K11n120

K11n119

K11n121

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n120 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X10,4,11,3 X5,15,6,14 X7,20,8,21 X2,10,3,9 X11,17,12,16 X13,18,14,19 X15,9,16,8 X17,1,18,22 X19,6,20,7 X21,12,22,13
Gauss code	1, -5, 2, -1, -3, 10, -4, 8, 5, -2, -6, 11, -7, 3, -8, 6, -9, 7, -10, 4, -11, 9
Dowker-Thistlethwaite code	4 10 -14 -20 2 -16 -18 -8 -22 -6 -12

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Chiral Unknotting number $\{1,2\}$ 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11n120/ThurstonBennequinNumber Hyperbolic Volume 12.7751 A-Polynomial See Data:K11n120/A-polynomial

[edit Notes for K11n120's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $4$ Rasmussen s-Invariant -2

[edit Notes for K11n120's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{4}+4t^{3}-7t^{2}+8t-7+8t^{-1}-7t^{-2}+4t^{-3}-t^{-4}$
Conway polynomial	$-z^{8}-4z^{6}-3z^{4}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 47, 2 }
Jones polynomial	$-q^{6}+3q^{5}-5q^{4}+7q^{3}-8q^{2}+8q-6+5q^{-1}-3q^{-2}+q^{-3}$
HOMFLY-PT polynomial (db, data sources)	$-z^{8}a^{-2}-6z^{6}a^{-2}+z^{6}a^{-4}+z^{6}-12z^{4}a^{-2}+5z^{4}a^{-4}+4z^{4}-10z^{2}a^{-2}+7z^{2}a^{-4}-z^{2}a^{-6}+4z^{2}-3a^{-2}+3a^{-4}-a^{-6}+2$
Kauffman polynomial (db, data sources)	$2z^{9}a^{-1}+2z^{9}a^{-3}+7z^{8}a^{-2}+3z^{8}a^{-4}+4z^{8}+3az^{7}-4z^{7}a^{-1}-6z^{7}a^{-3}+z^{7}a^{-5}+a^{2}z^{6}-29z^{6}a^{-2}-13z^{6}a^{-4}-15z^{6}-10az^{5}-4z^{5}a^{-1}+5z^{5}a^{-3}-z^{5}a^{-5}-3a^{2}z^{4}+40z^{4}a^{-2}+25z^{4}a^{-4}+3z^{4}a^{-6}+15z^{4}+6az^{3}+7z^{3}a^{-1}+3z^{3}a^{-3}+3z^{3}a^{-5}+z^{3}a^{-7}+a^{2}z^{2}-20z^{2}a^{-2}-16z^{2}a^{-4}-4z^{2}a^{-6}-7z^{2}-az-2za^{-1}-2za^{-3}-2za^{-5}-za^{-7}+3a^{-2}+3a^{-4}+a^{-6}+2$
The A2 invariant	$q^{8}-q^{6}+q^{4}+1+q^{-2}-2q^{-4}+2q^{-6}-2q^{-8}+2q^{-10}+q^{-16}-q^{-18}+q^{-20}-q^{-22}$
The G2 invariant	Data:K11n120/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n120 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["K11n120"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{4}+4t^{3}-7t^{2}+8t-7+8t^{-1}-7t^{-2}+4t^{-3}-t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

$-z^{8}-4z^{6}-3z^{4}+1$

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]=

{ 47, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-z^{8}a^{-2}-6z^{6}a^{-2}+z^{6}a^{-4}+z^{6}-12z^{4}a^{-2}+5z^{4}a^{-4}+4z^{4}-10z^{2}a^{-2}+7z^{2}a^{-4}-z^{2}a^{-6}+4z^{2}-3a^{-2}+3a^{-4}-a^{-6}+2$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$2z^{9}a^{-1}+2z^{9}a^{-3}+7z^{8}a^{-2}+3z^{8}a^{-4}+4z^{8}+3az^{7}-4z^{7}a^{-1}-6z^{7}a^{-3}+z^{7}a^{-5}+a^{2}z^{6}-29z^{6}a^{-2}-13z^{6}a^{-4}-15z^{6}-10az^{5}-4z^{5}a^{-1}+5z^{5}a^{-3}-z^{5}a^{-5}-3a^{2}z^{4}+40z^{4}a^{-2}+25z^{4}a^{-4}+3z^{4}a^{-6}+15z^{4}+6az^{3}+7z^{3}a^{-1}+3z^{3}a^{-3}+3z^{3}a^{-5}+z^{3}a^{-7}+a^{2}z^{2}-20z^{2}a^{-2}-16z^{2}a^{-4}-4z^{2}a^{-6}-7z^{2}-az-2za^{-1}-2za^{-3}-2za^{-5}-za^{-7}+3a^{-2}+3a^{-4}+a^{-6}+2$

"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["K11n120"];

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]=

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

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:

(0, 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 $0$ $8$ $0$ $48$ $24$ $0$ ${\frac {272}{3}}$ ${\frac {32}{3}}$ $40$ $0$ $32$ $0$ $0$ $168$ $72$ $-88$ $104$ $-40$

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 K11n120. 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 χ 13                   1 -1 11                 2   2 9               3 1   -2 7             4 2     2 5           4 3       -1 3         4 4         0 1       3 5           2 -1     2 3             -1 -3   1 3               2 -5   2                 -2 -7 1                   1

Integral Khovanov Homology (db, data source)

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

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.

K11n119 K11n121