K11a37

K11a36

K11a38

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a37 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X8493 X14,5,15,6 X2837 X16,9,17,10 X20,12,21,11 X18,14,19,13 X6,15,7,16 X22,17,1,18 X12,20,13,19 X10,22,11,21
Gauss code	1, -4, 2, -1, 3, -8, 4, -2, 5, -11, 6, -10, 7, -3, 8, -5, 9, -7, 10, -6, 11, -9
Dowker-Thistlethwaite code	4 8 14 2 16 20 18 6 22 12 10

A Braid Representative	{{{braid_table}}}
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:K11a37/ThurstonBennequinNumber Hyperbolic Volume 13.3384 A-Polynomial See Data:K11a37/A-polynomial

[edit Notes for K11a37'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 K11a37's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-2t^{3}+9t^{2}-21t+29-21t^{-1}+9t^{-2}-2t^{-3}$
Conway polynomial	$-2z^{6}-3z^{4}-3z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 93, 0 }
Jones polynomial	$-q^{7}+3q^{6}-5q^{5}+9q^{4}-12q^{3}+14q^{2}-15q+13-10q^{-1}+7q^{-2}-3q^{-3}+q^{-4}$
HOMFLY-PT polynomial (db, data sources)	$-z^{6}a^{-2}-z^{6}+a^{2}z^{4}-3z^{4}a^{-2}+2z^{4}a^{-4}-3z^{4}+2a^{2}z^{2}-5z^{2}a^{-2}+5z^{2}a^{-4}-z^{2}a^{-6}-4z^{2}+2a^{2}-3a^{-2}+4a^{-4}-a^{-6}-1$
Kauffman polynomial (db, data sources)	$z^{10}a^{-2}+z^{10}a^{-4}+4z^{9}a^{-1}+7z^{9}a^{-3}+3z^{9}a^{-5}+9z^{8}a^{-2}+6z^{8}a^{-4}+3z^{8}a^{-6}+6z^{8}+7az^{7}-3z^{7}a^{-1}-20z^{7}a^{-3}-9z^{7}a^{-5}+z^{7}a^{-7}+6a^{2}z^{6}-34z^{6}a^{-2}-35z^{6}a^{-4}-13z^{6}a^{-6}-6z^{6}+3a^{3}z^{5}-9az^{5}-3z^{5}a^{-1}+16z^{5}a^{-3}+3z^{5}a^{-5}-4z^{5}a^{-7}+a^{4}z^{4}-8a^{2}z^{4}+40z^{4}a^{-2}+50z^{4}a^{-4}+17z^{4}a^{-6}-2z^{4}-2a^{3}z^{3}+5az^{3}-z^{3}a^{-1}-7z^{3}a^{-3}+5z^{3}a^{-5}+4z^{3}a^{-7}-a^{4}z^{2}+6a^{2}z^{2}-22z^{2}a^{-2}-27z^{2}a^{-4}-8z^{2}a^{-6}+4z^{2}-az+2za^{-1}+3za^{-3}-za^{-5}-za^{-7}-2a^{2}+3a^{-2}+4a^{-4}+a^{-6}-1$
The A2 invariant	Data:K11a37/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a37/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a37 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["K11a37"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-2t^{3}+9t^{2}-21t+29-21t^{-1}+9t^{-2}-2t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 93, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{7}+3q^{6}-5q^{5}+9q^{4}-12q^{3}+14q^{2}-15q+13-10q^{-1}+7q^{-2}-3q^{-3}+q^{-4}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-z^{6}a^{-2}-z^{6}+a^{2}z^{4}-3z^{4}a^{-2}+2z^{4}a^{-4}-3z^{4}+2a^{2}z^{2}-5z^{2}a^{-2}+5z^{2}a^{-4}-z^{2}a^{-6}-4z^{2}+2a^{2}-3a^{-2}+4a^{-4}-a^{-6}-1$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{10}a^{-2}+z^{10}a^{-4}+4z^{9}a^{-1}+7z^{9}a^{-3}+3z^{9}a^{-5}+9z^{8}a^{-2}+6z^{8}a^{-4}+3z^{8}a^{-6}+6z^{8}+7az^{7}-3z^{7}a^{-1}-20z^{7}a^{-3}-9z^{7}a^{-5}+z^{7}a^{-7}+6a^{2}z^{6}-34z^{6}a^{-2}-35z^{6}a^{-4}-13z^{6}a^{-6}-6z^{6}+3a^{3}z^{5}-9az^{5}-3z^{5}a^{-1}+16z^{5}a^{-3}+3z^{5}a^{-5}-4z^{5}a^{-7}+a^{4}z^{4}-8a^{2}z^{4}+40z^{4}a^{-2}+50z^{4}a^{-4}+17z^{4}a^{-6}-2z^{4}-2a^{3}z^{3}+5az^{3}-z^{3}a^{-1}-7z^{3}a^{-3}+5z^{3}a^{-5}+4z^{3}a^{-7}-a^{4}z^{2}+6a^{2}z^{2}-22z^{2}a^{-2}-27z^{2}a^{-4}-8z^{2}a^{-6}+4z^{2}-az+2za^{-1}+3za^{-3}-za^{-5}-za^{-7}-2a^{2}+3a^{-2}+4a^{-4}+a^{-6}-1$

"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["K11a37"];

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]=

{ $-2t^{3}+9t^{2}-21t+29-21t^{-1}+9t^{-2}-2t^{-3}$, $-q^{7}+3q^{6}-5q^{5}+9q^{4}-12q^{3}+14q^{2}-15q+13-10q^{-1}+7q^{-2}-3q^{-3}+q^{-4}$ }

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, 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 $-12$ $0$ $72$ $146$ $62$ $0$ $32$ $0$ $0$ $-288$ $0$ $-1752$ $-744$ $-{\frac {18591}{10}}$ ${\frac {8426}{15}}$ $-{\frac {29582}{15}}$ ${\frac {1375}{6}}$ $-{\frac {4511}{10}}$

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=$0 is the signature of K11a37. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

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

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.

K11a36 K11a38