K11n57

K11n56

K11n58

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n57 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X8493 X5,15,6,14 X2837 X9,17,10,16 X11,19,12,18 X13,20,14,21 X15,7,16,6 X17,11,18,10 X19,1,20,22 X21,12,22,13
Gauss code	1, -4, 2, -1, -3, 8, 4, -2, -5, 9, -6, 11, -7, 3, -8, 5, -9, 6, -10, 7, -11, 10
Dowker-Thistlethwaite code	4 8 -14 2 -16 -18 -20 -6 -10 -22 -12

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number 3 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11n57/ThurstonBennequinNumber Hyperbolic Volume 5.86054 A-Polynomial See Data:K11n57/A-polynomial

[edit Notes for K11n57's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $4$ Rasmussen s-Invariant -6

[edit Notes for K11n57's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{4}+3t^{3}-2t^{2}-t+3-t^{-1}-2t^{-2}+3t^{-3}-t^{-4}$
Conway polynomial	$-z^{8}-5z^{6}-4z^{4}+2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 7, 6 }
Jones polynomial	$-q^{6}+q^{5}-q^{4}+2q^{3}-q^{2}+q$
HOMFLY-PT polynomial (db, data sources)	$-z^{8}a^{-6}+z^{6}a^{-4}-7z^{6}a^{-6}+z^{6}a^{-8}+6z^{4}a^{-4}-16z^{4}a^{-6}+6z^{4}a^{-8}+10z^{2}a^{-4}-16z^{2}a^{-6}+9z^{2}a^{-8}-z^{2}a^{-10}+5a^{-4}-7a^{-6}+4a^{-8}-a^{-10}$
Kauffman polynomial (db, data sources)	$z^{9}a^{-5}+z^{9}a^{-7}+z^{8}a^{-4}+3z^{8}a^{-6}+2z^{8}a^{-8}-6z^{7}a^{-5}-5z^{7}a^{-7}+z^{7}a^{-9}-7z^{6}a^{-4}-20z^{6}a^{-6}-13z^{6}a^{-8}+9z^{5}a^{-5}+3z^{5}a^{-7}-6z^{5}a^{-9}+16z^{4}a^{-4}+40z^{4}a^{-6}+24z^{4}a^{-8}-z^{3}a^{-5}+7z^{3}a^{-7}+8z^{3}a^{-9}-15z^{2}a^{-4}-29z^{2}a^{-6}-16z^{2}a^{-8}-2z^{2}a^{-10}-3za^{-5}-5za^{-7}-3za^{-9}-za^{-11}+5a^{-4}+7a^{-6}+4a^{-8}+a^{-10}$
The A2 invariant	$q^{-4}+q^{-6}+q^{-8}+2q^{-10}+q^{-12}-q^{-16}-q^{-18}-2q^{-20}+q^{-26}+q^{-32}-q^{-34}$
The G2 invariant	Data:K11n57/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n57 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["K11n57"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{4}+3t^{3}-2t^{2}-t+3-t^{-1}-2t^{-2}+3t^{-3}-t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

$-z^{8}-5z^{6}-4z^{4}+2z^{2}+1$

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]=

{ 7, 6 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{6}+q^{5}-q^{4}+2q^{3}-q^{2}+q$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-z^{8}a^{-6}+z^{6}a^{-4}-7z^{6}a^{-6}+z^{6}a^{-8}+6z^{4}a^{-4}-16z^{4}a^{-6}+6z^{4}a^{-8}+10z^{2}a^{-4}-16z^{2}a^{-6}+9z^{2}a^{-8}-z^{2}a^{-10}+5a^{-4}-7a^{-6}+4a^{-8}-a^{-10}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{9}a^{-5}+z^{9}a^{-7}+z^{8}a^{-4}+3z^{8}a^{-6}+2z^{8}a^{-8}-6z^{7}a^{-5}-5z^{7}a^{-7}+z^{7}a^{-9}-7z^{6}a^{-4}-20z^{6}a^{-6}-13z^{6}a^{-8}+9z^{5}a^{-5}+3z^{5}a^{-7}-6z^{5}a^{-9}+16z^{4}a^{-4}+40z^{4}a^{-6}+24z^{4}a^{-8}-z^{3}a^{-5}+7z^{3}a^{-7}+8z^{3}a^{-9}-15z^{2}a^{-4}-29z^{2}a^{-6}-16z^{2}a^{-8}-2z^{2}a^{-10}-3za^{-5}-5za^{-7}-3za^{-9}-za^{-11}+5a^{-4}+7a^{-6}+4a^{-8}+a^{-10}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {5_2,}

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["K11n57"];

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]=

{ $-t^{4}+3t^{3}-2t^{2}-t+3-t^{-1}-2t^{-2}+3t^{-3}-t^{-4}$, $-q^{6}+q^{5}-q^{4}+2q^{3}-q^{2}+q$ }

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]=

{5_2,}

Vassiliev invariants

V2 and V3:

(2, 3)

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 $8$ $24$ $32$ ${\frac {412}{3}}$ ${\frac {140}{3}}$ $192$ $720$ $192$ $184$ ${\frac {256}{3}}$ $288$ ${\frac {3296}{3}}$ ${\frac {1120}{3}}$ ${\frac {49711}{15}}$ $-{\frac {1124}{15}}$ ${\frac {90484}{45}}$ ${\frac {833}{9}}$ ${\frac {3631}{15}}$

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=$6 is the signature of K11n57. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -2 -1 0 1 2 3 4 5 6 χ 17               1 1 0 15             1 1   0 13           1 1 1   -1 11         1 2 1     0 9       1 1         0 7     1 1 1         1 5   1 2             1 3                   0 1 1                 1

Integral Khovanov Homology (db, data source)

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

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.

K11n56 K11n58