K11a321

K11a320

K11a322

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a321 at Knotilus!

Knot presentations

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

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a321/ThurstonBennequinNumber Hyperbolic Volume 16.2583 A-Polynomial See Data:K11a321/A-polynomial

[edit Notes for K11a321'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 K11a321's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-3t^{3}+15t^{2}-27t+31-27t^{-1}+15t^{-2}-3t^{-3}$
Conway polynomial	$-3z^{6}-3z^{4}+6z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{11,t+1\}$
Determinant and Signature	{ 121, -4 }
Jones polynomial	$1-3q^{-1}+7q^{-2}-12q^{-3}+17q^{-4}-19q^{-5}+20q^{-6}-17q^{-7}+13q^{-8}-8q^{-9}+3q^{-10}-q^{-11}$
HOMFLY-PT polynomial (db, data sources)	$-z^{2}a^{10}-2a^{10}+3z^{4}a^{8}+7z^{2}a^{8}+3a^{8}-2z^{6}a^{6}-6z^{4}a^{6}-5z^{2}a^{6}-2a^{6}-z^{6}a^{4}-z^{4}a^{4}+3z^{2}a^{4}+2a^{4}+z^{4}a^{2}+2z^{2}a^{2}$
Kauffman polynomial (db, data sources)	$z^{5}a^{13}-2z^{3}a^{13}+za^{13}+3z^{6}a^{12}-4z^{4}a^{12}+z^{2}a^{12}+6z^{7}a^{11}-10z^{5}a^{11}+8z^{3}a^{11}-4za^{11}+7z^{8}a^{10}-10z^{6}a^{10}+7z^{4}a^{10}-5z^{2}a^{10}+2a^{10}+5z^{9}a^{9}-z^{7}a^{9}-10z^{5}a^{9}+12z^{3}a^{9}-4za^{9}+2z^{10}a^{8}+7z^{8}a^{8}-18z^{6}a^{8}+16z^{4}a^{8}-8z^{2}a^{8}+3a^{8}+10z^{9}a^{7}-21z^{7}a^{7}+20z^{5}a^{7}-11z^{3}a^{7}+2za^{7}+2z^{10}a^{6}+5z^{8}a^{6}-19z^{6}a^{6}+20z^{4}a^{6}-11z^{2}a^{6}+2a^{6}+5z^{9}a^{5}-11z^{7}a^{5}+11z^{5}a^{5}-8z^{3}a^{5}+za^{5}+5z^{8}a^{4}-13z^{6}a^{4}+12z^{4}a^{4}-7z^{2}a^{4}+2a^{4}+3z^{7}a^{3}-8z^{5}a^{3}+5z^{3}a^{3}+z^{6}a^{2}-3z^{4}a^{2}+2z^{2}a^{2}$
The A2 invariant	Data:K11a321/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a321/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a321 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["K11a321"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-3t^{3}+15t^{2}-27t+31-27t^{-1}+15t^{-2}-3t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

$\{11,t+1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 121, -4 }

In[8]:=

Jones[K][q]

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

Out[8]=

$1-3q^{-1}+7q^{-2}-12q^{-3}+17q^{-4}-19q^{-5}+20q^{-6}-17q^{-7}+13q^{-8}-8q^{-9}+3q^{-10}-q^{-11}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{5}a^{13}-2z^{3}a^{13}+za^{13}+3z^{6}a^{12}-4z^{4}a^{12}+z^{2}a^{12}+6z^{7}a^{11}-10z^{5}a^{11}+8z^{3}a^{11}-4za^{11}+7z^{8}a^{10}-10z^{6}a^{10}+7z^{4}a^{10}-5z^{2}a^{10}+2a^{10}+5z^{9}a^{9}-z^{7}a^{9}-10z^{5}a^{9}+12z^{3}a^{9}-4za^{9}+2z^{10}a^{8}+7z^{8}a^{8}-18z^{6}a^{8}+16z^{4}a^{8}-8z^{2}a^{8}+3a^{8}+10z^{9}a^{7}-21z^{7}a^{7}+20z^{5}a^{7}-11z^{3}a^{7}+2za^{7}+2z^{10}a^{6}+5z^{8}a^{6}-19z^{6}a^{6}+20z^{4}a^{6}-11z^{2}a^{6}+2a^{6}+5z^{9}a^{5}-11z^{7}a^{5}+11z^{5}a^{5}-8z^{3}a^{5}+za^{5}+5z^{8}a^{4}-13z^{6}a^{4}+12z^{4}a^{4}-7z^{2}a^{4}+2a^{4}+3z^{7}a^{3}-8z^{5}a^{3}+5z^{3}a^{3}+z^{6}a^{2}-3z^{4}a^{2}+2z^{2}a^{2}$

"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["K11a321"];

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]=

{ $-3t^{3}+15t^{2}-27t+31-27t^{-1}+15t^{-2}-3t^{-3}$, $1-3q^{-1}+7q^{-2}-12q^{-3}+17q^{-4}-19q^{-5}+20q^{-6}-17q^{-7}+13q^{-8}-8q^{-9}+3q^{-10}-q^{-11}$ }

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:

(6, -16)

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 $24$ $-128$ $288$ $860$ $164$ $-3072$ $-{\frac {18656}{3}}$ $-{\frac {3200}{3}}$ $-1120$ $2304$ $8192$ $20640$ $3936$ ${\frac {225791}{5}}$ $-{\frac {28612}{15}}$ ${\frac {331804}{15}}$ ${\frac {1649}{3}}$ ${\frac {16111}{5}}$

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 K11a321. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

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

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.

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