K11n121

K11n120

K11n122

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n121 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X10,3,11,4 X5,16,6,17 X7,12,8,13 X18,9,19,10 X2,11,3,12 X13,20,14,21 X15,6,16,7 X22,18,1,17 X8,19,9,20 X21,14,22,15
Gauss code	1, -6, 2, -1, -3, 8, -4, -10, 5, -2, 6, 4, -7, 11, -8, 3, 9, -5, 10, 7, -11, -9
Dowker-Thistlethwaite code	4 10 -16 -12 18 2 -20 -6 22 8 -14

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11n121/ThurstonBennequinNumber Hyperbolic Volume 12.2709 A-Polynomial See Data:K11n121/A-polynomial

[edit Notes for K11n121's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11n121's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{3}+6t^{2}-10t+11-10t^{-1}+6t^{-2}-t^{-3}$
Conway polynomial	$-z^{6}+5z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 45, -4 }
Jones polynomial	$1-3q^{-1}+5q^{-2}-6q^{-3}+8q^{-4}-7q^{-5}+7q^{-6}-5q^{-7}+2q^{-8}-q^{-9}$
HOMFLY-PT polynomial (db, data sources)	$-z^{2}a^{8}-2a^{8}+2z^{4}a^{6}+5z^{2}a^{6}+2a^{6}-z^{6}a^{4}-3z^{4}a^{4}-z^{2}a^{4}+a^{4}+z^{4}a^{2}+2z^{2}a^{2}$
Kauffman polynomial (db, data sources)	$z^{3}a^{11}-za^{11}+2z^{4}a^{10}-z^{2}a^{10}+4z^{5}a^{9}-5z^{3}a^{9}+4za^{9}+z^{8}a^{8}-z^{6}a^{8}+2z^{4}a^{8}+2z^{2}a^{8}-2a^{8}+z^{9}a^{7}-2z^{7}a^{7}+6z^{5}a^{7}-10z^{3}a^{7}+6za^{7}+4z^{8}a^{6}-11z^{6}a^{6}+9z^{4}a^{6}-2z^{2}a^{6}-2a^{6}+z^{9}a^{5}+z^{7}a^{5}-8z^{5}a^{5}+3z^{3}a^{5}+za^{5}+3z^{8}a^{4}-9z^{6}a^{4}+6z^{4}a^{4}-3z^{2}a^{4}+a^{4}+3z^{7}a^{3}-10z^{5}a^{3}+7z^{3}a^{3}+z^{6}a^{2}-3z^{4}a^{2}+2z^{2}a^{2}$
The A2 invariant	Data:K11n121/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11n121/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n121 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["K11n121"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{3}+6t^{2}-10t+11-10t^{-1}+6t^{-2}-t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 45, -4 }

In[8]:=

Jones[K][q]

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

Out[8]=

$1-3q^{-1}+5q^{-2}-6q^{-3}+8q^{-4}-7q^{-5}+7q^{-6}-5q^{-7}+2q^{-8}-q^{-9}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-z^{2}a^{8}-2a^{8}+2z^{4}a^{6}+5z^{2}a^{6}+2a^{6}-z^{6}a^{4}-3z^{4}a^{4}-z^{2}a^{4}+a^{4}+z^{4}a^{2}+2z^{2}a^{2}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{3}a^{11}-za^{11}+2z^{4}a^{10}-z^{2}a^{10}+4z^{5}a^{9}-5z^{3}a^{9}+4za^{9}+z^{8}a^{8}-z^{6}a^{8}+2z^{4}a^{8}+2z^{2}a^{8}-2a^{8}+z^{9}a^{7}-2z^{7}a^{7}+6z^{5}a^{7}-10z^{3}a^{7}+6za^{7}+4z^{8}a^{6}-11z^{6}a^{6}+9z^{4}a^{6}-2z^{2}a^{6}-2a^{6}+z^{9}a^{5}+z^{7}a^{5}-8z^{5}a^{5}+3z^{3}a^{5}+za^{5}+3z^{8}a^{4}-9z^{6}a^{4}+6z^{4}a^{4}-3z^{2}a^{4}+a^{4}+3z^{7}a^{3}-10z^{5}a^{3}+7z^{3}a^{3}+z^{6}a^{2}-3z^{4}a^{2}+2z^{2}a^{2}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n14,}

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["K11n121"];

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]=

{ $-t^{3}+6t^{2}-10t+11-10t^{-1}+6t^{-2}-t^{-3}$, $1-3q^{-1}+5q^{-2}-6q^{-3}+8q^{-4}-7q^{-5}+7q^{-6}-5q^{-7}+2q^{-8}-q^{-9}$ }

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]=

{K11n14,}

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:

(5, -11)

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 $20$ $-88$ $200$ ${\frac {1558}{3}}$ ${\frac {290}{3}}$ $-1760$ $-{\frac {9904}{3}}$ $-{\frac {1696}{3}}$ $-568$ ${\frac {4000}{3}}$ $3872$ ${\frac {31160}{3}}$ ${\frac {5800}{3}}$ ${\frac {126895}{6}}$ $-{\frac {1894}{3}}$ ${\frac {89798}{9}}$ ${\frac {3925}{18}}$ ${\frac {8719}{6}}$

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 $s=$-4 is the signature of K11n121. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 1                   1 1 -1                 2   -2 -3               3 1   2 -5             4 3     -1 -7           4 2       2 -9         3 4         1 -11       4 4           0 -13     1 3             2 -15   1 4               -3 -17   1                 1 -19 1                   -1

Integral Khovanov Homology (db, data source)

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

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.

K11n120 K11n122