K11a234

K11a233

K11a235

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a234 at Knotilus!

Knot presentations

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

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number 4 3-genus 4 Bridge index 2 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a234/ThurstonBennequinNumber Hyperbolic Volume 7.44001 A-Polynomial See Data:K11a234/A-polynomial

[edit Notes for K11a234's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $4$ Rasmussen s-Invariant -8

[edit Notes for K11a234's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$2t^{4}-4t^{3}+5t^{2}-5t+5-5t^{-1}+5t^{-2}-4t^{-3}+2t^{-4}$
Conway polynomial	$2z^{8}+12z^{6}+21z^{4}+11z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 37, 8 }
Jones polynomial	$-q^{15}+2q^{14}-3q^{13}+4q^{12}-5q^{11}+5q^{10}-5q^{9}+4q^{8}-3q^{7}+3q^{6}-q^{5}+q^{4}$
HOMFLY-PT polynomial (db, data sources)	$z^{8}a^{-8}+z^{8}a^{-10}+7z^{6}a^{-8}+6z^{6}a^{-10}-z^{6}a^{-12}+16z^{4}a^{-8}+10z^{4}a^{-10}-5z^{4}a^{-12}+14z^{2}a^{-8}+3z^{2}a^{-10}-6z^{2}a^{-12}+4a^{-8}-2a^{-10}-a^{-12}$
Kauffman polynomial (db, data sources)	$z^{10}a^{-10}+z^{10}a^{-12}+z^{9}a^{-9}+3z^{9}a^{-11}+2z^{9}a^{-13}+z^{8}a^{-8}-5z^{8}a^{-10}-4z^{8}a^{-12}+2z^{8}a^{-14}-5z^{7}a^{-9}-16z^{7}a^{-11}-9z^{7}a^{-13}+2z^{7}a^{-15}-7z^{6}a^{-8}+6z^{6}a^{-10}+5z^{6}a^{-12}-6z^{6}a^{-14}+2z^{6}a^{-16}+5z^{5}a^{-9}+25z^{5}a^{-11}+14z^{5}a^{-13}-4z^{5}a^{-15}+2z^{5}a^{-17}+16z^{4}a^{-8}-7z^{4}a^{-12}+5z^{4}a^{-14}-2z^{4}a^{-16}+2z^{4}a^{-18}+3z^{3}a^{-9}-13z^{3}a^{-11}-11z^{3}a^{-13}+3z^{3}a^{-15}-z^{3}a^{-17}+z^{3}a^{-19}-14z^{2}a^{-8}-3z^{2}a^{-10}+6z^{2}a^{-12}-2z^{2}a^{-14}+z^{2}a^{-16}-2z^{2}a^{-18}-3za^{-9}+za^{-11}+2za^{-13}-za^{-15}-za^{-19}+4a^{-8}+2a^{-10}-a^{-12}$
The A2 invariant	$q^{-14}+2q^{-18}+q^{-20}+2q^{-22}+q^{-24}-2q^{-30}-q^{-34}-q^{-44}$
The G2 invariant	Data:K11a234/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a234 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["K11a234"];

In[4]:=

Alexander[K][t]

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

Out[4]=

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

In[5]:=

Conway[K][z]

Out[5]=

$2z^{8}+12z^{6}+21z^{4}+11z^{2}+1$

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]=

{ 37, 8 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{15}+2q^{14}-3q^{13}+4q^{12}-5q^{11}+5q^{10}-5q^{9}+4q^{8}-3q^{7}+3q^{6}-q^{5}+q^{4}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{8}a^{-8}+z^{8}a^{-10}+7z^{6}a^{-8}+6z^{6}a^{-10}-z^{6}a^{-12}+16z^{4}a^{-8}+10z^{4}a^{-10}-5z^{4}a^{-12}+14z^{2}a^{-8}+3z^{2}a^{-10}-6z^{2}a^{-12}+4a^{-8}-2a^{-10}-a^{-12}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{10}a^{-10}+z^{10}a^{-12}+z^{9}a^{-9}+3z^{9}a^{-11}+2z^{9}a^{-13}+z^{8}a^{-8}-5z^{8}a^{-10}-4z^{8}a^{-12}+2z^{8}a^{-14}-5z^{7}a^{-9}-16z^{7}a^{-11}-9z^{7}a^{-13}+2z^{7}a^{-15}-7z^{6}a^{-8}+6z^{6}a^{-10}+5z^{6}a^{-12}-6z^{6}a^{-14}+2z^{6}a^{-16}+5z^{5}a^{-9}+25z^{5}a^{-11}+14z^{5}a^{-13}-4z^{5}a^{-15}+2z^{5}a^{-17}+16z^{4}a^{-8}-7z^{4}a^{-12}+5z^{4}a^{-14}-2z^{4}a^{-16}+2z^{4}a^{-18}+3z^{3}a^{-9}-13z^{3}a^{-11}-11z^{3}a^{-13}+3z^{3}a^{-15}-z^{3}a^{-17}+z^{3}a^{-19}-14z^{2}a^{-8}-3z^{2}a^{-10}+6z^{2}a^{-12}-2z^{2}a^{-14}+z^{2}a^{-16}-2z^{2}a^{-18}-3za^{-9}+za^{-11}+2za^{-13}-za^{-15}-za^{-19}+4a^{-8}+2a^{-10}-a^{-12}$

"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["K11a234"];

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]=

{ $2t^{4}-4t^{3}+5t^{2}-5t+5-5t^{-1}+5t^{-2}-4t^{-3}+2t^{-4}$, $-q^{15}+2q^{14}-3q^{13}+4q^{12}-5q^{11}+5q^{10}-5q^{9}+4q^{8}-3q^{7}+3q^{6}-q^{5}+q^{4}$ }

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:

(11, 35)

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 $44$ $280$ $968$ ${\frac {6874}{3}}$ ${\frac {926}{3}}$ $12320$ ${\frac {62704}{3}}$ ${\frac {10816}{3}}$ $2328$ ${\frac {42592}{3}}$ $39200$ ${\frac {302456}{3}}$ ${\frac {40744}{3}}$ ${\frac {5880341}{30}}$ ${\frac {55866}{5}}$ ${\frac {2973682}{45}}$ ${\frac {18859}{18}}$ ${\frac {240821}{30}}$

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=$8 is the signature of K11a234. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ 0 1 2 3 4 5 6 7 8 9 10 11 χ 31                       1 -1 29                     1   1 27                   2 1   -1 25                 2 1     1 23               3 2       -1 21             2 2         0 19           3 3           0 17         1 2             -1 15       2 3               1 13     1 1                 0 11     2                   2 9 1 1                     0 7 1                       1

Integral Khovanov Homology (db, data source)

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

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.

K11a233 K11a235