K11n124

K11n123

K11n125

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n124 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X10,3,11,4 X16,6,17,5 X7,15,8,14 X9,19,10,18 X2,11,3,12 X13,9,14,8 X20,15,21,16 X22,18,1,17 X19,13,20,12 X6,21,7,22
Gauss code	1, -6, 2, -1, 3, -11, -4, 7, -5, -2, 6, 10, -7, 4, 8, -3, 9, 5, -10, -8, 11, -9
Dowker-Thistlethwaite code	4 10 16 -14 -18 2 -8 20 22 -12 6

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Chiral Unknotting number 1 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11n124/ThurstonBennequinNumber Hyperbolic Volume 13.6991 A-Polynomial See Data:K11n124/A-polynomial

[edit Notes for K11n124'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 K11n124's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{3}-6t^{2}+14t-17+14t^{-1}-6t^{-2}+t^{-3}$
Conway polynomial	$z^{6}-z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 59, 2 }
Jones polynomial	$-q^{6}+4q^{5}-7q^{4}+9q^{3}-10q^{2}+10q-8+6q^{-1}-3q^{-2}+q^{-3}$
HOMFLY-PT polynomial (db, data sources)	$z^{6}a^{-2}+3z^{4}a^{-2}-z^{4}a^{-4}-2z^{4}+a^{2}z^{2}+3z^{2}a^{-2}-z^{2}a^{-4}-4z^{2}+a^{2}+a^{-2}-1$
Kauffman polynomial (db, data sources)	$z^{9}a^{-1}+z^{9}a^{-3}+5z^{8}a^{-2}+2z^{8}a^{-4}+3z^{8}+3az^{7}+5z^{7}a^{-1}+3z^{7}a^{-3}+z^{7}a^{-5}+a^{2}z^{6}-7z^{6}a^{-2}-z^{6}a^{-4}-5z^{6}-9az^{5}-18z^{5}a^{-1}-6z^{5}a^{-3}+3z^{5}a^{-5}-3a^{2}z^{4}-4z^{4}a^{-2}+2z^{4}a^{-4}+4z^{4}a^{-6}-5z^{4}+7az^{3}+10z^{3}a^{-1}+z^{3}a^{-3}-z^{3}a^{-5}+z^{3}a^{-7}+3a^{2}z^{2}+4z^{2}a^{-2}-z^{2}a^{-4}-z^{2}a^{-6}+7z^{2}-az-za^{-1}-a^{2}-a^{-2}-1$
The A2 invariant	Data:K11n124/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11n124/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n124 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["K11n124"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{3}-6t^{2}+14t-17+14t^{-1}-6t^{-2}+t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 59, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{9}a^{-1}+z^{9}a^{-3}+5z^{8}a^{-2}+2z^{8}a^{-4}+3z^{8}+3az^{7}+5z^{7}a^{-1}+3z^{7}a^{-3}+z^{7}a^{-5}+a^{2}z^{6}-7z^{6}a^{-2}-z^{6}a^{-4}-5z^{6}-9az^{5}-18z^{5}a^{-1}-6z^{5}a^{-3}+3z^{5}a^{-5}-3a^{2}z^{4}-4z^{4}a^{-2}+2z^{4}a^{-4}+4z^{4}a^{-6}-5z^{4}+7az^{3}+10z^{3}a^{-1}+z^{3}a^{-3}-z^{3}a^{-5}+z^{3}a^{-7}+3a^{2}z^{2}+4z^{2}a^{-2}-z^{2}a^{-4}-z^{2}a^{-6}+7z^{2}-az-za^{-1}-a^{2}-a^{-2}-1$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {9_32, K11n52,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {K11n166,}

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["K11n124"];

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]=

{ $t^{3}-6t^{2}+14t-17+14t^{-1}-6t^{-2}+t^{-3}$, $-q^{6}+4q^{5}-7q^{4}+9q^{3}-10q^{2}+10q-8+6q^{-1}-3q^{-2}+q^{-3}$ }

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]=

{9_32, K11n52,}

In[6]:=

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

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

Out[6]=

{K11n166,}

Vassiliev invariants

V2 and V3:

(-1, 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 $-4$ $0$ $8$ ${\frac {34}{3}}$ ${\frac {14}{3}}$ $0$ $0$ $-32$ $32$ $-{\frac {32}{3}}$ $0$ $-{\frac {136}{3}}$ $-{\frac {56}{3}}$ $-{\frac {1231}{30}}$ $-{\frac {1778}{15}}$ ${\frac {4018}{45}}$ ${\frac {79}{18}}$ ${\frac {689}{30}}$

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 K11n124. 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                 3   3 9               4 1   -3 7             5 3     2 5           5 4       -1 3         5 5         0 1       4 6           2 -1     2 4             -2 -3   1 4               3 -5   2                 -2 -7 1                   1

Integral Khovanov Homology (db, data source)

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

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.

K11n123 K11n125