K11a327

K11a326

K11a328

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a327 at Knotilus!

Knot presentations

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

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:K11a327/ThurstonBennequinNumber Hyperbolic Volume 19.2356 A-Polynomial See Data:K11a327/A-polynomial

[edit Notes for K11a327'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 K11a327's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$3t^{3}-17t^{2}+44t-59+44t^{-1}-17t^{-2}+3t^{-3}$
Conway polynomial	$3z^{6}+z^{4}+3z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 187, 2 }
Jones polynomial	$q^{9}-5q^{8}+11q^{7}-19q^{6}+26q^{5}-30q^{4}+31q^{3}-26q^{2}+20q-12+5q^{-1}-q^{-2}$
HOMFLY-PT polynomial (db, data sources)	$z^{6}a^{-2}+2z^{6}a^{-4}+5z^{4}a^{-4}-3z^{4}a^{-6}-z^{4}-z^{2}a^{-2}+7z^{2}a^{-4}-4z^{2}a^{-6}+z^{2}a^{-8}+3a^{-4}-2a^{-6}$
Kauffman polynomial (db, data sources)	$4z^{10}a^{-4}+4z^{10}a^{-6}+13z^{9}a^{-3}+23z^{9}a^{-5}+10z^{9}a^{-7}+17z^{8}a^{-2}+26z^{8}a^{-4}+19z^{8}a^{-6}+10z^{8}a^{-8}+12z^{7}a^{-1}-8z^{7}a^{-3}-36z^{7}a^{-5}-11z^{7}a^{-7}+5z^{7}a^{-9}-26z^{6}a^{-2}-71z^{6}a^{-4}-61z^{6}a^{-6}-20z^{6}a^{-8}+z^{6}a^{-10}+5z^{6}+az^{5}-14z^{5}a^{-1}-16z^{5}a^{-3}-3z^{5}a^{-5}-11z^{5}a^{-7}-9z^{5}a^{-9}+14z^{4}a^{-2}+54z^{4}a^{-4}+50z^{4}a^{-6}+12z^{4}a^{-8}-z^{4}a^{-10}-3z^{4}+4z^{3}a^{-1}+15z^{3}a^{-3}+23z^{3}a^{-5}+16z^{3}a^{-7}+4z^{3}a^{-9}-4z^{2}a^{-2}-16z^{2}a^{-4}-15z^{2}a^{-6}-3z^{2}a^{-8}-za^{-1}-4za^{-3}-8za^{-5}-5za^{-7}+3a^{-4}+2a^{-6}$
The A2 invariant	Data:K11a327/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a327/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a327 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["K11a327"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$3t^{3}-17t^{2}+44t-59+44t^{-1}-17t^{-2}+3t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 187, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{9}-5q^{8}+11q^{7}-19q^{6}+26q^{5}-30q^{4}+31q^{3}-26q^{2}+20q-12+5q^{-1}-q^{-2}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$4z^{10}a^{-4}+4z^{10}a^{-6}+13z^{9}a^{-3}+23z^{9}a^{-5}+10z^{9}a^{-7}+17z^{8}a^{-2}+26z^{8}a^{-4}+19z^{8}a^{-6}+10z^{8}a^{-8}+12z^{7}a^{-1}-8z^{7}a^{-3}-36z^{7}a^{-5}-11z^{7}a^{-7}+5z^{7}a^{-9}-26z^{6}a^{-2}-71z^{6}a^{-4}-61z^{6}a^{-6}-20z^{6}a^{-8}+z^{6}a^{-10}+5z^{6}+az^{5}-14z^{5}a^{-1}-16z^{5}a^{-3}-3z^{5}a^{-5}-11z^{5}a^{-7}-9z^{5}a^{-9}+14z^{4}a^{-2}+54z^{4}a^{-4}+50z^{4}a^{-6}+12z^{4}a^{-8}-z^{4}a^{-10}-3z^{4}+4z^{3}a^{-1}+15z^{3}a^{-3}+23z^{3}a^{-5}+16z^{3}a^{-7}+4z^{3}a^{-9}-4z^{2}a^{-2}-16z^{2}a^{-4}-15z^{2}a^{-6}-3z^{2}a^{-8}-za^{-1}-4za^{-3}-8za^{-5}-5za^{-7}+3a^{-4}+2a^{-6}$

"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["K11a327"];

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]=

{ $3t^{3}-17t^{2}+44t-59+44t^{-1}-17t^{-2}+3t^{-3}$, $q^{9}-5q^{8}+11q^{7}-19q^{6}+26q^{5}-30q^{4}+31q^{3}-26q^{2}+20q-12+5q^{-1}-q^{-2}$ }

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:

(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 $12$ $40$ $72$ $174$ $26$ $480$ ${\frac {2608}{3}}$ ${\frac {448}{3}}$ $136$ $288$ $800$ $2088$ $312$ ${\frac {42751}{10}}$ $-{\frac {1106}{15}}$ ${\frac {9714}{5}}$ ${\frac {11}{2}}$ ${\frac {2751}{10}}$

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 K11a327. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -3 -2 -1 0 1 2 3 4 5 6 7 8 χ 19                       1 1 17                     4   -4 15                   7 1   6 13                 12 4     -8 11               14 7       7 9             16 12         -4 7           15 14           1 5         11 16             5 3       9 15               -6 1     4 12                 8 -1   1 8                   -7 -3   4                     4 -5 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=1$ $i=3$ $r=-3$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-1$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=0$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{9}$ $r=1$ ${\mathbb {Z} }^{15}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=2$ ${\mathbb {Z} }^{16}\oplus {\mathbb {Z} }_{2}^{15}$ ${\mathbb {Z} }^{15}$ $r=3$ ${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{16}$ ${\mathbb {Z} }^{16}$ $r=4$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{14}$ ${\mathbb {Z} }^{14}$ $r=5$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=6$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=7$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=8$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

K11a326 K11a328