K11n56

K11n55

K11n57

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n56 at Knotilus!

Knot presentations

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

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11n56/ThurstonBennequinNumber Hyperbolic Volume 10.3372 A-Polynomial See Data:K11n56/A-polynomial

[edit Notes for K11n56'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 K11n56's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{3}-4t^{2}+8t-9+8t^{-1}-4t^{-2}+t^{-3}$
Conway polynomial	$z^{6}+2z^{4}+z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 35, 2 }
Jones polynomial	$q^{5}-3q^{4}+4q^{3}-5q^{2}+6q-5+5q^{-1}-3q^{-2}+2q^{-3}-q^{-4}$
HOMFLY-PT polynomial (db, data sources)	$z^{6}-a^{2}z^{4}-2z^{4}a^{-2}+5z^{4}-3a^{2}z^{2}-6z^{2}a^{-2}+z^{2}a^{-4}+9z^{2}-2a^{2}-4a^{-2}+a^{-4}+6$
Kauffman polynomial (db, data sources)	$az^{9}+z^{9}a^{-1}+2a^{2}z^{8}+3z^{8}a^{-2}+5z^{8}+a^{3}z^{7}-az^{7}+z^{7}a^{-1}+3z^{7}a^{-3}-10a^{2}z^{6}-12z^{6}a^{-2}+z^{6}a^{-4}-23z^{6}-5a^{3}z^{5}-11az^{5}-17z^{5}a^{-1}-11z^{5}a^{-3}+15a^{2}z^{4}+17z^{4}a^{-2}+32z^{4}+7a^{3}z^{3}+19az^{3}+24z^{3}a^{-1}+15z^{3}a^{-3}+3z^{3}a^{-5}-9a^{2}z^{2}-13z^{2}a^{-2}-z^{2}a^{-4}+z^{2}a^{-6}-20z^{2}-3a^{3}z-8az-10za^{-1}-7za^{-3}-2za^{-5}+2a^{2}+4a^{-2}+a^{-4}+6$
The A2 invariant	Data:K11n56/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11n56/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n56 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["K11n56"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{3}-4t^{2}+8t-9+8t^{-1}-4t^{-2}+t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 35, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{6}-a^{2}z^{4}-2z^{4}a^{-2}+5z^{4}-3a^{2}z^{2}-6z^{2}a^{-2}+z^{2}a^{-4}+9z^{2}-2a^{2}-4a^{-2}+a^{-4}+6$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$az^{9}+z^{9}a^{-1}+2a^{2}z^{8}+3z^{8}a^{-2}+5z^{8}+a^{3}z^{7}-az^{7}+z^{7}a^{-1}+3z^{7}a^{-3}-10a^{2}z^{6}-12z^{6}a^{-2}+z^{6}a^{-4}-23z^{6}-5a^{3}z^{5}-11az^{5}-17z^{5}a^{-1}-11z^{5}a^{-3}+15a^{2}z^{4}+17z^{4}a^{-2}+32z^{4}+7a^{3}z^{3}+19az^{3}+24z^{3}a^{-1}+15z^{3}a^{-3}+3z^{3}a^{-5}-9a^{2}z^{2}-13z^{2}a^{-2}-z^{2}a^{-4}+z^{2}a^{-6}-20z^{2}-3a^{3}z-8az-10za^{-1}-7za^{-3}-2za^{-5}+2a^{2}+4a^{-2}+a^{-4}+6$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {8_16, 10_156, K11n15, K11n58,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {K11n58,}

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["K11n56"];

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]=

{ $t^{3}-4t^{2}+8t-9+8t^{-1}-4t^{-2}+t^{-3}$, $q^{5}-3q^{4}+4q^{3}-5q^{2}+6q-5+5q^{-1}-3q^{-2}+2q^{-3}-q^{-4}$ }

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]=

{8_16, 10_156, K11n15, K11n58,}

In[6]:=

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

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

Out[6]=

{K11n58,}

Vassiliev invariants

V2 and V3:

(1, -1)

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 $4$ $-8$ $8$ $-{\frac {34}{3}}$ $-{\frac {38}{3}}$ $-32$ $-{\frac {80}{3}}$ $-{\frac {32}{3}}$ $24$ ${\frac {32}{3}}$ $32$ $-{\frac {136}{3}}$ $-{\frac {152}{3}}$ ${\frac {751}{30}}$ ${\frac {58}{15}}$ ${\frac {422}{45}}$ $-{\frac {175}{18}}$ $-{\frac {689}{30}}$

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 K11n56. 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 χ 11                   1 1 9                 2   -2 7               2 1   1 5             3 2     -1 3           3 2       1 1         3 4         1 -1       2 2           0 -3     1 3             2 -5   1 2               -1 -7   1                 1 -9 1                   -1

Integral Khovanov Homology (db, data source)

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

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.

K11n55 K11n57