K11a52

K11a51

K11a53

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a52 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X8394 X14,5,15,6 X12,8,13,7 X2,9,3,10 X18,12,19,11 X22,13,1,14 X20,16,21,15 X10,18,11,17 X16,20,17,19 X6,21,7,22
Gauss code	1, -5, 2, -1, 3, -11, 4, -2, 5, -9, 6, -4, 7, -3, 8, -10, 9, -6, 10, -8, 11, -7
Dowker-Thistlethwaite code	4 8 14 12 2 18 22 20 10 16 6

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Three dimensional invariants

Symmetry type Chiral Unknotting number $\{1,2\}$ 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a52/ThurstonBennequinNumber Hyperbolic Volume 16.8243 A-Polynomial See Data:K11a52/A-polynomial

[edit Notes for K11a52's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $3$ Rasmussen s-Invariant 0

[edit Notes for K11a52's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-2t^{3}+13t^{2}-32t+43-32t^{-1}+13t^{-2}-2t^{-3}$
Conway polynomial	$-2z^{6}+z^{4}+2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 137, 0 }
Jones polynomial	$q^{6}-5q^{5}+10q^{4}-15q^{3}+20q^{2}-22q+22-18q^{-1}+13q^{-2}-7q^{-3}+3q^{-4}-q^{-5}$
HOMFLY-PT polynomial (db, data sources)	$-z^{6}a^{-2}-z^{6}+2a^{2}z^{4}-z^{4}a^{-2}+z^{4}a^{-4}-z^{4}-a^{4}z^{2}+3a^{2}z^{2}+z^{2}a^{-2}-z^{2}-a^{4}+2a^{2}+2a^{-2}-a^{-4}-1$
Kauffman polynomial (db, data sources)	$2z^{10}a^{-2}+2z^{10}+5az^{9}+12z^{9}a^{-1}+7z^{9}a^{-3}+6a^{2}z^{8}+14z^{8}a^{-2}+9z^{8}a^{-4}+11z^{8}+5a^{3}z^{7}-18z^{7}a^{-1}-8z^{7}a^{-3}+5z^{7}a^{-5}+3a^{4}z^{6}-5a^{2}z^{6}-41z^{6}a^{-2}-21z^{6}a^{-4}+z^{6}a^{-6}-27z^{6}+a^{5}z^{5}-6a^{3}z^{5}-9az^{5}-z^{5}a^{-1}-9z^{5}a^{-3}-10z^{5}a^{-5}-5a^{4}z^{4}-a^{2}z^{4}+27z^{4}a^{-2}+11z^{4}a^{-4}-z^{4}a^{-6}+19z^{4}-2a^{5}z^{3}+2a^{3}z^{3}+10az^{3}+10z^{3}a^{-1}+7z^{3}a^{-3}+3z^{3}a^{-5}+3a^{4}z^{2}+4a^{2}z^{2}-2z^{2}a^{-2}-z^{2}+a^{5}z-3az-3za^{-1}+za^{-5}-a^{4}-2a^{2}-2a^{-2}-a^{-4}-1$
The A2 invariant	Data:K11a52/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a52/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a52 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["K11a52"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-2t^{3}+13t^{2}-32t+43-32t^{-1}+13t^{-2}-2t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 137, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{6}-5q^{5}+10q^{4}-15q^{3}+20q^{2}-22q+22-18q^{-1}+13q^{-2}-7q^{-3}+3q^{-4}-q^{-5}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$2z^{10}a^{-2}+2z^{10}+5az^{9}+12z^{9}a^{-1}+7z^{9}a^{-3}+6a^{2}z^{8}+14z^{8}a^{-2}+9z^{8}a^{-4}+11z^{8}+5a^{3}z^{7}-18z^{7}a^{-1}-8z^{7}a^{-3}+5z^{7}a^{-5}+3a^{4}z^{6}-5a^{2}z^{6}-41z^{6}a^{-2}-21z^{6}a^{-4}+z^{6}a^{-6}-27z^{6}+a^{5}z^{5}-6a^{3}z^{5}-9az^{5}-z^{5}a^{-1}-9z^{5}a^{-3}-10z^{5}a^{-5}-5a^{4}z^{4}-a^{2}z^{4}+27z^{4}a^{-2}+11z^{4}a^{-4}-z^{4}a^{-6}+19z^{4}-2a^{5}z^{3}+2a^{3}z^{3}+10az^{3}+10z^{3}a^{-1}+7z^{3}a^{-3}+3z^{3}a^{-5}+3a^{4}z^{2}+4a^{2}z^{2}-2z^{2}a^{-2}-z^{2}+a^{5}z-3az-3za^{-1}+za^{-5}-a^{4}-2a^{2}-2a^{-2}-a^{-4}-1$

"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["K11a52"];

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]=

{ $-2t^{3}+13t^{2}-32t+43-32t^{-1}+13t^{-2}-2t^{-3}$, $q^{6}-5q^{5}+10q^{4}-15q^{3}+20q^{2}-22q+22-18q^{-1}+13q^{-2}-7q^{-3}+3q^{-4}-q^{-5}$ }

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:

(2, 0)

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 $8$ $0$ $32$ ${\frac {172}{3}}$ ${\frac {20}{3}}$ $0$ $0$ $32$ $-32$ ${\frac {256}{3}}$ $0$ ${\frac {1376}{3}}$ ${\frac {160}{3}}$ ${\frac {8551}{15}}$ ${\frac {1156}{15}}$ ${\frac {5404}{45}}$ ${\frac {377}{9}}$ $-{\frac {89}{15}}$

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=$0 is the signature of K11a52. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 6 χ 13                       1 1 11                     4   -4 9                   6 1   5 7                 9 4     -5 5               11 6       5 3             11 9         -2 1           11 11           0 -1         8 12             4 -3       5 10               -5 -5     2 8                 6 -7   1 5                   -4 -9   2                     2 -11 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-1$ $i=1$ $r=-5$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-2$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-1$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=0$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{11}$ $r=1$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=2$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=3$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=4$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=5$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=6$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

K11a51 K11a53