K11n123

K11n122

K11n124

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n123 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X10,3,11,4 X5,16,6,17 X7,15,8,14 X12,10,13,9 X2,11,3,12 X13,18,14,19 X15,21,16,20 X17,22,18,1 X19,8,20,9 X21,7,22,6
Gauss code	1, -6, 2, -1, -3, 11, -4, 10, 5, -2, 6, -5, -7, 4, -8, 3, -9, 7, -10, 8, -11, 9
Dowker-Thistlethwaite code	4 10 -16 -14 12 2 -18 -20 -22 -8 -6

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 2 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11n123/ThurstonBennequinNumber Hyperbolic Volume 13.524 A-Polynomial See Data:K11n123/A-polynomial

[edit Notes for K11n123's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $2$ Rasmussen s-Invariant 0

[edit Notes for K11n123's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$3t^{2}-14t+23-14t^{-1}+3t^{-2}$
Conway polynomial	$3z^{4}-2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 57, 0 }
Jones polynomial	$-q^{3}+4q^{2}-6q+9-10q^{-1}+9q^{-2}-8q^{-3}+6q^{-4}-3q^{-5}+q^{-6}$
HOMFLY-PT polynomial (db, data sources)	$a^{6}-3z^{2}a^{4}-2a^{4}+2z^{4}a^{2}+3z^{2}a^{2}+2a^{2}+z^{4}-z^{2}-1-z^{2}a^{-2}+a^{-2}$
Kauffman polynomial (db, data sources)	$a^{3}z^{9}+az^{9}+3a^{4}z^{8}+5a^{2}z^{8}+2z^{8}+3a^{5}z^{7}+5a^{3}z^{7}+3az^{7}+z^{7}a^{-1}+a^{6}z^{6}-5a^{4}z^{6}-8a^{2}z^{6}-2z^{6}-9a^{5}z^{5}-20a^{3}z^{5}-9az^{5}+2z^{5}a^{-1}-3a^{6}z^{4}-6a^{4}z^{4}-4a^{2}z^{4}+4z^{4}a^{-2}+3z^{4}+7a^{5}z^{3}+14a^{3}z^{3}+6az^{3}+z^{3}a^{-3}+3a^{6}z^{2}+8a^{4}z^{2}+7a^{2}z^{2}-2z^{2}a^{-2}-2a^{5}z-2a^{3}z-a^{6}-2a^{4}-2a^{2}-a^{-2}-1$
The A2 invariant	Data:K11n123/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11n123/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n123 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["K11n123"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$3t^{2}-14t+23-14t^{-1}+3t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 57, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

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

"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["K11n123"];

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]=

{ $3t^{2}-14t+23-14t^{-1}+3t^{-2}$, $-q^{3}+4q^{2}-6q+9-10q^{-1}+9q^{-2}-8q^{-3}+6q^{-4}-3q^{-5}+q^{-6}$ }

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:

(-2, 2)

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 $-8$ $16$ $32$ $-{\frac {28}{3}}$ $-{\frac {20}{3}}$ $-128$ $-{\frac {320}{3}}$ $-{\frac {128}{3}}$ $16$ $-{\frac {256}{3}}$ $128$ ${\frac {224}{3}}$ ${\frac {160}{3}}$ ${\frac {5729}{15}}$ $-{\frac {436}{15}}$ ${\frac {12596}{45}}$ $-{\frac {449}{9}}$ ${\frac {929}{15}}$

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=$0 is the signature of K11n123. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

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

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.

K11n122 K11n124