K11a294

K11a293

K11a295

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a294 at Knotilus!

Knot presentations

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

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number $\{1,2\}$ 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a294/ThurstonBennequinNumber Hyperbolic Volume 16.0735 A-Polynomial See Data:K11a294/A-polynomial

[edit Notes for K11a294's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $3$ Rasmussen s-Invariant 2

[edit Notes for K11a294's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$2t^{3}-11t^{2}+29t-39+29t^{-1}-11t^{-2}+2t^{-3}$
Conway polynomial	$2z^{6}+z^{4}+3z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 123, -2 }
Jones polynomial	$q^{3}-4q^{2}+8q-13+18q^{-1}-19q^{-2}+20q^{-3}-17q^{-4}+12q^{-5}-7q^{-6}+3q^{-7}-q^{-8}$
HOMFLY-PT polynomial (db, data sources)	$-z^{4}a^{6}-2z^{2}a^{6}-2a^{6}+z^{6}a^{4}+3z^{4}a^{4}+6z^{2}a^{4}+3a^{4}+z^{6}a^{2}+z^{4}a^{2}-2z^{4}-2z^{2}+z^{2}a^{-2}$
Kauffman polynomial (db, data sources)	$2a^{4}z^{10}+2a^{2}z^{10}+6a^{5}z^{9}+11a^{3}z^{9}+5az^{9}+8a^{6}z^{8}+9a^{4}z^{8}+7a^{2}z^{8}+6z^{8}+6a^{7}z^{7}-9a^{5}z^{7}-25a^{3}z^{7}-6az^{7}+4z^{7}a^{-1}+3a^{8}z^{6}-20a^{6}z^{6}-32a^{4}z^{6}-23a^{2}z^{6}+z^{6}a^{-2}-13z^{6}+a^{9}z^{5}-13a^{7}z^{5}+2a^{5}z^{5}+24a^{3}z^{5}-2az^{5}-10z^{5}a^{-1}-5a^{8}z^{4}+24a^{6}z^{4}+40a^{4}z^{4}+19a^{2}z^{4}-2z^{4}a^{-2}+6z^{4}-2a^{9}z^{3}+11a^{7}z^{3}+11a^{5}z^{3}-12a^{3}z^{3}-4az^{3}+6z^{3}a^{-1}-10a^{6}z^{2}-18a^{4}z^{2}-10a^{2}z^{2}+z^{2}a^{-2}-z^{2}-4a^{7}z-4a^{5}z+2a^{3}z+3az+za^{-1}+2a^{6}+3a^{4}$
The A2 invariant	Data:K11a294/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a294/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a294 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["K11a294"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$2t^{3}-11t^{2}+29t-39+29t^{-1}-11t^{-2}+2t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 123, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{3}-4q^{2}+8q-13+18q^{-1}-19q^{-2}+20q^{-3}-17q^{-4}+12q^{-5}-7q^{-6}+3q^{-7}-q^{-8}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$2a^{4}z^{10}+2a^{2}z^{10}+6a^{5}z^{9}+11a^{3}z^{9}+5az^{9}+8a^{6}z^{8}+9a^{4}z^{8}+7a^{2}z^{8}+6z^{8}+6a^{7}z^{7}-9a^{5}z^{7}-25a^{3}z^{7}-6az^{7}+4z^{7}a^{-1}+3a^{8}z^{6}-20a^{6}z^{6}-32a^{4}z^{6}-23a^{2}z^{6}+z^{6}a^{-2}-13z^{6}+a^{9}z^{5}-13a^{7}z^{5}+2a^{5}z^{5}+24a^{3}z^{5}-2az^{5}-10z^{5}a^{-1}-5a^{8}z^{4}+24a^{6}z^{4}+40a^{4}z^{4}+19a^{2}z^{4}-2z^{4}a^{-2}+6z^{4}-2a^{9}z^{3}+11a^{7}z^{3}+11a^{5}z^{3}-12a^{3}z^{3}-4az^{3}+6z^{3}a^{-1}-10a^{6}z^{2}-18a^{4}z^{2}-10a^{2}z^{2}+z^{2}a^{-2}-z^{2}-4a^{7}z-4a^{5}z+2a^{3}z+3az+za^{-1}+2a^{6}+3a^{4}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a178,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {K11a146,}

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["K11a294"];

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]=

{ $2t^{3}-11t^{2}+29t-39+29t^{-1}-11t^{-2}+2t^{-3}$, $q^{3}-4q^{2}+8q-13+18q^{-1}-19q^{-2}+20q^{-3}-17q^{-4}+12q^{-5}-7q^{-6}+3q^{-7}-q^{-8}$ }

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]=

{K11a178,}

In[6]:=

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

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

Out[6]=

{K11a146,}

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 $12$ $-40$ $72$ $174$ $26$ $-480$ $-{\frac {2704}{3}}$ $-{\frac {64}{3}}$ $-296$ $288$ $800$ $2088$ $312$ ${\frac {43711}{10}}$ $-{\frac {11906}{15}}$ ${\frac {13074}{5}}$ ${\frac {299}{2}}$ ${\frac {3871}{10}}$

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 $s=$-2 is the signature of K11a294. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

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

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.

K11a293 K11a295