K11n24

K11n23

K11n25

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n24 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X8493 X12,5,13,6 X2837 X14,9,15,10 X11,18,12,19 X6,13,7,14 X15,21,16,20 X17,1,18,22 X19,10,20,11 X21,17,22,16
Gauss code	1, -4, 2, -1, 3, -7, 4, -2, 5, 10, -6, -3, 7, -5, -8, 11, -9, 6, -10, 8, -11, 9
Dowker-Thistlethwaite code	4 8 12 2 14 -18 6 -20 -22 -10 -16

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:K11n24/ThurstonBennequinNumber Hyperbolic Volume 9.90549 A-Polynomial See Data:K11n24/A-polynomial

[edit Notes for K11n24's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $3$ Rasmussen s-Invariant 0

[edit Notes for K11n24's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{3}-3t^{2}+5t-5+5t^{-1}-3t^{-2}+t^{-3}$
Conway polynomial	$z^{6}+3z^{4}+2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 23, 2 }
Jones polynomial	$-q^{4}+2q^{3}-3q^{2}+4q-3+4q^{-1}-3q^{-2}+2q^{-3}-q^{-4}$
HOMFLY-PT polynomial (db, data sources)	$z^{6}-a^{2}z^{4}-z^{4}a^{-2}+5z^{4}-3a^{2}z^{2}-3z^{2}a^{-2}+8z^{2}-2a^{2}-2a^{-2}+5$
Kauffman polynomial (db, data sources)	$az^{9}+z^{9}a^{-1}+2a^{2}z^{8}+2z^{8}a^{-2}+4z^{8}+a^{3}z^{7}-2az^{7}-2z^{7}a^{-1}+z^{7}a^{-3}-10a^{2}z^{6}-10z^{6}a^{-2}-20z^{6}-5a^{3}z^{5}-8az^{5}-8z^{5}a^{-1}-5z^{5}a^{-3}+14a^{2}z^{4}+14z^{4}a^{-2}+28z^{4}+7a^{3}z^{3}+16az^{3}+16z^{3}a^{-1}+7z^{3}a^{-3}-8a^{2}z^{2}-8z^{2}a^{-2}-16z^{2}-3a^{3}z-7az-7za^{-1}-3za^{-3}+2a^{2}+2a^{-2}+5$
The A2 invariant	$-q^{12}-q^{8}+q^{4}+q^{2}+3+q^{-2}+q^{-4}-q^{-8}-q^{-12}$
The G2 invariant	Data:K11n24/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n24 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["K11n24"];

In[4]:=

Alexander[K][t]

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

Out[4]=

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

In[5]:=

Conway[K][z]

Out[5]=

$z^{6}+3z^{4}+2z^{2}+1$

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]=

{ 23, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$az^{9}+z^{9}a^{-1}+2a^{2}z^{8}+2z^{8}a^{-2}+4z^{8}+a^{3}z^{7}-2az^{7}-2z^{7}a^{-1}+z^{7}a^{-3}-10a^{2}z^{6}-10z^{6}a^{-2}-20z^{6}-5a^{3}z^{5}-8az^{5}-8z^{5}a^{-1}-5z^{5}a^{-3}+14a^{2}z^{4}+14z^{4}a^{-2}+28z^{4}+7a^{3}z^{3}+16az^{3}+16z^{3}a^{-1}+7z^{3}a^{-3}-8a^{2}z^{2}-8z^{2}a^{-2}-16z^{2}-3a^{3}z-7az-7za^{-1}-3za^{-3}+2a^{2}+2a^{-2}+5$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {8_7,}

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["K11n24"];

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]=

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

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_7,}

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:

(2, 0)

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 $8$ $0$ $32$ ${\frac {76}{3}}$ $-{\frac {28}{3}}$ $0$ $0$ $0$ $0$ ${\frac {256}{3}}$ $0$ ${\frac {608}{3}}$ $-{\frac {224}{3}}$ ${\frac {3031}{15}}$ ${\frac {1396}{15}}$ $-{\frac {4436}{45}}$ ${\frac {137}{9}}$ $-{\frac {809}{15}}$

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=$2 is the signature of K11n24. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -5 -4 -3 -2 -1 0 1 2 3 χ 9                 1 -1 7               1   1 5             2 1   -1 3           2 1     1 1         2 3       1 -1       2 2 1       1 -3     1 2           1 -5   1 2             -1 -7   1               1 -9 1                 -1

Integral Khovanov Homology (db, data source)

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

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.

K11n23 K11n25