K11a325

K11a324

K11a326

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a325 at Knotilus!

Knot presentations

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

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number $\{1,2\}$ 3-genus 2 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a325/ThurstonBennequinNumber Hyperbolic Volume 14.4126 A-Polynomial See Data:K11a325/A-polynomial

[edit Notes for K11a325's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a325's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-6t^{2}+24t-35+24t^{-1}-6t^{-2}$
Conway polynomial	$1-6z^{4}$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 95, -2 }
Jones polynomial	$q^{3}-3q^{2}+6q-9+13q^{-1}-15q^{-2}+15q^{-3}-13q^{-4}+10q^{-5}-6q^{-6}+3q^{-7}-q^{-8}$
HOMFLY-PT polynomial (db, data sources)	$-a^{8}+3z^{2}a^{6}+2a^{6}-2z^{4}a^{4}-z^{2}a^{4}-3z^{4}a^{2}-4z^{2}a^{2}-2a^{2}-z^{4}+z^{2}+2+z^{2}a^{-2}$
Kauffman polynomial (db, data sources)	$2a^{4}z^{10}+2a^{2}z^{10}+4a^{5}z^{9}+9a^{3}z^{9}+5az^{9}+4a^{6}z^{8}+a^{2}z^{8}+5z^{8}+4a^{7}z^{7}-8a^{5}z^{7}-32a^{3}z^{7}-17az^{7}+3z^{7}a^{-1}+3a^{8}z^{6}-3a^{6}z^{6}-9a^{4}z^{6}-21a^{2}z^{6}+z^{6}a^{-2}-17z^{6}+a^{9}z^{5}-5a^{7}z^{5}+10a^{5}z^{5}+45a^{3}z^{5}+20az^{5}-9z^{5}a^{-1}-6a^{8}z^{4}-6a^{6}z^{4}+17a^{4}z^{4}+37a^{2}z^{4}-3z^{4}a^{-2}+17z^{4}-2a^{9}z^{3}-2a^{7}z^{3}-6a^{5}z^{3}-21a^{3}z^{3}-11az^{3}+4z^{3}a^{-1}+3a^{8}z^{2}+6a^{6}z^{2}-6a^{4}z^{2}-19a^{2}z^{2}+z^{2}a^{-2}-9z^{2}+a^{9}z+a^{7}z+a^{5}z+3a^{3}z+2az-a^{8}-2a^{6}+2a^{2}+2$
The A2 invariant	Data:K11a325/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a325/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a325 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["K11a325"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-6t^{2}+24t-35+24t^{-1}-6t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

$1-6z^{4}$

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]=

{ 95, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$2a^{4}z^{10}+2a^{2}z^{10}+4a^{5}z^{9}+9a^{3}z^{9}+5az^{9}+4a^{6}z^{8}+a^{2}z^{8}+5z^{8}+4a^{7}z^{7}-8a^{5}z^{7}-32a^{3}z^{7}-17az^{7}+3z^{7}a^{-1}+3a^{8}z^{6}-3a^{6}z^{6}-9a^{4}z^{6}-21a^{2}z^{6}+z^{6}a^{-2}-17z^{6}+a^{9}z^{5}-5a^{7}z^{5}+10a^{5}z^{5}+45a^{3}z^{5}+20az^{5}-9z^{5}a^{-1}-6a^{8}z^{4}-6a^{6}z^{4}+17a^{4}z^{4}+37a^{2}z^{4}-3z^{4}a^{-2}+17z^{4}-2a^{9}z^{3}-2a^{7}z^{3}-6a^{5}z^{3}-21a^{3}z^{3}-11az^{3}+4z^{3}a^{-1}+3a^{8}z^{2}+6a^{6}z^{2}-6a^{4}z^{2}-19a^{2}z^{2}+z^{2}a^{-2}-9z^{2}+a^{9}z+a^{7}z+a^{5}z+3a^{3}z+2az-a^{8}-2a^{6}+2a^{2}+2$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {K11a7,}

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["K11a325"];

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]=

{ $-6t^{2}+24t-35+24t^{-1}-6t^{-2}$, $q^{3}-3q^{2}+6q-9+13q^{-1}-15q^{-2}+15q^{-3}-13q^{-4}+10q^{-5}-6q^{-6}+3q^{-7}-q^{-8}$ }

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]=

{}

In[6]:=

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

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

Out[6]=

{K11a7,}

Vassiliev invariants

V2 and V3:

(0, -2)

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 $0$ $-16$ $0$ $112$ $48$ $0$ $-{\frac {1024}{3}}$ $-{\frac {64}{3}}$ $-176$ $0$ $128$ $0$ $0$ $920$ $-{\frac {1424}{3}}$ $864$ $168$ $136$

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 K11a325. 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                     2   -2 3                   4 1   3 1                 5 2     -3 -1               8 4       4 -3             8 6         -2 -5           7 7           0 -7         6 8             2 -9       4 7               -3 -11     2 6                 4 -13   1 4                   -3 -15   2                     2 -17 1                       -1

Integral Khovanov Homology (db, data source)

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

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.

K11a324 K11a326