11             -3 7       5 11               6 5     3 8                 -5 3   1 6                   5 1   2                     -2 -1 1                       1

Integral Khovanov Homology (db, data source)

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

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.

K11a1 K11a3