K11n128

K11n127

K11n129

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n128 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X10,3,11,4 X5,19,6,18 X7,15,8,14 X9,16,10,17 X2,11,3,12 X13,7,14,6 X15,20,16,21 X17,1,18,22 X19,12,20,13 X21,9,22,8
Gauss code	1, -6, 2, -1, -3, 7, -4, 11, -5, -2, 6, 10, -7, 4, -8, 5, -9, 3, -10, 8, -11, 9
Dowker-Thistlethwaite code	4 10 -18 -14 -16 2 -6 -20 -22 -12 -8

A Braid Representative
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:K11n128/ThurstonBennequinNumber Hyperbolic Volume 12.2619 A-Polynomial See Data:K11n128/A-polynomial

[edit Notes for K11n128'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 K11n128's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{3}-5t^{2}+10t-11+10t^{-1}-5t^{-2}+t^{-3}$
Conway polynomial	$z^{6}+z^{4}-z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 43, 2 }
Jones polynomial	$q^{5}-3q^{4}+5q^{3}-6q^{2}+7q-7+6q^{-1}-4q^{-2}+3q^{-3}-q^{-4}$
HOMFLY-PT polynomial (db, data sources)	$z^{6}-a^{2}z^{4}-2z^{4}a^{-2}+4z^{4}-2a^{2}z^{2}-5z^{2}a^{-2}+z^{2}a^{-4}+5z^{2}-2a^{-2}+a^{-4}+2$
Kauffman polynomial (db, data sources)	$2az^{9}+2z^{9}a^{-1}+3a^{2}z^{8}+4z^{8}a^{-2}+7z^{8}+a^{3}z^{7}-6az^{7}-4z^{7}a^{-1}+3z^{7}a^{-3}-14a^{2}z^{6}-15z^{6}a^{-2}+z^{6}a^{-4}-30z^{6}-4a^{3}z^{5}-az^{5}-5z^{5}a^{-1}-8z^{5}a^{-3}+18a^{2}z^{4}+19z^{4}a^{-2}+z^{4}a^{-4}+36z^{4}+4a^{3}z^{3}+8az^{3}+8z^{3}a^{-1}+7z^{3}a^{-3}+3z^{3}a^{-5}-6a^{2}z^{2}-12z^{2}a^{-2}-2z^{2}a^{-4}+z^{2}a^{-6}-15z^{2}-a^{3}z-3az-3za^{-1}-2za^{-3}-za^{-5}+2a^{-2}+a^{-4}+2$
The A2 invariant	$-q^{12}+q^{10}+q^{6}+2q^{4}-q^{2}+1-q^{-2}+q^{-6}-q^{-8}+q^{-10}-q^{-12}+q^{-16}$
The G2 invariant	Data:K11n128/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n128 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["K11n128"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{3}-5t^{2}+10t-11+10t^{-1}-5t^{-2}+t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 43, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{5}-3q^{4}+5q^{3}-6q^{2}+7q-7+6q^{-1}-4q^{-2}+3q^{-3}-q^{-4}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$2az^{9}+2z^{9}a^{-1}+3a^{2}z^{8}+4z^{8}a^{-2}+7z^{8}+a^{3}z^{7}-6az^{7}-4z^{7}a^{-1}+3z^{7}a^{-3}-14a^{2}z^{6}-15z^{6}a^{-2}+z^{6}a^{-4}-30z^{6}-4a^{3}z^{5}-az^{5}-5z^{5}a^{-1}-8z^{5}a^{-3}+18a^{2}z^{4}+19z^{4}a^{-2}+z^{4}a^{-4}+36z^{4}+4a^{3}z^{3}+8az^{3}+8z^{3}a^{-1}+7z^{3}a^{-3}+3z^{3}a^{-5}-6a^{2}z^{2}-12z^{2}a^{-2}-2z^{2}a^{-4}+z^{2}a^{-6}-15z^{2}-a^{3}z-3az-3za^{-1}-2za^{-3}-za^{-5}+2a^{-2}+a^{-4}+2$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {9_22,}

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["K11n128"];

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]=

{ $t^{3}-5t^{2}+10t-11+10t^{-1}-5t^{-2}+t^{-3}$, $q^{5}-3q^{4}+5q^{3}-6q^{2}+7q-7+6q^{-1}-4q^{-2}+3q^{-3}-q^{-4}$ }

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]=

{9_22,}

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:

(-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 $-4$ $-8$ $8$ $-{\frac {14}{3}}$ $-{\frac {10}{3}}$ $32$ ${\frac {208}{3}}$ ${\frac {64}{3}}$ $24$ $-{\frac {32}{3}}$ $32$ ${\frac {56}{3}}$ ${\frac {40}{3}}$ ${\frac {3809}{30}}$ ${\frac {502}{15}}$ ${\frac {4858}{45}}$ $-{\frac {1025}{18}}$ $-{\frac {31}{30}}$

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 K11n128. 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               3 1   2 5             3 2     -1 3           4 3       1 1         4 4         0 -1       2 3           -1 -3     2 4             2 -5   1 2               -1 -7   2                 2 -9 1                   -1

Integral Khovanov Homology (db, data source)

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

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.

K11n127 K11n129